Optimal semiclassical spectral asymptotics for differential operators with non-smooth

Søren Mikkelsen
School of Mathematics, University of Bristol
BS8 1UG, United Kingdom

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and obtaining optimal spectral asymptotics again under certain regularity conditions. For the Weyl law we assume the coefficients are differentiable with Hölder continuous derivatives and for the Riesz means we assume the coefficients are two times differentiable with Hölder continuous derivatives.

§ INTRODUCTION

In this paper we will study (uniformly) elliptic self-adjoint differential operators $A(\hbar)$ acting in $L^2(\R^d)$ defined as Friedrichs extensions of sesquilinear forms given by

\begin{equation}\label{int_gen_sf}
\mathcal{A}_\hbar[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}(x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar),
\end{equation}

where $\hbar>0$ is the semiclassical parameter, $\mathcal{D}(   \mathcal{A}_\hbar)$ is the associated form domain and for $\alpha\in\N_0^d$, we have used the standard notation

\begin{equation*}
(\hbar D_x)^\alpha = \prod_{j=1}^d (-i\hbar\partial_{x_j})^{\alpha_j}.
\end{equation*}

Our exact assumptions on the functions $a_{\alpha\beta}$ will be stated later. For $\gamma$ in $[0,1]$ we will analyse the asymptotics as $\hbar$ tends to zero of the traces

\begin{equation}
\Tr[(A(\hbar))_{-}^\gamma ],
\end{equation}

where we have used the notation $(x)_{-} = \max(0,-x)$. In the case $\gamma=0$ we use the convention that $(x)_{-}^0 = \boldsymbol{1}_{(-\infty,0]}(x)$, where $\boldsymbol{1}_{(-\infty,0]}$ is the characteristic function for the set $(-\infty,0]$. There are no technical obstructions in considering higher values of $\gamma$, but more regularity of the coefficients will be needed to obtain sharp remainder estimates.

These kinds of traces are in the literature denoted as Riesz means, when $\gamma>0$, and counting functions when $\gamma=0$. Both objects have been studied extensively in the literature in different variations. For background on the Weyl law (asymptotic expansions of the counting function) and the connection to physics we refer the reader to the surveys  <cit.>.

In the case of smooth coefficients and with some regularity conditions it was first
proven in <cit.> by Helffer and Robert that an estimate of the following type

\begin{equation}\label{Helffer_Robert_res}
\big| \Tr[\boldsymbol{1}_{(-\infty,\lambda_0]}(A(\hbar))] - \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}}  \boldsymbol{1}_{(-\infty,\lambda_0]}(a_0(x,p)) \, dxdp \big| \leq C\hbar^{1-d},
\end{equation}

is true, where

\begin{equation}\label{int_princ_sym}
a_0(x,p) =  \sum_{\abs{\alpha},\abs{\beta}\leq m} a_{\alpha\beta}(x)p^{\alpha+\beta},
\end{equation}

and $\lambda_0$ is a number such that there exists a strictly larger number $\lambda$ for which
$a_0^{-1}((-\infty,\lambda])$ is compact. Moreover, $\lambda_0$ is assumed to be non-critical for
$a_0(x,p)$. A number $\lambda_0$ is a non-critical value when

\begin{equation*}
|\nabla a_0(x,p)| \geq c>0 \quad\text{for all } (x,p) \in a_0^{-1}(\{\lambda_0\}).
\end{equation*}

We will from here on always use that $a_0(x,p)$ is given by (<ref>) when we are working with a sesquilinear form $ \mathcal{A}_\hbar$ given by (<ref>).

Actually they proved (<ref>) for a larger class of
pseudo-differential operators. The high energy analog was proven by
Hörmander in <cit.>, where the operator was defined to act in $L^2(M)$, where $M$ is
a smooth compact manifold without a boundary. For the Riesz means an asymptotic expansion was obtained by Helffer and Robert in <cit.> under the same conditions as above and again for this larger class of pseudo-differential operators. For the high energy case they where studied by Hörmander in <cit.>.

When considering the assumptions above one question immediately arise:

*  What happens if the coefficients are not smooth? Can such results
still be proven with optimal errors?

A considerable body of literature have been devoted to this questions. Most prominent are the works of Ivrii <cit.> and Zielinski <cit.>. A more detailed review of the literature will be given in  section <ref> after presenting the result obtained in this work.

§.§ Assumptions and main results I

Before we state our assumptions and results on the Riesz means we need the following definition to clarify terminology.

For $k$ in $\N$ and $\mu$ in $(0,1]$ we denote by $C^{k,\mu}(\R^d)$
the subspace of $C^{k}(\R^d)$ defined by

\begin{equation*}
\begin{aligned}
C^{k,\mu}(\R^d) = \big\{ f\in C^{k}(\R^d) \, \big| \, \exists C>0:   |\partial_x^{\alpha} f(x) - \partial_x^{\alpha} f(y&) |  \leq C |x-y|^{\mu}
\\
& \forall \alpha \in\N^d \text{ with } \abs{\alpha}=k \text{ and } \forall x,y\in\R^d \big\}.
\end{aligned}
\end{equation*}

In the case $\mu=0$ we use the convention

\begin{equation*}
C^{k,0}(\R^d) = C^{k}(\R^d).
\end{equation*}

We will for both the Weyl law and the Riesz means make the following assumption on the sesquilinear form.

[Tempered variation model]
Let  $(k,\mu)$ be numbers in $\N\times[0,1]$ and $\mathcal{A}_\hbar$ be a sesquilinear form given by

\begin{equation*}
\mathcal{A}_\hbar[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}(x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar).
\end{equation*}

Assume the coefficients $a_{\alpha\beta}(x)$ are in $C^{k,\mu}(\R^d)$ and satisfy that $a_{\alpha\beta}(x) = \overline{a_{\beta\alpha}(x)}$ for all multi-indices $\alpha$ and $\beta$. For all multi-indices $\alpha$ and $\beta$ assume that

*  There is a $\zeta_0>0$ such that $\min_{x\in\R^d}(a_{\alpha\beta}(x))> - \zeta_0$.

*  There is a $\zeta_1>\zeta_0$ and $C_1,M>0$ such that

\begin{equation*}
a_{\alpha\beta}(x) + \zeta_1 \leq C_1(a_{\alpha\beta}(y)+\zeta_1)(1+|x-y|)^M,
\end{equation*}

for all $x,y$ in $\R^d$.

*  For all $\delta$ in $\N_0^d$ with $|\delta|\leq k$ there is a $c_\delta>0$ such that

\begin{equation*}
\abs{\partial_{x}^\delta a_{\alpha\beta}(x)} \leq  c_\delta(a_{\alpha\beta}(x) + \zeta_1).
\end{equation*}

Suppose there exists a constant $C_2$ such that

\begin{equation}\label{ellip_assumption}
\sum_{\abs{\alpha}=\abs{\beta}= m} a_{\alpha\beta}(x) p^{\alpha+\beta} \geq C_2 \abs{p}^{2m},
\end{equation}

for all $(x,p)$ in $\R^d_x\times\R_p^d$.

The assumption that the coefficients are in $C^{k,\mu}(\R^d)$ is that we need some regularity to get optimal asymptotic results. The assumption $a_{\alpha\beta}(x) = \overline{a_{\beta\alpha}(x)}$  is made to ensure the form is symmetric. The assumptions <ref>–<ref>  ensure that the coefficients are of tempered variation. Assumption <ref> is the condition that for all $\alpha$ and $\beta$ $a_{\alpha\beta}(x) + \zeta_1$ is a tempered weight. Note that due to the semiclassical structure of the problem these assumptions are needed to be true for all $\alpha$ and $\beta$ and not just $|\alpha|=|\beta|=m$.  The “ellipticity” assumption <ref> will help ensure the existence of eigenvalues.

We can now state our two main results. Firstly we have that

Let $A(\hbar)$ be the Friedrichs extension of a sesquilinear form $\mathcal{A}_\hbar$ which satisfies Assumption <ref> with the numbers $(1,\mu)$ where $\mu>0$. Suppose there is a $\nu>0$ such that the set $a_0^{-1}((-\infty,\nu])$ is compact and there is $c>0$ such that

\begin{equation}\label{non_critical_main_thm_1}
|\nabla_p a_0(x,p)| \geq c \quad\text{for all } (x,p)\in a_0^{-1}(\{0\}).
\end{equation}

Then we have

\begin{equation*}
\big |\Tr[\boldsymbol{1}_{(-\infty,0]}(A(\hbar))] - \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} \boldsymbol{1}_{(-\infty,0]}( a_{0}(x,p)) \,dx dp \big| \leq C \hbar^{1-d},
\end{equation*}

for all sufficiently small $\hbar$.

The assumption that there is a $\nu>0$ such that the set $a_0^{-1}((-\infty,\nu])$ is compact is needed to ensure that we only have pure point spectrum in $(-\infty,0]$. Due to the ellipticity assumption this is really an assumption on the coefficients $a_{\alpha\beta}(x)$.

Secondly for the Reisz means with $\gamma$ in $(0,1]$ we have

Assume $\gamma$ is in $(0,1]$ and let $A(\hbar)$ be the Friedrichs extension of a sesquilinear form $\mathcal{A}_\hbar$ which satisfies Assumption <ref> with the numbers $(2,\mu)$. If $\gamma=1$ we suppose $\mu>0$ and if $\gamma<1$ we may suppose $\mu=0$. Lastly, suppose there is a $\nu>0$ such that the set $a_0^{-1}((-\infty,\nu])$ is compact and there is $c>0$ such that

\begin{equation}\label{non_critical_main_thm}
|\nabla_p a_0(x,p)| \geq c \quad\text{for all } (x,p)\in a_0^{-1}(\{0\}).
\end{equation}

Then we have

\begin{equation*}
\big |\Tr[(A(\hbar))^\gamma_{-}] - \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} ( a_{0}(x,p))^{\gamma}_{-} +\hbar \gamma a_1(x,p) ( a_{0}(x,p))^{\gamma-1}_{-} \,dx dp \big| \leq C \hbar^{1+\gamma-d},
\end{equation*}

for all sufficiently small $\hbar$ where the symbol $a_1(x,p)$ is defined as

\begin{equation}
a_{1}(x,p) =  i \sum_{|\alpha|, |\beta|\leq m}\sum_{j=1}^d \frac{\beta_j-\alpha_j}{2} \partial_{x_j} a_{\alpha\beta} (x) p^{\alpha+\beta-\eta_j},
\end{equation}

where $\eta_j$ is the multi index with all entries zero except the j'th which is one.

In the case where $\gamma \leq \frac{1}{3}$ we may assume the coefficients are in $C^{1,\mu}(\R^d)$, with $\mu\geq \frac{2\gamma}{1-\gamma}$ and still obtain sharp estimates. Further details are given in Remark <ref>.

There are no technical obstruction in considering larger values of $\gamma$. For a $\gamma>1$ we will need the coefficients to have $\lceil \gamma \rceil$ derivatives if $\gamma$ is not an integer in general. If $\gamma$ is an integer we will need the coefficients to be in $C^{\gamma,\mu}(\R^d)$ for some $\mu>0$. The expansions will also consist of  $\lceil \gamma \rceil +1$ terms, where these terms can be calculated explicitly.

Let $A(x)$ be a $d\times d$–dimensional symmetric matrix and suppose that $A(x)$ is positive definite for all $x$ in $\R^d$. We assume that each entry in $a_{ij}(x)$ of $A(x)$  are in $C^{1,\mu}(\R^d)$, where $\mu>0$, and satisfies assumption <ref>–<ref> from Assumption <ref>. We will further assume that $\det(A(x)) \geq c(1+ |x|^2)^{d+1}$ for all $x$. Then one could be interested in the number of eigenvalues less than or equal to $E^2$ for the second order differential operator  with quadratic form

\begin{equation}
\langle  A(x)(-i\nabla_x) \varphi, (-i\nabla_x)\varphi \rangle_{L^2(\R^d)}.
\end{equation}

In this example we will denote this number by $\mathcal{N}(E^2)$. This is equivalent to counting the number of eigenvalues less than or equal to zero of the operator with quadratic form

\begin{equation}
\langle  A(x)(-iE^{-1}\nabla_x) \varphi, (-iE^{-1}\nabla_x)\varphi \rangle_{L^2(\R^d)} -\langle   \varphi, \varphi \rangle_{L^2(\R^d)} .
\end{equation}

Treating $E^{-1}$ as the semiclassical parameter. It is a simple calculation to check that all assumptions for Theorem <ref> are satisfied. This gives us that

\begin{equation*}
\big |\mathcal{N}(E^2) - \frac{1}{(2\pi)^d} \int_{\R^{2d}} \boldsymbol{1}_{(-\infty,E^2]}( \langle A(x)p,p\rangle) \,dx dp \big| \leq C E^{1-d},
\end{equation*}

for $E$ suffciently large. Note that the phase space integral has this more familiar form

\begin{equation*}
\frac{1}{(2\pi)^d} \int_{\R^{2d}} \boldsymbol{1}_{(-\infty,E^2]}( \langle A(x)p,p\rangle) \,dx dp =   E^d \frac{\Vol_d(B(0,1))}{(2\pi)^d} \int_{\R^d} \frac{1}{\sqrt{\det(A(x))}} \, dx,
\end{equation*}

where $\Vol_d(B(0,1))$ is the $d$-dimensional volume of the unit ball.

§.§ Assumptions and main results II

The main motivation and the novel results, to the best of my knowledge, in this work are the following results, where we consider spectral asymptotic of admissible operators perturbed by differential operators defined as Friedrichs extensions of sesquilinear forms with non-smooth coefficients. We will need some regularity conditions on the admissible operator, which will be the same as the assumption made in <cit.>. We will here recall them:

For $\hbar$ in $(0,\hbar_0]$ let $B(\hbar)$ be an admissible operator with tempered weight $m$ and symbol

\begin{equation}
b_\hbar(x,p) = \sum_{j\geq0} \hbar^j b_j(x,p).
\end{equation}

Assume that

*  The operator $B(\hbar)$ is symmetric on $\mathcal{S}(\R^d)$ for all  $\hbar$ in $(0,\hbar_0]$.

*  There is a $\zeta_0>0$ such that

\begin{equation*}
\min_{(x,p)\in\R^d\times\R^d}(b_0(x,p))> - \zeta_0.
\end{equation*}

*  There is a $\zeta_1>\zeta_0$ such that $b_0+\zeta_1$ is a tempered weight and $b_j$ is in $\Gamma_{0,1}^{b_0+\zeta_1} \left(\R^{2d}\right)$.

The definition of a tempered weight is recalled in Section <ref>, where the definition of the space $\Gamma_{0,1}^{b_0+\zeta_1} \left(\R^{2d}\right)$ is also given. We can now state the second set of main theorems.

Let $B(\hbar)$ be an admissible operator satisfing Assumption <ref> with $\hbar$ in $(0,\hbar_0]$ and $\mathcal{A}_\hbar$ be a sesquilinear form satisfying Assumption <ref> with the numbers $(1,\mu)$ with $\mu>0$.

Define the operator $\tilde{B}(\hbar)$ as the Friedrichs extension of the form sum $B(\hbar) + \mathcal{A}_\hbar$ and let

\begin{equation}
\tilde{b}_0(x,p) = b_0(x,p) +  \sum_{\abs{\alpha},\abs{\beta}\leq m} a_{\alpha\beta}(x)p^{\alpha+\beta}.
\end{equation}

We suppose there is a $\nu$ such that $\tilde{b}_0^{-1}((-\infty,\nu])$ is compact and there is $c>0$ such that

\begin{equation}\label{non_critical_main_thm_2}
|\nabla_p \tilde{b}_0(x,p)| \geq c \quad\text{for all } (x,p)\in \tilde{b}_0^{-1}(\{0\}).
\end{equation}

Then we have

\begin{equation*}
\big |\Tr[\boldsymbol{1}_{(-\infty,0]}(\tilde{B}(\hbar))] - \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} \boldsymbol{1}_{(-\infty,0]}( \tilde{b}_0(x,p)) \,dx dp \big| \leq C \hbar^{1-d},
\end{equation*}

for all sufficiently small $\hbar$.

Next we give the result for Riesz means for this type of operator.

Assume $\gamma$ is in $(0,1]$.
Let $B(\hbar)$ be an admissible operator satisfing Assumption <ref> with $\hbar$ in $(0,\hbar_0]$ and $\mathcal{A}_\hbar$ be a sesquilinear form satisfying Assumption <ref> with the numbers $(2,\mu)$. If $\gamma=1$ we suppose $\mu>0$ and if $\gamma<1$ we may suppose $\mu=0$.

Define the operator $\tilde{B}(\hbar)$ as the Friedrichs extension of the form sum $B(\hbar) + \mathcal{A}_\hbar$ and let

\begin{equation}
\tilde{b}_0(x,p) = b_0(x,p) +  \sum_{\abs{\alpha},\abs{\beta}\leq m} a_{\alpha\beta}(x)p^{\alpha+\beta}.
\end{equation}

We suppose there is a $\nu$ such that $\tilde{b}_0^{-1}((-\infty,\nu])$ is compact and there is $c>0$ such that

\begin{equation}\label{non_critical_main_thm_3}
|\nabla_p \tilde{b}_0(x,p)| \geq c \quad\text{for all } (x,p)\in \tilde{b}_0^{-1}(\{0\}),
\end{equation}

Then we have

\begin{equation*}
\big |\Tr[(\tilde{B}(\hbar))^\gamma_{-}] - \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} ( \tilde{b}_{0}(x,p))^{\gamma}_{-} +\hbar \gamma \tilde{b}_1(x,p) ( \tilde{b}_{0}(x,p))^{\gamma-1}_{-} \,dx dp \big| \leq C \hbar^{1+\gamma-d},
\end{equation*}

for all sufficiently small $\hbar$ where the symbol $\tilde{b}_1(x,p)$ is defined as

\begin{equation}
\tilde{b}_{1}(x,p) = b_1(x,p)+  i \sum_{|\alpha|, |\beta|\leq m}\sum_{j=1}^d \frac{\beta_j-\alpha_j}{2} \partial_{x_j} a_{\alpha\beta} (x) p^{\alpha+\beta-\eta_j}.
\end{equation}

§.§ Previous work

The first results with an optimal Weyl law without full regularity was
proven in the papers
<cit.> by Zielinski. In these papers
Zielinski obtained an optimal Weyl law under the assumption that
the coefficients are differentiable with Lipschitz continuous first
derivative.  Zielinski did not in those papers consider the
semiclassical setting. These results were generalised by Ivrii in the
semiclassical setting in <cit.>. Here the coefficients are
assumed to be differentiable and with a Hölder continuous first
derivative. This was further generalised by Bronstein and Ivrii
in <cit.>, where they reduced the assumptions further by
assuming the first derivative to have modulus of continuity
$\mathcal{O}(|\log(x-y)|^{-1})$. All these papers considered
differential operators acting in $L^2(M)$, where $M$ is
a compact manifold both with and without a boundary. In <cit.> Zielinski  considers the semiclassical setting with differential operators acting in $L^2(\R^d)$ and proves an optimal Weyl Law under the assumption that all coefficients are one time differentiable with a Hölder continuous derivative. Moreover, it is assumed that the coefficients and the derivatives is bounded. However, it is remarked that it is possible to consider unbounded coefficients in a framework of tempered variation models. As we have seen above this is indeed the case and thereby proven this minor technical generalisation.

Another question one could ask is can such results be obtained without a non-critical condition? This question have also been studied in the literature and it is possible for
Schrödinger operators to prove optimal Weyl laws without a non-critical condition by a multiscale argument  see
<cit.>, this
approach is also described in <cit.>.  This multiscale argument can
be seen as a discreet approach and a continuous version have been
proved and used in <cit.>.  The essence of this approach is
to localise and locally introduce a non-critical condition by
unitary conjugation. Then by an optimal Weyl law with a non-critical
condition one obtain the right asymptotics locally. The last step is
to average out the localisations.  Ivrii has also considered
multiscale analysis for higher order differential operators but to
treat these cases extra assumptions on the Hessian of the principal
symbol is needed see <cit.>. There is
also another approach by Zielinski see <cit.>,
where he proves optimal Weyl laws without a non-critical condition, but
with an extra assumption on a specific phase space volume.

In the case of Riesz means sharp remainder estimates for the asymptotic expansions has been obtained by Bronstein and Ivrii in <cit.>, for differential operators acting in $L^2(M)$, where $M$ is a compact closed manifold. However, it is mentioned in a remark that the results of the paper needs to be combined with the methods in section 4.3 in <cit.> to obtain the reminder estimate for the Riesz means. Result on Riesz means can also be found in  <cit.> by Ivrii. Again this is for differential operators acting in acting in $L^2(M)$, where $M$ is a compact closed manifold. In chapter 11 of <cit.> unbounded domains are also considered but in the large eigenvalue/high energy case. In this chapter all results are stated for “smooth” coefficients. However, it is remarked in the start that the case of non-smooth coefficients are left to the reader.
The results given here on the Reisz means for differential operators associated to quadratic forms are a small generalisation of previous known results.

§ PRELIMINARIES AND NOTATION

§.§ Notation and definitions

We will here mainly set up the non standard notation used and some
definitions. We will in the following use the notation

\begin{equation*}
\lambda(x) = (1+\abs{x}^2)^{\frac{1}{2}},
\end{equation*}

for $x$ in $\R^d$ and not the usual bracket notation. Moreover, for
more vectors $x_1,x_2,x_3$ from $\R^d$ we will use the convention

\begin{equation*}
\lambda(x_1,x_2,x_3) = (1+\abs{x_1}^2+\abs{x_2}^2+\abs{x_3}^2)^{\frac{1}{2}},
\end{equation*}

and similar in the case of $2$ or even more vectors. For the natural numbers we use the following conventions

\begin{equation*}
\N= \{1,2,3,\dots \} \quad\text{and}\quad \N_0 = \{0\} \cup \N.
\end{equation*}

When working with the Fourier transform we will use the following semiclassical
version for $\hbar>0$

\begin{equation*}
\mathcal{F}_\hbar[ \varphi ](p) \coloneqq \int_{\R^d} e^{-i\hbar^{-1} \langle x,p \rangle}\varphi(x) \, dx,
\end{equation*}

and with inverse given by

\begin{equation*}
\mathcal{F}_\hbar^{-1}[\psi] (x) \coloneqq \frac{1}{(2\pi\hbar)^d} \int_{\R^d} e^{i\hbar^{-1} \langle x,p \rangle}\psi(p) \, dp,
\end{equation*}

where $\varphi$ and $\psi$ are elements of $\mathcal{S}(\R^d)$.

We will by $\mathcal{L}(\mathcal{B}_1,\mathcal{B}_2)$ denote the linear bounded operators from the space $\mathcal{B}_1$ into $\mathcal{B}_1$ and $\mathcal{L}(\mathcal{B}_1)$ denotes the linear bounded operators from the space $\mathcal{B}_1$ into  itself. For an operator $A$ acting in a Hilbert space we will denote the spectrum of $A$ by

\begin{equation*}
\spec(A).
\end{equation*}

As we later will need stationary phase asymptotics, we will for sake of completeness recall it here.

Let $B$ be a invertible, symmetric real $n\times n$ matrix and
$(u,v)\rightarrow a(u,v;\hbar)$ be a function in
$C^\infty(\R_u^{d} \times \R_v^{n})$ for all $\hbar$ in
$(0,\hbar_0]$. We suppose $v\rightarrow a(u,v;\hbar)$ has compact
support for all $u$ in $\R_u^{d}$ and $\hbar$ in
$(0,\hbar_0]$. Moreover we let

\begin{equation*}
I(u;a,B,\hbar) = \int_{\R^{n}} e^{\frac{i}{2\hbar} \langle B v,v\rangle} a(u,v;\hbar) \, dv.
\end{equation*}

Then for each $N$ in $\mathbb{N}$ we have

\begin{equation*}
I(u;a,B,\hbar)  = (2\pi\hbar)^{\frac{n}{2}}
\frac{e^{i\frac{\pi}{4} \sgn(B)}}{\abs{\det(B)}^{\frac12}}
\sum_{j=0}^N \frac{\hbar^j}{j!} \Big(\frac{\langle B^{-1}
D_v,D_v\rangle}{2i}\Big)^j a(u,v;\hbar) \Big|_{v=0} +
\hbar^{N+1} R_{N+1}(u;\hbar),
\end{equation*}

where $\sgn(B)$ is the difference between the number of positive and
negative eigenvalues of $B$. Moreover there exists a constant $c_n$
only depending on the dimension such the error term $R_{N+1}$
satisfies the bound

\begin{equation*}
\abs{R_{N+1}(u;\hbar)} \leq c_n \bigg\lVert \frac{\langle B^{-1} D_v,D_v\rangle^{N+1}}{(N+1)!}  a(u,v;\hbar)  \bigg\rVert_{H^{[\frac{n}{2}]+1}(\R_v^n)},
\end{equation*}

where $ [\tfrac{n}{2}]$ is the integer part of $\tfrac{n}{2}$ and
$\norm{\cdot}_{H^{[\tfrac{n}{2}]+1}(\R_v^n)}$ is the Sobolev norm.

A proof of the proposition can be found in e.g. <cit.> or
<cit.>. For the sesquilinear forms we consider we will use the following terminology

For a  sesquilinear form $\mathcal{A}_\hbar$ given by

\begin{equation*}
\mathcal{A}_\hbar[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}(x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar),
\end{equation*}

we call a number $E$ in $\R$ non-critical if there exists $c>0$ such that

\begin{equation*}
|\nabla_p a_0(x,p)| \geq c \quad\text{for all } (x,p)\in a_0^{-1}(\{E\}),
\end{equation*}

\begin{equation*}
a_0(x,p) =  \sum_{\abs{\alpha},\abs{\beta}\leq m} a_{\alpha\beta}(x)p^{\alpha+\beta}.
\end{equation*}

A  sesquilinear form $\mathcal{A}_\hbar$ given by

\begin{equation*}
\mathcal{A}_\hbar[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}(x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar),
\end{equation*}

is called elliptic if there exists a strictly positive constant $C$ such that

\begin{equation}
\sum_{\abs{\alpha}=\abs{\beta}= m} a_{\alpha\beta}(x) p^{\alpha+\beta} \geq C \abs{p}^{2m},
\end{equation}

for all $(x,p)$ in $\R^d_x\times\R_p^d$.

§.§ Approximation of quadratic forms

We will here construct our approximating (framing) quadratic forms and prove that some properties of the original quadratic form can be inherited by the approximations. The construction is similar to the ones used in <cit.>  to construct their framing operators. The first example of approximation non smooth coefficients in this way appeared in <cit.>, to the authors knowledge. The crucial part in this construction is  Proposition <ref>, for which a proof can be found in <cit.>.

Let $f$ be in $C^{k,\mu}(\R^d)$ for a $\mu$ in $[0,1]$. Then for
every $\varepsilon >0$ there exists a function $f_\varepsilon$ in
$C^\infty(\R^d)$ such that

\begin{equation}\label{B.smoothning_of_func_bounds}
\begin{aligned}
\abs{\partial_x^\alpha f_\varepsilon(x) -\partial_x^\alpha f(x)
} \leq{}& C_\alpha \varepsilon^{k+\mu-\abs{\alpha}} \qquad
\abs{\alpha}\leq k,
\\
\abs{\partial_x^\alpha f_\varepsilon(x)} \leq{}& C_\alpha
\varepsilon^{k+\mu-\abs{\alpha}} \qquad \abs{\alpha}\geq k+1,
\end{aligned}
\end{equation}

where the constants is independent of $\varepsilon$.

The function $f_\varepsilon$ is a smoothing (mollification) of
$f$. Usually this is done by convolution with a compactly supported
smooth function. However here one uses a Schwartz function in the
convolution in order to ensure the stated error terms. The convolution
with a compactly supported smooth function will in most cases only
give an error of order $\varepsilon$.

The ideas used to construct the approximating or framing quadratic forms are of the same type as the ideas used to construct the framing operators in

Let $\mathcal{A}_\hbar$ be an elliptic sesquilinear form given by

\begin{equation*}
\mathcal{A}_\hbar[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}(x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar).
\end{equation*}

Assume the coefficients $a_{\alpha\beta}(x)$ are in $C^{k,\mu}(\R^d)$, for a pair $(k,\mu)$ in $\N\times[0,1]$, bounded from below and satisfies that $a_{\alpha\beta}(x) = \overline{a_{\beta\alpha}(x)}$ for all multi indices $\alpha$ and $\beta$. Then for all $\varepsilon>0$ there exists a pair of sesquilinear forms  $\mathcal{A}_{\hbar,\varepsilon}^{+}$ and  $\mathcal{A}_{\hbar,\varepsilon}^{-}$ such that

\begin{equation}
\mathcal{A}_{\hbar,\varepsilon}^{-}[\varphi,\varphi]\leq \mathcal{A}_{\hbar}[\varphi,\varphi]\leq\mathcal{A}_{\hbar,\varepsilon}^{+}[\varphi,\varphi]
\end{equation}

for all $\varphi$ in $\mathcal{D}(   \mathcal{A}_\hbar)$. Moreover, for all sufficiently small $\varepsilon$ the sesquilinear forms  $\mathcal{A}_{\hbar,\varepsilon}^{\pm}$ will be elliptic. If  $E$ is a non-critical value of $ \mathcal{A}_{\hbar}$ then $E$ will also be non-critical for  $\mathcal{A}_{\hbar,\varepsilon}^{\pm}$ for all $\varepsilon$ sufficiently small. The sesquilinear forms  $\mathcal{A}_{\hbar,\varepsilon}^{\pm}$ is explicit given by

\begin{equation}\label{B.def_approx_quad_form}
\begin{aligned}
\mathcal{A}_{\hbar,\varepsilon}^{\pm}[\varphi,\psi] = {}& \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}^\varepsilon (x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx
\\
& \pm C_1 \varepsilon^{k+\mu}  \sum_{\abs{\alpha}\leq m}  \int_{\R^d} (\hbar D_x)^\alpha\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx, \qquad \varphi,\psi \in \mathcal{D}(   \mathcal{A}_\hbar),
\end{aligned}
\end{equation}

where $a_{\alpha\beta}^\varepsilon(x)$  are the smoothed functions of $a_{\alpha\beta}(x)$ according to Proposition <ref> and $C_1$ is some positive constant.

We start by considering the form $ \mathcal{A}_{\hbar,\varepsilon}$  given by

\begin{equation*}
\mathcal{A}_{\hbar,\varepsilon}[\varphi,\psi] =  \sum_{\abs{\alpha},\abs{\beta}\leq m}  \int_{\R^d} a_{\alpha\beta}^\varepsilon (x) (\hbar D_x)^\beta\varphi(x) \overline{(\hbar D_x)^\alpha\psi(x)} \, dx,
\end{equation*}

where we have replaced the coefficients of $ \mathcal{A}_{\hbar}$ with smooth functions made according to
Proposition <ref>. For $\varphi$ in
$\mathcal{D}( \mathcal{A}_{\hbar}) \cap \mathcal{D}( \mathcal{A}_{\hbar,\varepsilon})$ we
have by Cauchy-Schwarz inequality

\begin{equation}\label{B.diff_framing_1}
\begin{aligned}
|  \mathcal{A}_{\hbar}[\varphi,\varphi]
-  \mathcal{A}_{\hbar,\varepsilon}[\varphi,\varphi] | \leq {}&
\sum_{\abs{\alpha},\abs{\beta}\leq m}| \langle
(a_{\alpha\beta}-a_{\alpha\beta}^\varepsilon)(\hbar D)^\beta
\varphi, (\hbar D)^\alpha \varphi \rangle |
\\
\leq{}& \sum_{\abs{\alpha},\abs{\beta}\leq m}
\frac{1}{2\varepsilon^{k+\mu}}
\norm{(a_{\alpha\beta}-a_{\alpha\beta}^\varepsilon)(\hbar
D)^\beta \varphi}_{L^2(\R^d)}^2 +
\frac{\varepsilon^{k+\mu}}{2} \norm{(\hbar D)^\alpha
\varphi}_{L^2(\R^d)}^2
\\
\leq{}& c \varepsilon^{k+\mu} \sum_{\abs{\alpha}\leq m} \langle
(\hbar D)^{\alpha} \varphi , (\hbar D)^{\alpha} \varphi\rangle,
\end{aligned}
\end{equation}

where we in the last inequality have used Proposition <ref>. From this inequality we get that  $\mathcal{D}( \mathcal{A}_{\hbar})= \mathcal{D}( \mathcal{A}_{\hbar,\varepsilon})$. We recognise the last bound in (<ref>) as the
quadratic form associated to $(I-\hbar^2\Delta)^m$. Hence for
sufficiently choice of constant $C_1$ we can choose the approximating forms $ \mathcal{A}_{\hbar,\varepsilon}^{\pm}$ to be given by (<ref>) such that

\begin{equation}
\mathcal{A}_{\hbar,\varepsilon}^{-}[\varphi,\varphi]\leq \mathcal{A}_{\hbar}[\varphi,\varphi]\leq\mathcal{A}_{\hbar,\varepsilon}^{+}[\varphi,\varphi]
\end{equation}

for all $\varphi$ in $\mathcal{D}(   \mathcal{A}_\hbar)$.

In order to show that we can choose the forms $ \mathcal{A}_{\hbar,\varepsilon}^{\pm}$ elliptic we note that

\begin{equation}\label{B.diff_framing_2}
\begin{aligned}
\sum_{\abs{\alpha}=\abs{\beta}= m}& a_{\alpha\beta}^\varepsilon
(x) p^{\alpha+\beta} \pm C_1\varepsilon^{k+\mu}\abs{p}^{2m}
\\
={}&\sum_{\abs{\alpha}=\abs{\beta}= m}
(a_{\alpha\beta}^\varepsilon (x) -a_{\alpha\beta} (x) )
p^{\alpha+\beta} +\sum_{\abs{\alpha}=\abs{\beta}= m}
a_{\alpha\beta} (x) p^{\alpha+\beta} \pm
C_1 \varepsilon^{k+\mu}\abs{p}^{2m}
\\
\geq& (C-(C_1-c)\varepsilon^{k+\mu}) \abs{p}^{2m}
\geq \tilde{C} \abs{p}^{2m},
\end{aligned}
\end{equation}

for sufficiently small $\varepsilon$ and all $(x,p)$ in
$\R_x^d\times \R_p^d$. This gives us that both forms $ \mathcal{A}_{\hbar,\varepsilon}^{\pm}$ are elliptic.

For the last part we assume $E$ is a non-critical value for the
form $ \mathcal{A}_{\hbar}$. That is there exist a $c>0$ such that

\begin{equation*}
|\nabla_p a_0(x,p)| \geq c \quad\text{for all } (x,p)\in a_0^{-1}(\{E\}),
\end{equation*}

\begin{equation*}
a_0(x,p) =  \sum_{\abs{\alpha},\abs{\beta}\leq m} a_{\alpha\beta}(x)p^{\alpha+\beta}.
\end{equation*}

In order to prove that $E$ is a non-critical value for the framing
forms we need to find an expression for
$a_{\varepsilon,0}^{-1}(\{E\})$ for the framing operators, where we
have omitted the $+$ and $-$ in the notation. By the ellipticity we
can in the following calculations with out loss of generality assume
$p$ belongs to a bounded set. We have

\begin{equation}\label{B.diff_framing_3}
\begin{aligned}
= \sum_{\abs{\alpha},\abs{\beta}\leq m}
(a_{\alpha\beta}^\varepsilon(x) -
a_{\alpha\beta}(x))p^{\alpha+\beta} \pm C_1
\varepsilon^{k+\mu}(1+p^2)^m +
\sum_{\abs{\alpha},\abs{\beta}\leq m}
\end{aligned}
\end{equation}

Since we can assume $p$ to be in a compact set we have that

\begin{equation*}
\Big| \sum_{\abs{\alpha},\abs{\beta}\leq m} (a_{\alpha\beta}^\varepsilon(x) - a_{\alpha\beta}(x))p^{\alpha+\beta}  \pm C_1 \varepsilon^{k+\mu}(1+p^2)^m \Big| \leq C \varepsilon^{k+\mu}.
\end{equation*}

This combined with (<ref>) implies the inclusion

\begin{equation*}
\{ (x,p) \in\R^{2d} \, |\,  a_{\varepsilon,0}(x,p) =E \} \subseteq \{ (x,p) \in\R^{2d} \, |\, | a_{0}(x,p) -E | \leq C \varepsilon^{k+\mu} \}.
\end{equation*}

Hence for a sufficiently small $\varepsilon$ we have the inclusion

\begin{equation}\label{B.diff_framing_4}
\{ (x,p) \in\R^{2d} \, |\,  a_{\varepsilon,0}(x,p) =E \} \subseteq \{ (x,p) \in\R^{2d} \, |\,  \abs{\nabla_p a_{0}(x,p)} \geq \frac{c}{2} \}.
\end{equation}

by continuity. For a point $(x,p)$ in
$\{ (x,p) \in\R^{2d} \, |\, a_{\varepsilon,0}(x,p) =E \}$ we have

\begin{equation}\label{B.diff_framing_5}
\nabla_p a_{\varepsilon,0}(x,p)
= \sum_{\abs{\alpha},\abs{\beta}\leq m}
(a_{\alpha\beta}^\varepsilon(x) - a_{\alpha\beta}(x)) \nabla_p
p^{\alpha+\beta} \pm C_1 \varepsilon^{k+\mu} \nabla_p (1+p^2)^m
+ \nabla_p a_{0}(x,p).
\end{equation}

Again since we can assume $p$ to be contained in a compact set we

\begin{equation*}
\Big| \sum_{\abs{\alpha},\abs{\beta}\leq m} (a_{\alpha\beta}^\varepsilon(x) - a_{\alpha\beta}(x)) \nabla_p p^{\alpha+\beta}  \pm C_1 \varepsilon^{k+\mu}  \nabla_p (1+p^2)^m \Big| \leq C \varepsilon^{k+\mu}.
\end{equation*}

Combining this with (<ref>) and
(<ref>) we get

\begin{equation}
|\nabla_p a_{\varepsilon,0}(x,p)| \geq  |\nabla_p a_{0}(x,p)| - C \varepsilon^{k+\mu} \geq \frac{c}{2} -   C \varepsilon^{k+\mu} \geq \frac{c}{4}.
\end{equation}

where the last inequality is for $\varepsilon$ sufficiently
small. This inequality proves $E$ is also a non-critical value of
the framing forms.

The framing forms constructed in the previous proposition are
forms with smooth coefficients. But when we take derivatives of
these coefficients we start to get negative powers of $\varepsilon$
from some point. Hence we can not associate a “classic”  pseudo-differential
operator to the form. We will instead  in the following sections consider a “rough” theory for pseudo-differential
operators. Here we will  see that it is in fact possible to
verify most of the results from classic theory of pseudo-differential
operators in this rough theory. After this has been developed we will return to these
framing forms.

As a last remark of this section note that there is not a unique way to construct these framing forms.

§ DEFINITIONS AND PROPERTIES OF ROUGH PSEUDO-DIFFERENTIAL

In this section we will inspired by the approximation results in the
previous section define a class of pseudo-differential operators with
rough symbols. We will further state and prove some of the properties for these
operators. The definitions are very
similar to the definitions in the monograph <cit.> and we will see that the properties of these operators can be deduced from the results in the monograph <cit.>.

§.§ Definition of rough operators and rough symbols

We start for the sake of completeness by recalling the definition of a tempered weight

A tempered weight function on $\R^D$ is a continuous function

\begin{equation*}
m:\R^D \rightarrow [0,\infty[,
\end{equation*}

for which there exists positive constants $C_0$, $N_0$ such that for
all points $x_1$ in $\R^D$ the estimate

\begin{equation*}
m(x) \leq C_0 m(x_1) \lambda(x_1-x)^{N_0},
\end{equation*}

holds for all points $x$ in $\R^D$.

For our purpose here we will consider the cases where $D=2d$ or
$D=3d$.  These
types of functions are in the literature sometimes called order
functions. This is the case in the monographs
<cit.>. But we have chosen the name tempered
weights to align with the terminology in the monographs
<cit.>. We can now define the symbols we will be working with.

Let $\Omega \subseteq \R_x^d \times \R_p^d \times \R_y^d$ be open,
$\rho$ be in $[0,1]$, $\varepsilon>0$, $\tau$ be in $\Z$ and $m$ a
tempered weight function on $\R_x^d \times \R_p^d \times \R_y^d$. We
call a function $a_\varepsilon$ a rough symbol of regularity $\tau$
with weights $(m,\rho, \varepsilon)$ if $a_\varepsilon$ is in
$C^{\infty}(\Omega)$ and satisfies that

\begin{equation}\label{B.symbolb_est_def}
\begin{aligned}
|\partial_x^\alpha\partial_p^\beta \partial_y^\gamma
\leq
\varepsilon^{\min(0,\tau-\abs{\alpha}-\abs{\gamma})} m (x,p,y)
\lambda(x,p,y)^{-\rho (\abs{\alpha}+\abs{\beta}+\abs{\gamma})}
\end{aligned}
\end{equation}

for all $(x,p,y)$ in $\Omega$ and $\alpha$, $\beta$, $\gamma$ in
$\N^d$, where the constants $C_{\alpha\beta\gamma}$'s do not depend on $\varepsilon$. The space of these functions is denoted
$\Gamma_{\rho,\varepsilon}^{m,\tau} \left(\Omega\right)$.

The space $\Gamma_{\rho,\varepsilon}^{m,\tau} \left(\Omega\right)$ can be turned into a Fréchet space with semi norms associated to the estimates in (<ref>).
It is important to note that the semi norms on  $\Gamma_{\rho,\varepsilon}^{m,\tau} \left(\Omega\right)$ should be chosen weighted such that the norms associated to a set of numbers $\alpha,\beta,\gamma$ will be bounded by the constant $C_{\alpha\beta\gamma}$ and hence independent of $\varepsilon$.

If $\varepsilon$ is equal to $1$, then these symbols are the same as
the symbols defined in the monograph <cit.>
(Definition II-$10$).  We will always assume $\varepsilon\leq1$ as we are interested in the cases of very small $\varepsilon$.

We will later call a function $a_\varepsilon(x,p)$ or $b_\varepsilon(p,y)$ a
rough symbol if it satisfies the above definition in the two
variables $x$ and $p$ or $p$ and $y$. This more general definition
is made in order to define the different forms of quantisation and
the interpolation between them.
If we say a symbol of regularity $\tau$ with tempered weight $m$ we implicit assume that $\rho=0$. This type of rough symbols is contained in the class of rough
symbols consider in <cit.>.

We will later assume that a rough symbol is a tempered weight. When this is done we will implicit assume that the constants from the definition of a tempered weight is independent of $\varepsilon$. This is an important assumption since we need the estimates we make to be uniform for $\hbar$ in $(0,\hbar_0]$ with $\hbar_0>0$ sufficiently small and then for a choice of $\delta$ in $(0,1)$ we need the estimates to be uniform for $\varepsilon$ in $[\hbar^{1-\delta},1]$.

Essentially the constants will be uniform for both  $\hbar$ in $(0,\hbar_0]$ and $\varepsilon$ in $(0,1]$, but if $\varepsilon\leq\hbar$ then the estimates will diverge in the semiclassical parameter. Hence we will assume the lower bound on $\varepsilon$.

The assumption that $\varepsilon\geq\hbar^{1-\delta}$ is in <cit.> called a microlocal uncertainty principal. In <cit.> there is two parameter instead of just one. This other parameter can be used to scale in the $p$-variable.

As we are interested in asymptotic expansions in the semiclassical
parameter we will define $\hbar$-$\varepsilon$-admissible symbols,
which is the symbols depending on the semiclassical parameter $\hbar$
for which we can make an expansion in $\hbar$.

With the notation from Definition <ref>. We call a symbol $a_\varepsilon(\hbar)$
$\hbar$-$\varepsilon$-admissible of regularity $\tau$ with weights
$(m,\rho, \varepsilon)$ in $\Omega$, if for fixed $\varepsilon$ and a $\hbar_0>0$ the map that takes $\hbar$
into $a_\varepsilon(\hbar)$ is smooth from $(0,\hbar_0]$ into
$\Gamma_{\rho,\varepsilon}^{m,\tau} \left(\Omega\right)$ such that
there exists a $N_0$ in $\N$ such for all $N\geq N_0$ we have

\begin{equation*}
a_\varepsilon(x,p,y;\hbar) = a_{\varepsilon,0}(x,p,y) + \hbar a_{\varepsilon,1}(x,p,y) + \dots + \hbar^N a_{\varepsilon,N}(x,p,y) + \hbar^{N+1} r_N(x,p,y;\hbar),
\end{equation*}

where $a_{\varepsilon,j}$ is in
$\Gamma_{\rho,\varepsilon,-2j}^{m,\tau_j} \left(\Omega\right)$ with
the notation $\tau_j=\tau-j$ and $r_N$ is a symbol satisfying the

\begin{equation*} \begin{aligned}
\hbar^{N+1} |\partial_x^\alpha& \partial_p^\beta \partial_y^\gamma
\\
\leq{}& C_{\alpha\beta\gamma} \hbar^{\kappa_1(N)}
\varepsilon^{-\abs{\alpha}-\abs{\gamma}
}m(x,y,p) \lambda(x,y,p)^{-\rho(\kappa_2(N)+
\abs{\alpha}+\abs{\beta} +\abs{\gamma})},
\end{aligned} \end{equation*}

where $\kappa_1$ is a positive strictly increasing function and
$\kappa_2$ is non-decreasing function. For $k$ in $\Z$
$\Gamma_{\rho,\varepsilon,k}^{m,\tau} \left(\Omega\right)$ is the space
of rough symbols of regularity $\tau$ with weights
$(m(1+|(x,y,p)|)^{k\rho},\rho, \varepsilon)$.

We will also use the terminology $\hbar$-$\varepsilon$-admissible
for symbols in two variables, where the definition is the same just
in two variables. This definition is slightly different to the
usual definition of an $\hbar$-admissible symbol
<cit.>. One
difference is in the error term.

The functions $\kappa_1$ and $\kappa_2$
will in most cases be dependent on the tempered weight function through the constants in the definition of a tempered weight, the
regularity $\tau$ and the dimension $d$.  It should be noted that
the function $\kappa_2$ might be constant negative.

We will now define the pseudo-differential operators associated to the
rough symbols. We will call them rough pseudo-differential operators.

Let $m$ be a tempered weight function on
$\R^d_x\times\R^d_p\times\R_y^d$, $\rho$ in $[0,1]$, $\varepsilon>0$
and $\tau$ in $\Z$. For a rough symbol $a_\varepsilon$ in
we associate the operator $\Op(a_\varepsilon)$ defined by

\begin{equation*}
\Op(a_\varepsilon)\psi(x) = \frac{1}{(2\pi \hbar)^{d}} \int_{\R^{2d}} e^{i \hbar^{-1} \langle{x-y},{p}\rangle }
a_\varepsilon( x,p,y) \psi(y) \, d y \, d p,
\end{equation*}

for $\psi$ in $\mathcal{S}(\R^d)$.

With the notation from
Definition <ref>. We remark that
the integral in the definition of $\Op(a_\varepsilon)\psi(x)$ shall
be considered as an oscillating integral. By applying the techniques
for oscillating integrals it can be proven that $\Op(a_\varepsilon)$
is a continuous linear operator from $\mathcal{S}(\R^d)$ into
itself. The proof of this is analogous to the proof in
<cit.> in the non-rough case. By duality it can
also be defined as an operator from $\mathcal{S}'(\R^d)$ into

We call an operator $A_\varepsilon(\hbar)$ from
$\hbar$-$\varepsilon$-admissible of regularity $\tau \geq0$ with
tempered weight $m$ if for fixed $\varepsilon$ and a $\hbar_0>0$ the map

\begin{equation*}
A_\varepsilon: (0,\hbar_0] \rightarrow\mathcal{L}(\mathcal{S}(\R^d),L^2(\R^d))
\end{equation*}

is smooth and there exists a sequence $a_{\varepsilon,j}$ in
where $\tau_0=\tau$ and $\tau_{j+1}= \tau_j-1$ and a sequence $R_N$
in $\mathcal{L}(L^2(\R^d))$ such that for $N\geq N_0$, $N_0$
sufficient large,

\begin{equation}\label{B:op_for_prin_sym}
A_\varepsilon(\hbar)=\sum_{j=0}^N \hbar^j  \Op( a_{\varepsilon,j}) + \hbar^{N+1}  R_N(\varepsilon,\hbar),
\end{equation}

\begin{equation*}
\hbar^{N+1}  \norm{R_N(\varepsilon,\hbar)}_{\mathcal{L}(L^2(\R^d))} \leq \hbar^{\kappa(N)} C_N ,
\end{equation*}

for a strictly positive increasing function $\kappa$.

By the results in Theorem <ref> we have that if the
tempered weight function $m$ is in $L^\infty(\R^d)$. Then for a
$\hbar$-$\varepsilon$-admissible symbol $a_\varepsilon(\hbar)$ of
regularity $\tau\geq0$ with tempered weight $m$ the operator
$A_\varepsilon(\hbar)=\Op(a_\varepsilon(\hbar))$ is a
$\hbar$-$\varepsilon$-admissible operator of regularity $\tau$.

When we have an operator $  A_\varepsilon(\hbar)$  with an expansion

\begin{equation*}
A_\varepsilon(\hbar)=\sum_{j\geq0} \hbar^j  \Op( a_{\varepsilon,j}),
\end{equation*}

where the sum is understood as a formal sum and in the sense that for all $N$ sufficiently large there exists $R_N$ in $\mathcal{L}(L^2(\R^d))$ such that the operator is of the same form as in (<ref>). Then we call the symbol $a_{\varepsilon,0}$ the principal symbol and the symbol $a_{\varepsilon,1}$ the subprincipal symbol.

Let $A_\varepsilon(\hbar)$ be a $\hbar$-$\varepsilon$-admissible of
regularity $\tau$ with tempered weight $m$. For any $t$ in $[0,1]$
we call all $\hbar$-$\varepsilon$-admissible symbols
$b_\varepsilon(\hbar)$ in
$\Gamma_{0,\varepsilon}^{m,\tau}(\R^d_x\times\R^d_p)$ such,

\begin{equation*}
A_\varepsilon(\hbar) \psi(x) =  \frac{1}{(2\pi \hbar)^{d}} \int_{\R^{2d}} e^{i \hbar^{-1} \langle{x-y},{p}\rangle } b_\varepsilon( (1-t)x +ty,p;\hbar) \psi(y) \, d y \, d p,
\end{equation*}

for all $\psi\in\mathcal{S}(\R^d)$ and all $\hbar\in]0,\hbar_0]$, where $\hbar_0$ is a strictly positive number,
rough $t$-$\varepsilon$-symbols of regularity $\tau$ associated to

In general for a symbol $b_\varepsilon(\hbar)$ in
$\Gamma_{\rho,\varepsilon}^{m,\tau}(\R^d_x\times\R^d_p)$ and $\psi$
in $\mathcal{S}(\R^d)$ we will use the notation

\begin{equation*}
\Opt(b_\varepsilon)\psi(x) = \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}}  e^{i\hbar^{-1} \langle x-y,p\rangle} b_\varepsilon( (1-t)x +ty,p;\hbar) \psi(y) \, d y \, d p
\end{equation*}

We have the special case of Weyl quantisation when $t=\frac12$,
which is the one we will work the most with. In this case we write

\begin{equation*}
\OpN{\frac12}(b_\varepsilon) = \OpW(b_\varepsilon).
\end{equation*}

For some application we will need stronger assumptions than
$\hbar$-$\varepsilon$-admissibility of our operators. The operators
satisfying these stronger assumptions will be called strongly
$\hbar$-$\varepsilon$-admissible operators with some regularity. As an
example we could consider a symbol $a_\varepsilon(x,p)$ in
$\Gamma_{\rho,\varepsilon}^{m,\tau} (\R_x^{d}\times\R^d_p)$. For this
symbol define
$\tilde{a}_\varepsilon(x,p,y) =a_\varepsilon(tx+(1-t)y,p) $ and ask if
this symbol is in
(\R_x^{d}\times\R^d_p\times\R^d_y)$, where
$\tilde{m}(x,p,y)=m(tx+(1-t)y,p)$. The answer will not in general be
positive. Hence in general we can not ensure decay in the variables
$(x,p,y)$ when viewing a function of $(x,p)$ as a function of
$(x,p,y)$. With this in mind we define a new class of symbols and
strongly $\hbar$-$\varepsilon$-admissible operators.

A symbol $a_\varepsilon$ belongs to the class
(\R_x^{d}\times\R^d_p\times\R^d_y)$ if $a_\varepsilon$ is in
$\Gamma_{0,\varepsilon}^{m,\tau} (\R_x^{d}\times\R^d_p\times\R^d_y)$
and there exists a positive $\nu$ such that

\begin{equation*}
a_\varepsilon \in \Gamma_{\rho,\varepsilon}^{m,\tau} (\Omega_\nu),
\end{equation*}

where $\Omega_\nu=\{(x,p,y)\in \R^{3d} \,|\, \abs{x-y}<\nu\}$.

We call the family of operators
$A_\varepsilon(\hbar)=\Op(a_\varepsilon(\hbar))$ strongly
$\hbar$-$\varepsilon$-admissible of regularity $\tau$ if
$a_\varepsilon(\hbar)$ is an $\hbar$-$\varepsilon$-admissible symbol
of regularity $\tau$ with respect to the weights $(m,0,\varepsilon)$
on $ \R_x^{d}\times\R^d_p\times\R^d_y$ and the weights
$(m,\rho,\varepsilon)$ on
$\Omega_\nu=\{(x,p,y)\in \R^{3d} \,|\, \abs{x-y}<\nu\}$ for a
positive $\nu$.

We note that a strongly $\hbar$-$\varepsilon$-admissible
operator is also $\hbar$-$\varepsilon$-admissible but as a
consequence of the definition the error term of a strongly
$\hbar$-$\varepsilon$-admissible operator will be a
pseudo-differential operator and not just a bounded operator as for
the $\hbar$-$\varepsilon$-admissible operators.

Before we start proving/stating results about these operators we make the following observation.

Let $m$ be a tempered weight function on
$\R^d_x\times\R^d_p\times\R_y^d$, $\rho$ in $[0,1]$, $0<\varepsilon\leq1$
and $\tau$ in $\N_0$. Consider  a rough symbol $a_\varepsilon$ in
$\Gamma_{\rho,\varepsilon}^{m,\tau}(\R^d_x\times\R^d_p\times\R_y^d)$. We suppose that there is a $\delta$ in $(0,1)$ such $\varepsilon \geq \hbar^{1-\delta}$ and consider the operator $\Op(a_\varepsilon)$ associated with $a_\varepsilon$. We define the unitary dilation operator $\mathcal{U}_{\varepsilon}$ by

\begin{equation*}
\mathcal{U}_{\varepsilon} f(x) = \varepsilon^{d/2} f(\varepsilon x),
\end{equation*}

for $f$ in $L^2(\R^d)$. We observe that with this operator we obtain the following equality

\begin{equation*}
\mathcal{U}_{\varepsilon} \Op(a_\varepsilon(x,p,y)) \mathcal{U}_{\varepsilon}^{*} = \Opdef_{\hbar^{\delta}}(a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p,\varepsilon y)).
\end{equation*}

\begin{equation*}
a_{\varepsilon}^{\#}(x,p,y) = a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p,\varepsilon y).
\end{equation*}

Since we have that $a_\varepsilon$ is in $\Gamma_{\rho,\varepsilon}^{m,\tau}(\R^d_x\times\R^d_p\times\R_y^d)$ we get for all $\alpha, \beta, \gamma$ in $\N^d_0$ that

\begin{equation*}
\begin{aligned}
\MoveEqLeft  |\partial_x^\alpha\partial_p^\beta \partial_y^\gamma a_{\varepsilon}^{\#}|
\\
C_{\alpha\beta\gamma} \varepsilon^{\min(0,\tau-\abs{\alpha}-\abs{\gamma})+\abs{\alpha}+\abs{\gamma}} \left(\tfrac{\hbar^{1-\delta}}{\varepsilon}\right)^{\abs{\beta}}  m (\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p,\varepsilon y)\lambda(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p,\varepsilon y)^{-\rho (\abs{\alpha}+\abs{\beta}+\abs{\gamma})}
\\
C_{\alpha\beta\gamma}C_0 \varepsilon^{\min(0,\tau-\abs{\alpha}-\abs{\gamma})+\abs{\alpha}+\abs{\gamma}} \left(\tfrac{\hbar^{1-\delta}}{\varepsilon}\right)^{\abs{\beta}}  m ( x,  p, y)\lambda((1-\varepsilon) x,(1- \tfrac{\hbar^{1-\delta}}{\varepsilon}) p,(1-\varepsilon) y)^{N_0}
\\
&\leq  \tilde{C}_{\alpha\beta\gamma}   \tilde{m} (x,p,y),
\end{aligned}
\end{equation*}

where we have used that  $  \hbar^{1-\delta} \leq \varepsilon\leq1$ and used the notation $\tilde{m} = m\lambda^{N_0}$. This shows that $a_{\varepsilon}^{\#}$ is in $\Gamma_{0,1}^{\tilde{m},0}(\R^d_x\times\R^d_p\times\R_y^d)$ by definition of the set. As mentioned earlier this is the type of symbols usually considered in the literature.

In what follows we will use this observation to establish the symbolic calculus for the rough pseudo-differential operators. We will here mainly focus on the  results we will need later in the proofs of our main theorems.

We will now prove a connection between operators with symbols in the
(\R_x^{d}\times\R^d_p\times\R^d_y)$ and $t$-quantised operators.

Let $a_\varepsilon$ be a symbol in
(\R_x^{d}\times\R^d_p\times\R^d_y)$ of regularity $\tau\geq0$ with
weights $(m,\rho,\varepsilon)$ and

\begin{equation*}
A_\varepsilon(\hbar) \psi(x) = \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} e^{i\hbar^{-1}\langle x-y , p \rangle} a_\varepsilon(x,p,y) \psi(y) \, dy \,dp.
\end{equation*}

We suppose there is a $\delta$ in $(0,1)$ such
$\varepsilon \geq \hbar^{1-\delta}$. Then for every $t$ in $[0,1]$
we can associate a unique $t$-$\varepsilon$-symbol $b_t$ of
regularity $\tau$ with weights $(\tilde{m},\rho,\varepsilon)$, where
$\tilde{m}(x,p)=m(x,x,p)$. The $t$-$\varepsilon$-symbol $b_t$ is
defined by the oscillating integral

\begin{equation*}
b_t(x,p,\hbar) = \frac{1}{(2\pi\hbar)^d}  \int_{\R^{2d}}  e^{i\hbar^{-1}\langle u,q \rangle} a_\varepsilon (x+tu,p+q,x-(1-t)u) \, dq \,du
\end{equation*}

and symbol $b_t$ has the following asymptotic expansion

\begin{equation*}
b_t(x,p;\hbar) = \sum_{j=0}^N \hbar^j a_{\varepsilon,j}(x,p) + \hbar^{N+1} r_{\varepsilon,N+1}(x,p;\hbar),
\end{equation*}

\begin{equation*}
a_{\varepsilon,j} (x,p) =  \frac{(-i)^j}{j!} \langle D_u , D_p\rangle^j a_\varepsilon(x+tu, p,x-(1-t)u) \Big|_{u=0},
\end{equation*}

and the error term satisfies that

\begin{equation*}
\hbar^{N+1} |\partial_x^\alpha\partial_p^\beta r_{N+1}(x,p,\hbar) |
\leq   C_{d,N,\alpha,\beta}  \hbar^{N +1} \varepsilon^{-(\tau-N-2-d-\abs{\alpha})_{-}}  m(x,p,x)  \lambda(x,p)^{\rho N_0 } ,
\end{equation*}

for all $\alpha$ and $\beta$ in $\N^d$. In particular we have that

\begin{equation*} \begin{aligned}
a_{\varepsilon,0}(x,p) &= a_\varepsilon(x,p,x)
\\
a_{\varepsilon,1} (x,p) &= (1-t) (\nabla_y D_p
a_\varepsilon)(x,p,x) - t (\nabla_x D_p a_\varepsilon) (x,p,x).
\end{aligned} \end{equation*}

It can be noted
that in order for the error term not to diverge, when
the semiclassical parameter tends to zero, one needs to take $N$
such that

\begin{equation*}
\tau -1-d + \delta(N+2+d) \geq 0.
\end{equation*}

If the symbol is a polynomial in one of the variables or both then
the asymptotic expansion will be exact and a finite sum. This is in
particular the case when ordinary differential operators are

Let $\mathcal{U}_{\varepsilon}$ be the unitary dilation operator as defined in Observation <ref> and define the operator $ A_\varepsilon^{\#}(\hbar)$ by conjugation with $\mathcal{U}_{\varepsilon}$. This new operator is defined by

\begin{equation*}
A_\varepsilon^{\#}(\hbar) \psi(x) = \frac{1}{(2\pi\hbar^{\delta})^d} \int_{\R^{2d}} e^{i\hbar^{-\delta}\langle x-y , p \rangle} a_\varepsilon^{\#}(x,p,y) \psi(y) \, dy \,dp,
\end{equation*}

where $\psi$ is a Schwartz function.	From Observation <ref> we have that $a_\varepsilon^{\#}$ is in $\tilde{\Gamma}_{0,1}^{m,0}(\R^d_x\times\R^d_p\times\R_y^d)$. So by <cit.> the result is true for the operator $ A_\varepsilon^{\#}(\hbar)$. Due to the identity

\begin{equation*}
A_\varepsilon(\hbar)  = \mathcal{U}_{\varepsilon}^{*} A_\varepsilon^{\#}(\hbar) \mathcal{U}_{\varepsilon},
\end{equation*}

what remains is to conjugate the terms in the representation of  $A_\varepsilon^{\#}(\hbar)$ by $\mathcal{U}_{\varepsilon}^{*}$. First we observe that

\begin{equation}\label{proof_connection_t_sym_1}
\begin{aligned}
\MoveEqLeft b_t^{\#}(\varepsilon^{-1} x, \tfrac{\varepsilon}{\hbar^{1-\delta}}  p;\hbar^{\delta})
\\
={}& \frac{1}{(2\pi\hbar^{\delta})^d}  \int_{\R^{2d}}   e^{i\hbar^{-\delta}\langle u,q \rangle} a_\varepsilon^{\#} (\varepsilon^{-1} x+tu,\tfrac{\varepsilon}{\hbar^{1-\delta}}p+q,\varepsilon^{-1}x-(1-t)u) \, dq \,du
\\
={}& \frac{1}{(2\pi\hbar^{\delta})^d}  \int_{\R^{2d}}  e^{i\hbar^{-\delta}\langle u,q \rangle} a_\varepsilon (x+t \varepsilon u,p+ \tfrac{\hbar^{1-\delta}}{\varepsilon}q,x-(1-t) \varepsilon u) \, dq \,du
\\
={}& \frac{1}{(2\pi\hbar)^d}   \int_{\R^{2d}}  e^{i\hbar^{-1}\langle u,q \rangle} a_\varepsilon (x+t u,p+ q,x-(1-t)  u) \, dq \,du
\\
={}&    b_t(x,p,\hbar).
\end{aligned}
\end{equation}

Next  by using the identity obtained in (<ref>) we get that

\begin{equation}\label{proof_connection_t_sym_1.5}
\begin{aligned}
A_\varepsilon(\hbar)  = {}& \mathcal{U}_{\varepsilon}^{*} A_\varepsilon^{\#}(\hbar) \mathcal{U}_{\varepsilon}
= \mathcal{U}_{\varepsilon}^{*}   \Opdef_{\hbar^{\delta},t}(b_t^{\#}) \mathcal{U}_{\varepsilon}
\\
={}&  \Opt (b_t^{\#}(\varepsilon^{-1} x, \tfrac{\varepsilon}{\hbar^{1-\delta}}  p;\hbar^{\delta}) ) = \Opt (b_t)
\end{aligned}
\end{equation}

From  the  asymptotic expansion of $ b_t^{\#}(x,p;\hbar^\delta) $ obtained in <cit.>  we have that

\begin{equation}\label{proof_connection_t_sym_2}
b_t^{\#}(x,p;\hbar^\delta) = \sum_{j=0}^N (\hbar^\delta)^j a_{\varepsilon,j}^{\#}(x,p) + (\hbar^\delta)^{N+1} r_{\varepsilon,N+1}^{\#}(x,p;\hbar^{\delta}).
\end{equation}

In order to arrive at the stated expansion of $ b_t(x,p,\hbar)$ we have to find an expression of $a_{\varepsilon,j}^{\#}(\varepsilon^{-1}x,\tfrac{\varepsilon}{\hbar^{1-\delta}} p)$ in terms of $a_\varepsilon$. By definition of $a_{\varepsilon,j}^{\#}$ we have that

\begin{equation} \label{proof_connection_t_sym_3}
a_{\varepsilon,j}^{\#} (\varepsilon^{-1}x, \tfrac{\varepsilon}{\hbar^{1-\delta}}p) =  \frac{(-i)^j}{j!} \hbar^{(1-\delta) j}   \langle D_u , D_p\rangle^j a_\varepsilon ( x+t u, p, x-(1-t)u) \Big|_{u=0}.
\end{equation}

What remains is to prove that the error term follows the desired estimate. The error term is give by

\begin{equation} \label{proof_connection_t_sym_4}
r_{\varepsilon,N+1}(x,p;\hbar) = r_{\varepsilon,N+1}^{\#}(\varepsilon^{-1}x,\tfrac{\varepsilon}{\hbar^{1-\delta}}p;\hbar^{\delta}).
\end{equation}

In order to see that this function satisfied the stated estimate one needs to go back to the proof of <cit.> and consider the exact definition of the function. But from here it is a straight forward argument to see that that the desired bound is indeed true. Combining (<ref>), (<ref>) (<ref>), (<ref>) and (<ref>) we obtain the desired result.

Alternatively one can also follow the proof in <cit.> and do the full stationary phase argument. This gives a slightly longer proof but one obtains the same results.

From this Theorem we immediate obtain the following Corollary.

Let $t_1$ be in $[0,1]$ and $b_{t_1}$ be a
$t_1$-$\varepsilon$-symbol of regularity $\tau\geq0$ with weights
$(m,\rho,\varepsilon)$ and suppose $\varepsilon\geq\hbar^{1-\delta}$
for a $\delta$ in $(0,1)$. Let $A_\varepsilon(\hbar)$ be the
associated operator acting on a Schwarzt function by the formula

\begin{equation*}
A_\varepsilon(\hbar) \psi(x) = \frac{1}{(2\pi\hbar)^d}  \int_{\R^{2d}} e^{i\hbar^{-1}\langle x-y, p\rangle} b_{t_1}((1-t_1)x+t_1y,p) \psi(y) \,dy \,dp.
\end{equation*}

Then for every $t_2$ in $[0,1]$ we can associate an admissible
$t_2$-$\varepsilon$-symbol given by the expansion

\begin{equation*}
b_{t_2}(\hbar) = \sum_{j=0}^N \hbar^j b_{t_2,j}+  \hbar^{N+1} r_{\varepsilon,N+1}(x,p;\hbar),
\end{equation*}

\begin{equation*}
b_{t_2,j}(x,p) = \frac{( t_1-t_2)^j}{j!} (\nabla_x D_p)^j b_{t_1}(x,p),
\end{equation*}

and the error term satisfies that

\begin{equation*}
\hbar^{N+1} |\partial_x^\alpha\partial_p^\beta r_{N+1}(x,p,\hbar) |
\leq   C_{d,N,\alpha,\beta}  \hbar^{N +1} \varepsilon^{-(\tau-N-2-d-\abs{\alpha})_{-}}  m(x,p)  \lambda(x,p)^{\rho N_0 },
\end{equation*}

for all $\alpha$ and $\beta$ in $\N^d$, the number $N_0$ is the
number connected to the tempered weight $m$.

This corollary can also be proven directly by considering the kernel
as an oscillating integral and the integrant as a function in the
variable $t_1$. To obtain the corollary do a Taylor expansion in $t_1$
at the point $t_2$, then preform integration by parts a number of times and
then one would recover the result.

§.§ Composition of rough pseudo-differential operators

With the rough pseudo-differential operators defined and the ability
to interpolate between the different quantisations our next aim is
results concerning composition of rough pseudo-differential
operators. This is done in the following theorem. We will here omit the proof as it is analogous to the proof of Theorem <ref>. The idea is to conjugate the operators with $\mathcal{U}_\varepsilon$ use the results from e.g. <cit.> and then conjugate with $\mathcal{U}_\varepsilon^{*}$.

Let $A_\varepsilon(\hbar)$ and $B_\varepsilon(\hbar)$ be two
t-quantised operators given by

\begin{equation*}
A_\varepsilon(\hbar) \psi(x) = \frac{1}{(2\pi\hbar)^d}  \int_{\R^{2d}}e^{i\hbar^{-1}\langle x-z, p\rangle} a_\varepsilon((1-t)x+tz,p) \psi(z) \,dz \,dp
\end{equation*}

\begin{equation*}
B_\varepsilon(\hbar) \psi(z) = \frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}}e^{i\hbar^{-1}\langle z-y, q\rangle} b_\varepsilon((1-t)z+ty,q) \psi(y) \,dy \,dq,
\end{equation*}

where $a_\varepsilon$ and $b_\varepsilon$ be two rough symbols of
regularity $\tau_1,\tau_2 \geq 0$ with weights
$(m_1, \rho , \varepsilon)$ and $(m_2, \rho , \varepsilon)$
respectively. We suppose there exists a number $\delta\in(0,1)$ such that
$\varepsilon\geq\hbar^{1-\delta}$.  Then the operator
$C_\varepsilon(\hbar) = A_\varepsilon(\hbar) \circ
B_\varepsilon(\hbar)$ is strongly $\hbar$-$\varepsilon$-admissible
and $C_ \varepsilon(\hbar) = \Opt(c_\varepsilon)$, where
$c_\varepsilon$ is a rough admissible symbol of regularity
$\tau = \min(\tau_1,\tau_2)$ with weights
$(m_1m_2, \rho , \varepsilon)$. The symbol $c_\varepsilon$ satisfies
the following: For every $N \geq N_\delta$ we have

\begin{equation*}
c_\varepsilon(\hbar) = \sum_{j=0}^N \hbar^j c_{\varepsilon,j} + \hbar^{N+1} r_{\varepsilon,N+1}(a_\varepsilon,b_\varepsilon;\hbar)
\end{equation*}

\begin{equation*}
c_{\varepsilon,j}(x,p) = \frac{(i\sigma(D_u,D_\mu;D_v,D_\nu))^j}{j!} [\tilde{a}_\varepsilon(x,p;u,v,\mu,\nu)\tilde{b}_\varepsilon(x,p;u,v,\mu,\nu)]\Big|_{\substack{u=v=0\\\mu=\nu=0}},
\end{equation*}

\begin{equation*}
\begin{aligned}
\sigma(u,\mu;v,\nu) &= \langle v,\mu \rangle - \langle u, \nu
\rangle
\\
\tilde{a}_\varepsilon(x,p;u,v,\mu,\nu) &= a_\varepsilon(x+tv
+t(1-t)u,\nu+(1-t)\mu +p)
\\
\tilde{b}_\varepsilon(x,p;u,v,\mu,\nu) &= b_\varepsilon(x+(1-t)v
- t(1-t)u,\nu-t\mu +p).
\end{aligned}
\end{equation*}

Moreover the error term
$r_{\varepsilon,N+1}(a_\varepsilon,b_\varepsilon;\hbar)$ satisfies
that for every multi indices $\alpha,\beta$ in $\N^d$ there exists a
constant $C(N,\alpha,\beta)$ independent of $a_\varepsilon$ and
$b_\varepsilon$ and a natural number $M$ such that:

\begin{equation*}
\begin{aligned}
\hbar^{N+1}|\partial^\alpha_x\partial^\beta_p
\leq {}& C \varepsilon^{-\abs{\alpha}}\hbar^{\delta(\tau-N-2d-2)_{-}
+ \tau -2d-1}
\mathcal{G}_{M,\tau}^{\alpha,\beta}(a_\varepsilon,m_1,b_\varepsilon,m_2)
\\
&\times m_1(x,\xi) m_2(x,\xi) \lambda(x,\xi)^{-\rho (\tilde{N}(M) + \abs{\alpha}+ \abs{\beta})},
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\mathcal{G}_{M,\tau}^{\alpha,\beta}(a_\varepsilon,m_1,b_\varepsilon,m_2)
= \sup_{{\substack{\abs{\gamma_1 +
\gamma_2}+\abs{\eta_1 + \eta_2}\leq M \\ (x,\xi) \in
\R^{2d}}}} & \varepsilon^{(\tau-M)_{-}+\abs{\alpha}}   \lambda(x,\xi)^{\rho(\abs{\gamma_1 + \gamma_2}+\abs{\eta_1 + \eta_2})}
\\
&\times  \frac{ \abs{\partial_x^\alpha\partial_\xi^\beta
( \partial_{x}^{\gamma_1} \partial_{\xi}^{\eta_1}a_\varepsilon(x,\xi) \partial_{x}^{\gamma_2} \partial_{\xi}^{\eta_2}b_\varepsilon(x,\xi))}}{m_1(x,\xi)
\end{aligned}
\end{equation*}

The function $\tilde{N}(M)$ is also depending on the weights $m_1$,
$m_2$ and the dimension $d$.

The number $N_\delta$ is explicit and it is the smallest number such

\begin{equation*}
\delta(N_\delta +2d + 2-\tau) +\tau > 2d+1.
\end{equation*}

This restriction is made in order to ensure that the error term is
estimated by the semiclassical parameter raised to a positive
power.  The main difference between this result and the classical analog is that for this new class there is a minimum of terms
in the expansion of the symbol for the composition in order to
obtain an error that does not diverge as $\hbar\rightarrow0$.

The form of the $c$'s we obtain in the theorem is sometimes expressed as

\begin{equation*} \begin{aligned}
c_\varepsilon(x,\xi;\hbar) = &e^{i\hbar
\sigma(D_u,D_\mu;D_v,D_\nu)} [ a_\varepsilon(x+tv+t(1-t)u,\nu
+(1- t)\mu + \xi)
\\
\hbox{}&\times b_\varepsilon(x+(1-t)v - t(1-t)u,\nu - t\mu + \xi)
] \Big|_{\substack{u=v=0 \\ \mu=\nu=0}}.
\end{aligned} \end{equation*}

We will see the 3 most important cases for this presentation of the composition for
$t$-quantised operators. We suppose the assumptions of
Theorem <ref> is

$\boldsymbol{t=0}$: In this case the amplitude will be independent of $u$
hence we have

\begin{equation*}
c_\varepsilon(x,p;\hbar) = e^{i\hbar \langle D_y,D_q,\rangle} [ a_\varepsilon(x,q) b_\varepsilon(y,p) ] \Big|_{\substack{y=x \\ p=q}}.
\end{equation*}

This gives the formula

\begin{equation*}
c_{\varepsilon,j}(x,p) = \sum_{\abs{\alpha}=j} \frac{1}{\alpha!} \partial_p^\alpha a_\varepsilon(x,p) D_x^\alpha b_\varepsilon(x,p).
\end{equation*}

$\boldsymbol{t=1}$: This case is similar to the one above, except a change
of signs. The composition formula is given by

\begin{equation*}
c_\varepsilon(x,p;\hbar) = e^{-i\hbar \langle D_y,D_q,\rangle} [ a_\varepsilon(y,p) b_\varepsilon(x,q) ] \Big|_{\substack{y=x \\ p=q}}.
\end{equation*}

This gives the formula

\begin{equation*}
c_{\varepsilon,j}(x,p) = (-1)^j \sum_{\abs{\alpha}=j} \frac{1}{\alpha!} D_x^\alpha a_\varepsilon(x,p) \partial_p^\alpha b_\varepsilon(x,p).
\end{equation*}

$\boldsymbol{t=}\tfrac{\boldsymbol{1}}{\boldsymbol{2}}$ (Weyl-quatisation):
After a bit of additional work we can arrive at the usual formula in this case as well. That is we obtain that

\begin{equation*}
c_\varepsilon(x,p;\hbar) = e^{i\frac{\hbar}{2} \sigma(D_x,D_p;D_y,D_q)} [ a_\varepsilon(x,p) b_\varepsilon(y,q) ] \Big|_{\substack{y=x \\ p=q}}
\end{equation*}

\begin{equation*}
c_{\varepsilon,j}(x,p) = \Big(\frac{i}{2} \Big)^j \frac{1}{j!} [\sigma(D_x,D_p;D_y,D_q)]^j  a_\varepsilon(x,p) b_\varepsilon(y,q) \Big|_{\substack{y=x \\ p=q}}.
\end{equation*}

The last equation can be rewritten by some algebra to the classic

\begin{equation*}
c_{\varepsilon,j}(x,p) = \sum_{\abs{\alpha}+\abs{\beta}=j} \frac{1}{\alpha!\beta!}\Big(\frac{1}{2} \Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}} (\partial_p^\alpha D_x^\beta a_\varepsilon) (\partial_p^\beta D_x^\alpha b_\varepsilon)(x,p).
\end{equation*}

In all three cases we can note that the symbols for the compositions of operators is the same as in the non-rough case.

We now have composition of operators given by a single symbol. The
next result generalises the previous to composition of strongly
$\hbar$-$\varepsilon$-admissible operators. Moreover it verifies that
the strongly $\hbar$-$\varepsilon$-admissible operators form an
algebra. More precisely we have.

Let $A_\varepsilon(\hbar)$ and $B_\varepsilon(\hbar)$ be two
strongly $\hbar$-$\varepsilon$-admissible operators of regularity
$\tau_a\geq0$ and $\tau_b\geq0$. with weights
$(m_1, \rho , \varepsilon)$ and $(m_2, \rho , \varepsilon)$
respectively and of the form

\begin{equation*}
A_\varepsilon(\hbar) = \OpW(a_\varepsilon) \quad\text{and}\quad B_\varepsilon(\hbar) = \OpW(b_\varepsilon).
\end{equation*}

We suppose $\varepsilon \geq \hbar^{1-\delta}$ for a $\delta$ in
$(0,1)$ and let $\tau=\min(\tau_a,\tau_b)$. Then is
B_\varepsilon(\hbar)$ a strongly $\hbar$-$\varepsilon$-admissible
operators of regularity $\tau\geq0$ with weights
$(m_1 m_2, \rho , \varepsilon)$. The symbol
$c_\varepsilon(x,p;\hbar)$ of $C_\varepsilon(\hbar)$ has for
$N\geq N_\delta$ the expansion

\begin{equation*}
c_\varepsilon(x,p;\hbar) = \sum_{j=0}^N \hbar^j c_{\varepsilon,j}(x,p) + \hbar^{N+1} \mathcal{R}_\varepsilon(a_\varepsilon(\hbar), b_\varepsilon(\hbar);\hbar),
\end{equation*}

\begin{equation*}
c_{\varepsilon,j}(x,p) = \sum_{\abs{\alpha}+\abs{\beta}+k+l=j} \frac{1}{\alpha!\beta!}\Big(\frac{1}{2} \Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}} (\partial_p^\alpha D_x^\beta a_{\varepsilon,k}) (\partial_p^\beta D_x^\alpha b_{\varepsilon,l})(x,p).
\end{equation*}

The symbols $a_{\varepsilon,k}$ and $b_{\varepsilon,l}$ are from the
expansion of $a_\varepsilon$ and $b_\varepsilon$ respectively. Let

\begin{equation*}
a_\varepsilon(x,p) = \sum_{k=0}^N \hbar^j a_{\varepsilon,j}(x,p) + \hbar^{N+1} r_{\varepsilon,N+1}(a_\varepsilon,x,p;\hbar)
\end{equation*}

and equvalint for $b_\varepsilon(x,p)$. Then for every multi indices
$\alpha$, $\beta$ there exists a constant $C(\alpha,\beta,N)$
independent of $a_\varepsilon$ and $b_\varepsilon$ and an integer
$M$ such that

\begin{equation*}
\begin{aligned}
\hbar&^{N+1} |\partial_x^\alpha \partial_p^\beta
\mathcal{R}_\varepsilon(a_\varepsilon(\hbar),
b_\varepsilon(\hbar);x,p;\hbar) |
\\
&\leq C(\alpha,\beta,N) \hbar^{\delta(\tau-N-2d-2)_{-}+ \tau - 2d
- 1} \varepsilon^{-\abs{\alpha}} m_1(x,p) m_2(x,p)
\lambda(x,p)^{-\rho(\tilde{N}(M)+\abs{\alpha}+\abs{\beta})}
\\
&\phantom{\leq}{} \times \Big[ \sum_{j=0}^N \{
\mathcal{G}^{\alpha,\beta}_{M,\tau}(a_{\varepsilon,j},m_1,
r_{\varepsilon,N+1}(b_\varepsilon(\hbar)),m_2) +
\mathcal{G}^{\alpha,\beta}_{M,\tau}(r_{\varepsilon,N+1}(a_{\varepsilon}(\hbar)),m_1,b_{\varepsilon,j},m_2)\}
\\
&\phantom{\leq}{} + \; \sum_{\mathclap{N\leq j+k \leq 2N}} \;
\mathcal{G}^{\alpha,\beta}_{M,\tau}(a_{\varepsilon,j},m_1,b_{\varepsilon,k},m_2)
\mathcal{G}^{\alpha,\beta}_{M,\tau}(r_{\varepsilon,N+1}(a_{\varepsilon}(\hbar)),m_1,
r_{\varepsilon,N+1}(b_\varepsilon(\hbar)),m_2) \Big],
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\mathcal{G}_{M,\tau}^{\alpha,\beta}(a_\varepsilon,m_1,b_\varepsilon,m_2)
= \sup_{{\substack{\abs{\gamma_1 +
\gamma_2}+\abs{\eta_1 + \eta_2}\leq M \\ (x,\xi) \in
\R^{2d}}}} & \varepsilon^{(\tau-M)_{-}+\abs{\alpha}} \frac{
\abs{\partial_x^\alpha\partial_\xi^\beta
( \partial_{x}^{\gamma_1} \partial_{\xi}^{\eta_1}a_\varepsilon(x,\xi) \partial_{x}^{\gamma_2} \partial_{\xi}^{\eta_2}b_\varepsilon(x,\xi))}}{m_1(x,\xi)
\\
\hbox{}&\times \lambda(x,\xi)^{\rho(\abs{\gamma_1 +
\gamma_2}+\abs{\eta_1 + \eta_2})}.
\end{aligned}
\end{equation*}

The function $\tilde{N}(M)$ is also depending on the weights $m_1$,
$m_2$ and the dimension $d$.

The proof of this theorem is an application of
Theorem <ref> a number of times and
recalling that the error operator of a strongly
$\hbar$-$\varepsilon$-admissible operator of some regularity is a
quantised pseudo-differential operator.

§.§ Rough pseudo-differential operators as operators on $L^2(\R^d)$

So far we have only considered operators acting on $\mathcal{S}(\R^d)$
or $\mathcal{S}'(\R^d)$. Hence they can be viewed as unbounded
operators acting in $L^2(\R^d)$ with domain $\mathcal{S}(\R^d)$. The
question is then when is this a bounded operator? The first theorem of
this section gives a criteria for when the operator can be extended to
a bounded operator. This theorem is a Calderon-Vaillancourt type
theorem and the proof uses the Calderon-Vaillancourt Theorem for the
non-rough pseudo-differential operators. We will not recall this
theorem but refer to <cit.>.

Let $a_\varepsilon$ be in
$\Gamma_{0,\varepsilon}^{m,\tau}(\R^d_x\times\R_p^d)$, where $m$ is
a bounded tempered weight function, $\tau\geq0$ and suppose there exists a
$\delta$ in $(0,1)$ such that
$\varepsilon\geq\hbar^{1-\delta}$. Then there exists a constant
$C_d$ and an integer $k_d$ only depending on the dimension such that

\begin{equation*}
\norm{\OpW(a_\varepsilon)\psi}_{L^2(\R^d)} \leq C_d \sup_{\substack{\abs{\alpha},\abs{\beta}\leq k_d \\ (x,p)\in \R^{2d}}} \varepsilon^{\abs{\alpha}} \abs{\partial_x^\alpha \partial_p^\beta a_\varepsilon(x,p)} \norm{\psi}_{L^2(\R^d)},
\end{equation*}

for all $\psi$ in $\mathcal{S}(\R^d)$. Especially can
$\OpW(a_\varepsilon)$ be extended to a bounded operator on

Let $\mathcal{U}_\varepsilon$ be the unitary dilation operator as defined in Observation <ref>. We have that

\begin{equation}\label{cal-vai-proof-1}
\begin{aligned}
\norm{\OpW(a_\varepsilon)\psi}_{L^2(\R^d)}  &=   \norm{\mathcal{U}_\varepsilon^{*}\mathcal{U}_\varepsilon\OpW(a_\varepsilon) \mathcal{U}_\varepsilon^{*}\mathcal{U}_\varepsilon\psi}_{L^2(\R^d)}
=   \norm{\OpW(a_\varepsilon^{\#}) \mathcal{U}_\varepsilon\psi}_{L^2(\R^d)}
\end{aligned}
\end{equation}

By our assumptions and Observation <ref> we note that the symbol $a_\varepsilon^{\#}$ satisfies the assumptions of the classical Calderon-Vaillancourt theorem. This gives us a constant $C_d$ and an integer $k_d$ only depending on the dimension such that

\begin{equation}\label{cal-vai-proof-2}
\begin{aligned}
\norm{\OpW(a_\varepsilon^{\#}) \mathcal{U}_\varepsilon\psi}_{L^2(\R^d)} \leq C_d \sup_{\substack{\abs{\alpha},\abs{\beta}\leq k_d \\ (x,p)\in \R^{2d}}}  \abs{\partial_x^\alpha \partial_p^\beta a_\varepsilon^{\#}(x,p)} \norm{\psi}_{L^2(\R^d)},
\end{aligned}
\end{equation}

where we have used the unitarity of $\mathcal{U}_\varepsilon$. By the definition of $a_\varepsilon^{\#}$ we have that

\begin{equation}\label{cal-vai-proof-3}
\begin{aligned}
\sup_{\substack{\abs{\alpha},\abs{\beta}\leq k_d \\ (x,p)\in \R^{2d}}}  \abs{\partial_x^\alpha \partial_p^\beta a_\varepsilon^{\#}(x,p)} ={}&  \sup_{\substack{\abs{\alpha},\abs{\beta}\leq k_d \\ (x,p)\in \R^{2d}}}  \abs{\partial_x^\alpha \partial_p^\beta a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p)}
\leq \sup_{\substack{\abs{\alpha},\abs{\beta}\leq k_d \\ (x,p)\in \R^{2d}}} \varepsilon^{\abs{\alpha}}  \abs{\partial_x^\alpha \partial_p^\beta a_\varepsilon( x,  p)}
\end{aligned}
\end{equation}

Combining (<ref>), (<ref>) and (<ref>) we arrive at the desired result. This completes the proof.

We can now give a criteria for the rough pseudo-differential operators
to be trace class. The criteria will be sufficient but not necessary.
Hence it does not provide a full characteristic for the set of rough
pseudo-differential operators which are trace class.

There exists a constant $C(d)$ only depending on the dimension such

\begin{equation*}
\norm{\OpW(a_\varepsilon)}_{\Tr} \leq  \frac{ C(d)} {\hbar^d} \sum_{\abs{\alpha}+\abs{\beta}\leq 2d+2}  \varepsilon^{\abs{\alpha}}   \hbar^{\delta\abs{\beta}}  \int_{\R^{2d}} |\partial_x^\alpha \partial_p^\beta a_\varepsilon(x,p)|  \,dxdp.
\end{equation*}

for every $a_\varepsilon$ in
$\Gamma_{0,\varepsilon}^{m,\tau}(\R^d_x\times\R_p^d)$ with

Let $\mathcal{U}_\varepsilon$ be the unitary dilation operator as defined in Observation <ref>. We have by the unitary invariance of the trace norm that

\begin{equation*}
\norm{\OpW(a_\varepsilon)}_{\Tr} =   \norm{\mathcal{U}_\varepsilon \OpW(a_\varepsilon) \mathcal{U}_\varepsilon^{*}}_{\Tr} =  \norm{ \OpW(a_\varepsilon^{\#}) }_{\Tr}.
\end{equation*}

From our assumptions and Observation <ref> we get that $a_\varepsilon^{\#}$ satisfies the assumption for <cit.>. From this theorem we get the exsistence of a constant $C(d)$ only depending on the dimension such that

\begin{equation}\label{B.thm_est_tr_eq_1}
\norm{ \OpW(a_\varepsilon^{\#}) }_{\Tr} \leq  \frac{ C(d)} {\hbar^{\delta d}} \sum_{\abs{\alpha}+\abs{\beta}\leq 2d+2} \hbar^{\delta \abs{\beta}}  \int_{\R^{2d}} |\partial_x^\alpha \partial_p^\beta a_\varepsilon^{\#}(x,p)]  \,dxdp.
\end{equation}

By the definition of $\partial_x^\alpha \partial_p^\beta a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p)$ we have that

\begin{equation}\label{B.thm_est_tr_eq_2}
\begin{aligned}
\int_{\R^{2d}} |\partial_x^\alpha \partial_p^\beta a_\varepsilon^{\#}(x,p)|  \,dxdp &=   \int_{\R^{2d}}|\partial_x^\alpha \partial_p^\beta a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p)|  \,dxdp
\\
&= \varepsilon^{\abs{\alpha}}  \left(\tfrac{\hbar^{1-\delta}}{\varepsilon}\right)^{\abs{\beta}} \int_{\R^{2d}} |[\partial_x^\alpha \partial_p^\beta a_\varepsilon](\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p)|  \,dxdp
\\
&= \hbar^{(\delta-1)d} \varepsilon^{\abs{\alpha}}  \left(\tfrac{\hbar^{1-\delta}}{\varepsilon}\right)^{\abs{\beta}}  \int_{\R^{2d}} |[\partial_x^\alpha \partial_p^\beta a_\varepsilon](x,  p)|  \,dxdp
\end{aligned}
\end{equation}

combining (<ref>) and (<ref>) we get that

\begin{equation*}
\norm{\OpW(a_\varepsilon)}_{\Tr} \leq  \frac{ C(d)} {\hbar^d} \sum_{\abs{\alpha}+\abs{\beta}\leq 2d+2}  \varepsilon^{\abs{\alpha}}   \hbar^{\delta\abs{\beta}}  \int_{\R^{2d}} |\partial_x^\alpha \partial_p^\beta a_\varepsilon(x,p)|  \,dxdp.
\end{equation*}

This is the desired estimate and this concludes the proof.

The previous theorem gives us a sufficient condition for the rough
pseudo-differential operators to be trace class. The next theorem
gives the form of the trace for the rough pseudo-differential

Let $a_\varepsilon$ be in
$\Gamma_{0,\varepsilon}^{m,\tau}(\R^d_x\times\R_p^d)$ with
$\tau\geq0$ and suppose
$\partial_x^\alpha \partial_p^\beta a_\varepsilon(x,p)$ is an
element of $L^1(\R^d_x\times\R_p^d)$ for all
$\abs{\alpha}+\abs{\beta}\leq 2d+2$. Then $\OpW(a_\varepsilon)$ is
trace class and

\begin{equation*}
\Tr(\OpW(a_\varepsilon))=\frac{1}{(2\pi\hbar)^d} \int_{\R^{2d}} a_\varepsilon(x,p) \,dxdp.
\end{equation*}

That our operator is trace class under the assumptions follows from Theorem <ref>. To obtain the formula for the trace let $\mathcal{U}_\varepsilon$ be the unitary dilation operator as defined in Observation <ref>. Then by <cit.> and Observation <ref> we get that

\begin{equation*}
\begin{aligned}
\Tr(\OpW(a_\varepsilon)) &= \Tr(\mathcal{U}_\varepsilon\OpW(a_\varepsilon)\mathcal{U}_\varepsilon^{*}) = \Tr(\OpW(a_\varepsilon^{\#}))
\\
&= \frac{1}{(2\pi\hbar^\delta)^d} \int_{\R^{2d}} a_\varepsilon(\varepsilon x, \tfrac{\hbar^{1-\delta}}{\varepsilon} p) \,dxdp = \frac{1}{(2\pi\hbar)^d}  \int_{\R^{2d}} a_\varepsilon(x,p) \,dxdp.
\end{aligned}
\end{equation*}

This identity concludes the proof.

The last thing for this section is a sharp Gårdinger inequality, which we will need later.

Let $a_\varepsilon$ be a bounded rough symbol of regularity
$\tau\geq0$ which satisfies

\begin{equation*}
a_\varepsilon (x,p) \geq 0 \quad\text{for all } (x,p) \in \R_x^d\times\R^d_p,
\end{equation*}

and suppose there exists $\delta\in(0,1)$ such that
$\varepsilon>\hbar^{1-\delta}$. Then there exists a $C_0>0$ and
$\hbar_0>0$ such

\begin{equation*}
\langle \OpW(a_\varepsilon) g,g \rangle \geq -\hbar^{\delta} C \norm{g}_{L^2(\R^d)}
\end{equation*}

for all $g$ in $L^2(\R^d)$ and $\hbar$ in $(0,\hbar_0]$.

Again let $\mathcal{U}_\varepsilon$ be the unitary dilation operator as defined in Observation <ref>. After conjugation with this operator we will be able to use the usual semiclassical sharp Gårdinger
inequality (see e.g. <cit.>) with the semiclassical parameter
$\hbar^\delta$. We have that

\begin{equation}\label{B.sharp_gaar_eq_1}
\begin{aligned}
\langle \OpW(a_\varepsilon) g,g \rangle
&= \langle \mathcal{U}_\varepsilon\OpW(a_\varepsilon)\mathcal{U}_\varepsilon^{*}\mathcal{U}_\varepsilon g,\mathcal{U}_\varepsilon g \rangle
=  \langle \OpW(a_\varepsilon^{\#})\mathcal{U}_\varepsilon g,\mathcal{U}_\varepsilon g \rangle
\\
&\geq - C_0 \hbar^{\delta} \norm{\mathcal{U}_\varepsilon g}_{L^2(\R^d)} =  - C_0 \hbar^{\delta} \norm{g}_{L^2(\R^d)}
\end{aligned}
\end{equation}

The existence of the numbers $C_0$ and $\hbar_0$ is ensured by the usual semiclassical sharp Gårdinger
inequality. This is the desired estimate and this ends the proof.

§ SELF-ADJOINTNESS AND FUNCTIONAL CALCULUS FOR ROUGH PSEUDO-DIFFERENTIAL OPERATOR

In this section we will establish a functional calculus for rough pseudo-differential operators. In order for this to be well defined we will also give a set of assumptions that ensures essential self-adjointness of the operators. We could as in the previous section use unitary conjugations to obtain non-rough operators and the use the results for this type of operators. We have chosen not to do this, since in the construction we will need to control polynomials in the derivatives of our symbols. So to make it transparent that the regularity of these polynomials are as desired we have chosen to do the full construction. The construction is based on the results of Helffer–Robert in <cit.> and this approach is also described in <cit.>. The method they used is based on the Mellin transform, where we will use the Helffer-Sjöstrand formula instead. The construction of a functional calculus using this formula can be found in the monographs <cit.>. There is also a construction of the functional calculus using Fourier theory in <cit.>, but we have not tried to adapt this to the case studied here.

§.§ Essential self-adjointness of rough pseudo-differential operators

First we will give criteria for the operator to be lower semi-bounded
and essential self-adjoint.

Let $A_\varepsilon(\hbar)$ be a $\hbar$-$\varepsilon$-admissible
operator of regularity $\tau$ and suppose that

*  $A_\varepsilon(\hbar)$ is symmetric on
$\mathcal{S}(\R^n)$ for all $\hbar$ in $]0,\hbar_0]$.

*  The principal symbol $a_{\varepsilon,0}$
satisfies that

\begin{equation*}
\min_{(x,p) \in \R^{2n}} a_{\varepsilon,0}(x,p) = \zeta_0 > -\infty.
\end{equation*}

*  Let $\zeta_1 < \zeta_0$ and $\zeta_1 \leq
0$. Then $a_{\varepsilon,0} - \zeta_1$ is a tempered weight
function with constants independent of $\varepsilon$ and

\begin{equation*}
a_{\varepsilon,j} \in \Gamma_{0,\varepsilon}^{a_{\varepsilon,0} - \zeta_1,\tau-j} \left(\R_x^{d}\times\R^d_p\right),
\end{equation*}

for all $j$ in $\N$.

Let $A_\varepsilon(\hbar)$, for $\hbar$ in $(0,\hbar_0]$, be a
$\hbar$-$\varepsilon$-admissible operator of regularity $\tau\geq 1$
with tempered weight $m$ and symbol

\begin{equation*}
a_\varepsilon(\hbar) = \sum_{j\geq 0} \hbar^j  a_{\varepsilon,j}.
\end{equation*}

Suppose that $A_\varepsilon(h)$ satisfies
Assumption <ref>. Then there exists $\hbar_1$
in $(0,\hbar_0]$ such that for all $\hbar$ in $(0,\hbar_1]$
$A_\varepsilon(\hbar)$ is essential self-adjoint and lower

We let $t<\zeta_0$, where $\zeta_0$ is the number from
Assumption <ref>. For this $t$ we define the

\begin{equation*}
b_{\varepsilon,t}(x,p) = \frac{1}{a_{\varepsilon,0}(x,p) -t}.
\end{equation*}

By assumption we have that
$b_{\varepsilon,t} \in \Gamma_{0,\varepsilon}^{(a_{\varepsilon,0} -
\gamma_1)^{-1},\tau}(\R_x^d\times\R^d_p)$. For $N$ sufficiently
large we get by assumption that

\begin{equation*}
\begin{aligned}
(A_\varepsilon(\hbar) - t) \OpW(b_{\varepsilon,t}) ={}& \sum_{j=0}^N
\hbar^j \OpW( a_{\varepsilon,j}) \OpW(b_{\varepsilon,t}) + \hbar^{N+1} R_N(\hbar)\OpW(b_{\varepsilon,t})
\\
={}& I + \hbar S_N(\varepsilon,\hbar).
\end{aligned}
\end{equation*}

The formula for composition of operators and the Calderon-Vaillancourt theorem gives us that the   operator $S_N$ satisfies the estimate

\begin{equation}
\sup_{\hbar\in(0,\hbar_0]}  \norm{S_N(\varepsilon,\hbar)}_{\mathcal{L}(L^2(\R^d))}<\infty
\end{equation}

We note that if
$\hbar$ is chosen such that
$\hbar\norm{S_N(\varepsilon,\hbar)}_{\mathcal{L}(L^2(\R^d))}<1$ then
the operator $ I + \hbar S_N(\varepsilon,\hbar)$ will be

The Calderon-Vaillancourt theorem gives us that
$ \OpW(b_{\varepsilon,t})$ is a bounded operator. This implies that
the expression
$ \OpW(b_{\varepsilon,t}) ( I + \hbar S_N(\varepsilon,\hbar))^{-1}$
is a well defined bounded operator. Hence we have that the operator
$(A_\varepsilon(\hbar) - t) $ maps its domain surjective onto all of
$L^2(\R^d)$. By <cit.> this implies that
$A_\varepsilon(\hbar)$ is essential self-adjoint.

Since we have for all $t<\zeta_0$ that
$(A_\varepsilon(\hbar) - t) $ maps its domain surjective onto all of
$L^2(\R^d)$ they are all in the resolvent set and hence the operator
has to be lower semi-bounded.

§.§ The resolvent of a rough pseudo-differential operator

A main part in the construction of the functional calculus is to prove that the resolvent of a $\hbar$-$\varepsilon$-admissible
operator of regularity $\tau$, which satisfies Assumption <ref>, is an operator of the same type. this is the content of the following Theorem.

Let $A(\hbar)$, for $\hbar$ in $(0,\hbar_0]$, be a
$\hbar$-$\varepsilon$-admissible operator of regularity
$\tau \geq 1$ with tempered weight $m$ and symbol

\begin{equation*}
a_\varepsilon(\hbar) = \sum_{j\geq 0} \hbar^j a_{\varepsilon,j}.
\end{equation*}

Suppose that $A_\varepsilon(h)$ satisfies
Assumption <ref> with the numbers $\zeta_0$
and $\zeta_1$. For $z$ in $\C\setminus[\zeta_1,\infty)$ we define
the sequence of symbols

\begin{equation}\label{B.res.symbolform_1}
\begin{aligned}
b_{\varepsilon,z,0} &= (a_{\varepsilon,0} - z)^{-1}
\\
b_{\varepsilon,z,j+1} &= - b_{\varepsilon,z,0} \cdot
\sum_{\substack{l + \abs{\alpha} + \abs{\beta} + k = j+1\\ 0\leq
l \leq j }} \frac{1}{\alpha ! \beta !}
\frac{1}{2^{\abs{\alpha}}} \frac{1}{(-2)^{\abs{\beta}}}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,l}),
\end{aligned}
\end{equation}

for $j\geq1$. Moreover we define

\begin{equation*}
B_{\varepsilon,z,M}(\hbar) = \sum_{j=0}^M \hbar^j b_{\varepsilon,z,j}.
\end{equation*}

Then for $N$ in $\N$ we have that

\begin{equation}
(A_\varepsilon(h) - z ) \OpW B_{\varepsilon,z,N} = I + h^{N+1} \Delta_{z,N+1}(h),
\end{equation}

\begin{equation}
\hbar^{N+1}\norm{ \Delta_{z,N+1}(h)}_{\mathcal{L}(L^2(\R^d))} \leq C \hbar^{\kappa(N)} \left( \frac{\abs{z}}{\dist(z,[\zeta_1,\infty))} \right)^{q(N)},
\end{equation}

where $\kappa$ is a positive strictly increasing function and $q(N)$
is a positive integer depending on $N$. In particular we have for
all $z$ in $\C\setminus[\zeta_1,\infty)$ and all $\hbar$ in
$(0,\hbar_1]$ ($\hbar_1$ sufficient small and independent of $z$),
that $(A_\varepsilon(h) - z)^{-1}$ is a
$\hbar$-$\varepsilon$-admissible operator with respect to the
tempered weight $(a_{\varepsilon,0}-\zeta_1)^{-1}$ and of
regularity $\tau$ with symbol:

\begin{equation}
B_z(h) = \sum_{j\geq0} h^j b_{\varepsilon,z,j}.
\end{equation}

Before we can prove the theorem we will need some lemma's with the
same setting. It is in these lemma's we will see that the symbol for the resolvent has the same regularity. From these lemmas we can also find the explicit formula's for every symbol in the expansion.

Let the setting be as in Theorem <ref>. For
every $j$ in $\mathbb{N}$ we have

\begin{equation}\label{B.res.symbolform_2}
b_{\varepsilon,z,j} = \sum_{k=1}^{2j-1} d_{\varepsilon, j ,k} b_{\varepsilon,z,0}^{k+1},
\end{equation}

where $d_{\varepsilon, j ,k}$ are universal polynomials in
$\partial_p^\alpha\partial_x^\beta a_{\varepsilon,l}$ for
$\abs{\alpha}+\abs{\beta}+l\leq j$ and
$d_{\varepsilon, j ,k} \in
\Gamma_{0,\varepsilon}^{(a_0-\zeta_1)^k,\tau-j}$ for all $k$,
$1\leq k \leq 2j-1$.  In particular we have that

\begin{equation*}
b_{\varepsilon,z,1}=-a_{\varepsilon,1} b_{\varepsilon,z,0}^2.
\end{equation*}

In order to prove this Lemma we will be needing the following Lemma:

Let the setting be as in
Lemma <ref>. For any $j$ and $k$ in
$\mathbb{N}$ we let
$d_{\varepsilon, j ,k} b_{\varepsilon,z,0}^{k+1}$ be one of the
elements in the expansion of $b_{\varepsilon,z,j}$. Then for all
multi indices $\alpha$ and $\beta$ it holds that

\begin{equation*}
\partial_p^\beta \partial_x^\alpha d_{\varepsilon, j ,k} b_{\varepsilon,z,0}^{k+1}  = \sum_{n=0}^{\abs{\alpha}+\abs{\beta}} \tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta} b_{\varepsilon,z,0}^{k+1+n},
\end{equation*}

where $ \tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta}$ are
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j +\abs{\alpha}+\abs{\beta}$ of
degree at most $k+n$ and they are of regularity at least
$\tau-j-\abs{\alpha}$. They are determined only by $\alpha$, $\beta$
and $d_{\varepsilon, j ,k}$.

The proof is an application of Theorem <ref> (Faà
di Bruno formula) and the Corollary <ref> to the
formula. For our $\alpha,\beta$ we have by the Leibniz's formula that:

\begin{equation*}
\begin{aligned}
\MoveEqLeft  \partial_p^\beta \partial_x^\alpha d_{\varepsilon, j ,k}
= \partial_p^\beta \big\{
(\partial_x^{\alpha} d_{\varepsilon, j ,k})
b_{\varepsilon,z,0}^{k+1} + \sum_{\substack{\abs{\gamma}=1\\
\gamma\leq\alpha}}^{\abs{\alpha}}
\binom{\alpha}{\gamma} \partial_x^{\alpha-\gamma} d_{\varepsilon,
j ,k} \partial_x^\gamma b_{\varepsilon,z,0}^{k+1} \big\}
\\
={} & (\partial_p^\beta \partial_x^{\alpha} d_{\varepsilon, j
,k})b_{\varepsilon,z,0}^{k+1} +
\sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}\binom{\beta}{\zeta} \partial_p^{\beta-\zeta} \partial_x^{\alpha}
d_{\varepsilon, j ,k} \partial_p^\zeta b_{\varepsilon,z,0}^{k+1}
+ \sum_{\substack{\abs{\gamma}=1\\
\gamma\leq\alpha}}^{\abs{\alpha}} \binom{\alpha}{\gamma}
(\partial_p^\beta\partial_x^{\alpha-\gamma} d_{\varepsilon, j
,k}) \partial_x^\gamma b_{\varepsilon,z,0}^{k+1}
\\
& +
\sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}
\sum_{\substack{\abs{\gamma}=1\\ \gamma\leq\alpha}}^{\abs{\alpha}}
\binom{\beta}{\zeta} \binom{\alpha}{\gamma}
d_{\varepsilon, j ,k}) \partial_p^\zeta \partial_x^\gamma
b_{\varepsilon,z,0}^{k+1} .
\end{aligned}
\end{equation*}

We will here consider each of the three sums separately for the
first we get by the Faà di Bruno formula
(Theorem <ref>)

\begin{equation*}
\begin{aligned}
\MoveEqLeft \sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}
\binom{\beta}{\zeta} \partial_p^{\beta-\zeta} \partial_x^{\alpha}
d_{\varepsilon, j ,k} \partial_p^\zeta b_{\varepsilon,z,0}^{k+1}
\\
& =
\sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}\binom{\beta}{\zeta} \partial_p^{\beta-\zeta} \partial_x^{\alpha}
d_{\varepsilon, j ,k} \sum_{{n}=1}^{\abs{\zeta}} (-1)^{n}
\frac{(k+n)!}{k!}  b_{\varepsilon,z,0}^{k+1+ n} \;\;
\sum_{\mathclap{\substack{\zeta_1+\cdots+\zeta_n=\zeta\\\abs{\zeta_i}>0}}}
\;\; c_{\zeta_1,\dots,\zeta_n} \partial_p^{\zeta_1}a_0
\cdots \partial_p^{\zeta_n}a_0
\\
&= \sum_{n_\beta=1}^{\abs{\beta}} \Big\{
\sum_{\substack{\abs{\zeta}\geq n_\beta \\ \zeta\leq
\beta}}^{\abs{\beta}}
c_{k,n_\beta,\beta,\zeta} \partial_p^{\beta-\zeta} \partial_x^{\alpha}
d_{\varepsilon, j ,k}
\sum_{\substack{\zeta_1+\cdots+\zeta_{n_\beta}=\zeta\\\abs{\zeta_i}>0}}
c_{\zeta_1,\dots,\zeta_n} \partial_p^{\zeta_1}a_0
\cdots \partial_p^{\zeta_{n_\beta}}a_0
\Big\}b_{\varepsilon,z,0}^{k+1+ n_\beta}
\\
&= \sum_{n_\beta=1}^{\abs{\beta}}
\tilde{d}_{\varepsilon,j,k,\alpha,\beta,n_\beta}
b_{\varepsilon,z,0}^{k+1+ n_\beta}.
\end{aligned}
\end{equation*}

This calculation gives that we have a polynomial structure, where the coefficients are
the polynomials $ \tilde{d}_{\varepsilon,j,k,\alpha,\beta,n_\beta}$, which themselfs
are polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j +\abs{\alpha}+\abs{\beta}$ and
of regularity $\tau-j-\abs{\alpha}$.

For the second sum we again use Faà di Bruno formula
(Theorem <ref>) and get

\begin{equation*}
\begin{aligned}
\MoveEqLeft  \sum_{\substack{\abs{\gamma}=1\\
\gamma\leq\alpha}}^{\abs{\alpha}} \binom{\alpha}{\gamma}
(\partial_p^\beta\partial_x^{\alpha-\gamma} d_{\varepsilon, j
,k}) \partial_x^\gamma b_{\varepsilon,z,0}^{k+1}
\\
\sum_{\substack{\abs{\gamma}=1\\\gamma\leq\alpha}}^{\abs{\alpha}}\binom{\alpha}{\gamma} \partial_p^{\beta} \partial_x^{\alpha-\gamma}
d_{\varepsilon, j ,k} \sum_{{n}=1}^{\abs{\gamma}} (-1)^{n}
\frac{(k+n)!}{k!}  b_{\varepsilon,z,0}^{k+1+ n} \quad
\sum_{\mathclap{\substack{\gamma_1+\cdots+\gamma_n=\gamma\\\abs{\gamma_i}>0}}}
\;c_{\gamma_1,\dots,\gamma_n} \partial_x^{\gamma_1}a_0
\cdots \partial_x^{\gamma_n}a_0
\\
={}& \sum_{n_\alpha=1}^{\abs{\alpha}} \Big\{
\sum_{\substack{\abs{\gamma}\geq n_\alpha \\ \gamma\leq
\alpha}}^{\abs{\alpha}}
c_{k,n_\alpha,\alpha,\gamma} \partial_p^{\beta} \partial_x^{\alpha-\gamma}
d_{\varepsilon, j ,k} \quad
\sum_{\mathclap{\substack{\gamma_1+\cdots+\gamma_{n_\alpha}=\gamma\\\abs{\gamma_i}>0}}}\;
c_{\gamma_1,\dots,\gamma_n} \partial_x^{\gamma_1}a_0
\cdots \partial_x^{\gamma_{n_\alpha}}a_0
\Big\}b_{\varepsilon,z,0}^{k+1+ n_\alpha}
\\
={}& \sum_{n_\alpha=1}^{\abs{\alpha}}
\tilde{d}_{\varepsilon,j,k,\alpha,\beta,n_\alpha}
b_{\varepsilon,z,0}^{k+1+ n_\alpha}.
\end{aligned}
\end{equation*}

Again we have the polynomial structure,where the coefficients are
the polynomials $ \tilde{d}_{\varepsilon,j,k,\alpha,\beta,n_\alpha}$ again
are polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j +\abs{\alpha}+\abs{\beta}$ and
they are of at least regularity $\tau-j-\abs{\alpha}$.

For the last sum we need a slightly modified version of the Faà di
Bruno formula which is Corollary <ref>. If we
use this we get that

\begin{equation*}
\begin{aligned}
\MoveEqLeft  \sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}
\sum_{\substack{\abs{\gamma}=1\\ \gamma\leq\alpha}}^{\abs{\alpha}}
\binom{\beta}{\zeta} \binom{\alpha}{\gamma}
d_{\varepsilon, j ,k}) \partial_p^\zeta \partial_x^\gamma
\\
={}& \sum_{\substack{\abs{\zeta}=1\\\zeta\leq\beta}}^{\abs{\beta}}
\sum_{\substack{\abs{\gamma}=1\\ \gamma\leq\alpha}}^{\abs{\alpha}}
\binom{\beta}{\zeta} \binom{\alpha}{\gamma}
d_{\varepsilon, j ,k}) \sum_{n=1}^{\abs{\zeta}+\abs{\gamma}} c_n
\sum_{\mathcal{I}_n(\gamma,\zeta)}c_{\gamma_1\cdots\gamma_k}^{\zeta_1\cdots\zeta_k} \partial_p^{\zeta_1} \partial_x^{\gamma_1}
\cdots \partial_p^{\zeta_n} \partial_x^{\gamma_n}
\\
={}& \sum_{n=1}^{\abs{\alpha}+\abs{\beta}}
\Big\{\sum_{\substack{\abs{\zeta} + \abs{\gamma} \geq n
\\\zeta\leq\beta,\
\gamma\leq\alpha}}^{\abs{\beta}+\abs{\alpha}}
\sum_{\mathcal{I}_n(\gamma,\zeta)}
d_{\varepsilon, j ,k})
\partial_p^{\zeta_1} \partial_x^{\gamma_1} a_{\varepsilon,0}
\cdots \partial_p^{\zeta_n} \partial_x^{\gamma_n}
a_{\varepsilon,0} \Big\} b_{\varepsilon,z,0}^{k+1+n}
\\
={}& \sum_{n=1}^{\abs{\alpha}+\abs{\beta}}
\tilde{d}_{\varepsilon,j,k,n,\alpha,\beta}
\end{aligned}
\end{equation*}

where $\tilde{d}_{\varepsilon,j,k,n,\alpha,\beta}$ are polynomials
in $\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j +\abs{\alpha}+\abs{\beta}$ of
degree at most $k+n$ and they are of regularity at least
$\tau-j-\abs{\alpha}$. If we combine all of the above calculations
we get the desired result:

\begin{equation*}
\partial_p^\beta \partial_x^\alpha d_{\varepsilon, j ,k} b_{\varepsilon,z,0}^{k+1}  = \sum_{n=0}^{\abs{\alpha}+\abs{\beta}} \tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta} b_{\varepsilon,z,0}^{k+1+n},
\end{equation*}

where $ \tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta}$ are
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j +\abs{\alpha}+\abs{\beta}$ of
degree at most $k+n$ and they are of regularity at least
$\tau-j-\abs{\alpha}$. The form of the polynomials is entirely
determined by the multi-indices $\alpha,\beta$ and the polynomial
$ d_{\varepsilon, j ,k} $.

The proof will be induction in the parameter $j$. We start by
considering the case $j=1$, where we by definition of
$b_{\varepsilon,z,1}$ have

\begin{equation*}
\begin{aligned}
b_{\varepsilon,z,1} &= - b_{\varepsilon,z,0} \cdot \sum_{
\abs{\alpha} + \abs{\beta} + k = 1} \frac{1}{\alpha ! \beta !}
\frac{1}{2^{\abs{\alpha}}} \frac{1}{(-2)^{\abs{\beta}}}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,0})
\\
&= - b_{\varepsilon,z,0} \big(
a_{\varepsilon,1}b_{\varepsilon,z,0} -\frac{i}{2}
\sum_{n=1}^d \partial_{p_n} a_{\varepsilon,0} \partial_{x_n}
a_{\varepsilon,0}b_{\varepsilon,z,0}^2 - \partial_{x_n}
a_{\varepsilon,0}\partial_{p_n} a_{\varepsilon,0}
b_{\varepsilon,z,0}^2 \big)
\\
&= - a_{\varepsilon,1}b_{\varepsilon,z,0}^2.
\end{aligned}
\end{equation*}

This calculation verifies the form of $b_{\varepsilon,z,1}$ stated
in the lemma and that it has the form given by
(<ref>) with
$d_{\varepsilon,1,1}=-a_{\varepsilon,1}$ which is in the symbol
class $\Gamma_{0,\varepsilon}^{(a_0-\zeta_1),\tau-1}$ by assumption.

Assume the lemma to be correct for $b_{\varepsilon,z,j}$ and
consider $b_{\varepsilon,z,j+1}$. By the definition of
$b_{\varepsilon,z,j+1}$ and our assumption we have

\begin{equation*}
\begin{aligned}
b_{\varepsilon,z,j+1} ={}& - b_{\varepsilon,z,0} \cdot
\sum_{\substack{l + \abs{\alpha} + \abs{\beta} + k = j+1\\ 0\leq l
\leq j }} \frac{1}{\alpha ! \beta !}
\frac{1}{2^{\abs{\alpha}}} \frac{1}{(-2)^{\abs{\beta}}}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,l})
\\
={}& - b_{\varepsilon,z,0} \Big\{\sum_{\abs{\alpha} + \abs{\beta} +
k = j+1} c_{\alpha,\beta} (\partial_p^{\alpha} D_x^{\beta}
a_{\varepsilon,k}) (\partial_p^{\beta} D_x^{\alpha}
\\
\hbox{}&\qquad+ \sum_{l=1}^j \sum_{m=1}^{2l-1}
\sum_{l+\abs{\alpha} + \abs{\beta} + k = j+1} c_{\alpha,\beta}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha}d_{\varepsilon, l ,m}
\end{aligned}
\end{equation*}

We will consider each of the sums separately. To calculate the first
sum we get by applying Corollary <ref>

\begin{equation*}
\begin{aligned}
\MoveEqLeft \sum_{\abs{\alpha} + \abs{\beta} + k = j+1} c_{\alpha,\beta}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,0}) - c_{\alpha,\beta} a_{\varepsilon,j+1} b_{\varepsilon,z,0}
\\
={}&     \sum_{\substack{\abs{\alpha} + \abs{\beta} + k = j+1 \\
\abs{\alpha} + \abs{\beta}\geq 1}} \Big\{c_{\alpha,\beta}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
\sum_{n=1}^{\abs{\alpha}+\abs{\beta}} c_n
b_{\varepsilon,z,0}^{n+1}  \sum_{\mathcal{I}_n(\alpha,\beta)}c_{\alpha_1\cdots\alpha_n}^{\beta_1\cdots\beta_n} \partial_p^{\beta_1}
D_x^{\alpha_1} a_{\varepsilon,0} \cdots \partial_p^{\beta_n}
D_x^{\alpha_n} a_{\varepsilon,0}\Big\}
\\
\sum_{n=1}^{j+1} \Big\{ \sum_{\substack{\abs{\alpha} + \abs{\beta}
+ k = j+1 \\ \abs{\alpha} + \abs{\beta}\geq n}}
\sum_{\mathcal{I}_n(\alpha,\beta)} c_{\alpha,\beta,n}
(\partial_p^{\alpha} D_x^{\beta}
a_{\varepsilon,k}) \partial_p^{\beta_1} D_x^{\alpha_1}
a_{\varepsilon,0} \cdots \partial_p^{\beta_n} D_x^{\alpha_n}
a_{\varepsilon,0} \Big\} b_{\varepsilon,z,0}^{n+1}
\\
={}& \sum_{n=1}^{j+1} \tilde{d}_{\varepsilon,j,n,\alpha,\beta}
b_{\varepsilon,z,0}^{n+1} - c_{\alpha,\beta} a_{\varepsilon,j+1} b_{\varepsilon,z,0} ,
\end{aligned}
\end{equation*}

where $ \tilde{d}_{\varepsilon, j , n, \alpha , \beta}$ are
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j + 1$ of degree $n+1$ and of
regularity at least $\tau-j-1$. The index set $\mathcal{I}_n(\alpha,\beta)$ is defined by

\begin{equation*} \begin{aligned}
\mathcal{I}_n(\alpha,\beta) =
\{(\alpha_1,\dots,\alpha_n,&\beta_1,\dots,\beta_n) \in \N^{2nd} \
\\
& | \ \sum_{l=1}^n \alpha_l=\alpha, \, \sum_{l=1}^n \beta_l=\beta,
\, \max(\abs{\alpha_l},\abs{\beta_l}) \geq1 \, \forall l \}.
\end{aligned} \end{equation*}

The form of the polynomials is
determined by the multi indices $\alpha$ and $\beta$. Moreover we
have that the polynomials
$ \tilde{d}_{\varepsilon, j , n, \alpha , \beta}$ are elements of

If we now consider the triple sum and apply
Lemma <ref> we get

\begin{equation*}
\begin{aligned}
\MoveEqLeft \sum_{l=1}^j \sum_{m=1}^{2l-1} \sum_{l+\abs{\alpha} + \abs{\beta}
+ k = j+1} c_{\alpha,\beta} (\partial_p^{\alpha} D_x^{\beta}
a_{\varepsilon,k}) (\partial_p^{\beta} D_x^{\alpha}d_{\varepsilon,
l ,m} b_{\varepsilon,z,0}^{m+1})
\\
={}& \sum_{l=1}^j \sum_{m=1}^{2l-1} \sum_{l+\abs{\alpha} +
\abs{\beta} + k = j+1} c_{\alpha,\beta} (\partial_p^{\alpha}
D_x^{\beta} a_{\varepsilon,k}) (-i)^{\abs{\alpha}} \sum_{n=0}^{
\abs{\alpha}+\abs{\beta}} \tilde{d}_{\varepsilon, l ,m, n,
\alpha,\beta} b_{\varepsilon,z,0}^{m+1+n}
\\
={}& \sum_{m=1}^{2j-1} \sum_{l =\lceil \frac{m-1}{2} \rceil +1 }^j
\sum_{l+\abs{\alpha} + \abs{\beta} + k = j+1} \sum_{n=0}^{
\abs{\alpha}+\abs{\beta}} c_{\alpha,\beta} (\partial_p^{\alpha}
D_x^{\beta} a_{\varepsilon,k}) (-i)^{\abs{\alpha}}
\tilde{d}_{\varepsilon, l ,m, n, \alpha,\beta}
\\
={}&\sum_{m=1}^{2j-1} \sum_{l =\lceil \frac{m-1}{2} \rceil +1 }^j
\sum_{n=0}^{ j+1-l} \sum_{\abs{\alpha} + \abs{\beta} = n}
c_{\alpha,\beta} (\partial_p^{\alpha} D_x^{\beta}
a_{\varepsilon,j+1-l-n}) (-i)^{\abs{\alpha}}
\tilde{d}_{\varepsilon, l ,m, n, \alpha,\beta}
\end{aligned}
\end{equation*}

where the $\tilde{d}_{\varepsilon, l ,m, n \alpha,\beta}$ are the
polynomials from Lemma <ref>. The way
we have expressed the sums ensures $k$ always is the fixed value
$j+1-l-n$. From Lemma <ref> we have
that $\tilde{d}_{\varepsilon, l ,m, n, \alpha,\beta}$ are
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,m}$ with
$\abs{\alpha'}+\abs{\beta'}+m \leq l+n\leq j+1$ of degree $n+m$ and
with regularity at least $\tau-l-\abs{\alpha}$. Hence the factors
$c_{\alpha,\beta} (\partial_p^{\alpha} D_x^{\beta}
a_{\varepsilon,j+1-l-n}) (-i)^{\abs{\alpha}} \tilde{d}_{\varepsilon,
l ,m, n, \alpha,\beta}$ will be polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,m}$ with
$\abs{\alpha'}+\abs{\beta'}+m\leq j+1$ of degree $n+m+1$. The
regularity of the terms will be at least

\begin{equation*}
\tau- l-\abs{\alpha} - (j+1-l-n) - \abs{\beta} = \tau-(j+1),
\end{equation*}

where most terms will have more regularity. By rewriting and
renaming some of the terms we get the following equality

\begin{equation*}
\sum_{l=1}^j \sum_{m=1}^{2l-1} \sum_{l+\abs{\alpha} + \abs{\beta} + k = j+1} c_{\alpha,\beta}  (\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k}) (\partial_p^{\beta} D_x^{\alpha}d_{\varepsilon, l ,m} b_{\varepsilon,z,0}^{m+1})  = \sum_{n=1}^{2j}  \tilde{d}_{\varepsilon,j,n,\alpha,\beta}  b_{\varepsilon,z,0}^{n+1},
\end{equation*}

where $\tilde{d}_{\varepsilon,j,n,\alpha,\beta} $ again are
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j+1$ of degree $n+1$ of
regularity at least $\tau-(j+1)$. By combing these calculation we
arrive at the expression

\begin{equation*}
\begin{aligned}
b_{\varepsilon,z,j+1} ={}& - b_{\varepsilon,z,0} \cdot
\sum_{\substack{l + \abs{\alpha} + \abs{\beta} + k = j+1\\ 0\leq l
\leq j }} \frac{1}{\alpha ! \beta !}
\frac{1}{2^{\abs{\alpha}}} \frac{1}{(-2)^{\abs{\beta}}}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,l})
\\
={}& - b_{\varepsilon,z,0} \Big\{ \sum_{n=0}^{j+1}
\tilde{d}_{\varepsilon,j,n,\alpha,\beta} b_{\varepsilon,z,0}^{n+1}
+ \sum_{n=1}^{2j} \tilde{d}_{\varepsilon,j,n,\alpha,\beta}
\\
={}& \sum_{k=1}^{2j+1} d_{\varepsilon,j+1,k}
\end{aligned}
\end{equation*}

where the polynomials $d_{\varepsilon,j+1,k}$ are universal
polynomials in
$\partial_x^{\alpha'} \partial_p^{\beta'} a_{\varepsilon,k}$ with
$\abs{\alpha'}+\abs{\beta'}+k \leq j+1$ of degree $k$ and with
regularity at least $\tau-j-1$. Hence they are elements of
$\Gamma_{0,\varepsilon}^{(a_0-\zeta_1)^k,\tau-(j+1)}$.  This ends
the proof.

Let the setting be as in Theorem <ref>. For
every $j$ in $\N$ and $\alpha$, $\beta$ in $\N^d$ there exists a
number $c_{j,\alpha,\beta}>0$ such that

\begin{equation*}
\abs{\partial_p^\beta\partial_x^\alpha b_{\varepsilon,z,j}} \leq c_{j,\alpha,\beta} \varepsilon^{-(\tau-j-\abs{\alpha})_{-}}  (a_{\varepsilon,0}-\zeta_1)^{-1} \left( \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{2j+\abs{\alpha}+\abs{\beta}},
\end{equation*}

for all $z \in \C\setminus[\zeta_1,\infty)$ and all
$(x,p) \in \R^d_x\times\R^d_p$.

We start by considering the fraction
and we will divide it into two cases according to the real part of
$z$. If $\re(z)<\zeta_1$ then

\begin{equation*}
\frac{\abs{a_{\varepsilon,0}-\zeta_1}}{\abs{a_{\varepsilon,0}-z}} \leq 1 \leq  \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}.
\end{equation*}

If instead $\re(z)\geq\zeta_1$ and $\abs{\im{z}}>0$ we have by the
law of sines that

\begin{equation*}
\frac{\abs{a_{\varepsilon,0} - z}}{\sin(\phi_1)} = \frac{\abs{a_{\varepsilon,0}-\zeta_1}}{\sin(\phi_2)} \geq \abs{a_{\varepsilon,0}-\zeta_1},
\end{equation*}

where $\phi_1$ and $\phi_2$ are angles in the triangle with vertices $\zeta_1$, $a_{\varepsilon,0}$ and $z$.  We have in the above estimate used that $0< \sin(\phi_2)\leq 1$. If we apply this
inequality and the law of sines again we arrive at the following

\begin{equation*}
\frac{\abs{a_{\varepsilon,0}-\zeta_1}}{\abs{a_{\varepsilon,0}-z}} \leq \frac{1}{\sin(\phi_1)} = \frac{\abs{z-\zeta_1}}{\abs{\im(z)}}.
\end{equation*}

Combining these two cases we get the estimate

\begin{equation}\label{B.res_aprox4.1}
\frac{\abs{a_{\varepsilon,0}-\zeta_1}}{\abs{a_{\varepsilon,0}-z}} \leq \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))} \quad\quad\text{For all $z \in\C\setminus[\zeta_1,\infty)$.}
\end{equation}

If we now consider a
given $b_{\varepsilon,z,j}$ and $\alpha,\beta$ in
$\N^d$. Lemma <ref> and
Lemma <ref> gives us

\begin{equation*}
\partial_p^\beta\partial_x^\alpha b_{\varepsilon,z,j} = \sum_{k=1}^{2j-1} \partial_p^\beta\partial_x^\alpha (d_{\varepsilon,j,k}  b_{\varepsilon,z,0}^{k+1}) = \sum_{k=1}^{2j-1}  \sum_{n=0}^{\abs{\alpha}+\abs{\beta}} \tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta} b_{\varepsilon,z,0}^{k+1+n},
\end{equation*}

with $\tilde{d}_{\varepsilon,j,k,\alpha,\beta}$ in
$\Gamma_{0,\varepsilon}^{(a_0-\zeta_1)^{k+n},\tau-j-\abs{\alpha}}$. By
taking absolute value and applying (<ref>) we get

\begin{equation*}
\begin{aligned}
\abs{\partial_p^\beta\partial_x^\alpha b_{\varepsilon,z,j} }
\leq{}&
\sum_{k=1}^{2j-1} \sum_{n=0}^{\abs{\alpha}+\abs{\beta}}
\abs{\tilde{d}_{\varepsilon, j ,k, n, \alpha , \beta}
\\
\leq{}& \abs{b_{\varepsilon,z,0}} \sum_{k=1}^{2j-1}
\sum_{n=0}^{\abs{\alpha}+\abs{\beta}}
\varepsilon^{-(\tau-j-\abs{\alpha})_{-}} c_{j,k,\alpha,\beta}
\left(
\frac{\abs{a_{\varepsilon,0}-\zeta_1}}{\abs{a_{\varepsilon,0}-z}}\right)^{k+n}
\\
\leq{}& c_{j,\alpha,\beta} \varepsilon^{-(\tau-j-\abs{\alpha})_{-}}
\left(\frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{2j+\abs{\alpha}+\abs{\beta}},
\end{aligned}
\end{equation*}

where we have use that

\begin{equation*}
\abs{b_{\varepsilon,z,0}} = \frac{(a_{\varepsilon,0}-\zeta_1)}{|a_{\varepsilon,0}-z|(a_{\varepsilon,0}-\zeta_1)}  \leq \frac{1}{(a_{\varepsilon,0}-\zeta_1)}  \left(\frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right).
\end{equation*}

We have now obtained the desired estimate and this ends the proof.

By Lemma <ref> the
symbols $b_{\varepsilon,z,j}$ are in the class
$\Gamma_{0,\varepsilon}^{(a_0-\zeta_1)^{-1},\tau-j}$ for every $j$
in $\N$, where $b_{\varepsilon,z,j}$ is defined
(<ref>). Hence we have that

\begin{equation*}
B_{\varepsilon,z,N}(\hbar) = \sum_{j=0}^N \hbar^j b_{\varepsilon,z,j}.
\end{equation*}

is a well defined symbol for every $N$ in $\N$. Moreover as
$(a_0-\zeta_1)^{-1}$ is a bounded function we have by
Theorem <ref> that $\OpW(B_{\varepsilon,z,N}(\hbar))$
is a bounded operator. Now for $N$ sufficiently large we have by

\begin{equation*}
A_\varepsilon(\hbar) -z = \OpW( a_{\varepsilon,0}-z)+ \sum_{k=1}^N \hbar^k \OpW( a_{\varepsilon,k}) + \hbar^{N+1}  R_N(\varepsilon,\hbar),
\end{equation*}

where the error term satisfies

\begin{equation*}
\hbar^{N+1}  \norm{R_N(\varepsilon,\hbar)}_{\mathcal{L}(L^2(\R^d))} \leq \hbar^{\kappa(N)} C_N
\end{equation*}

for a positive strictly increasing function $\kappa$. If we consider
the composition of $A_\varepsilon(\hbar)$ and
$\OpW(B_{\varepsilon,z,N}(\hbar))$ we get

\begin{equation}\label{B.res_con_1}
\begin{aligned}
\MoveEqLeft A_\varepsilon(\hbar)\OpW(B_{\varepsilon,z,N}(\hbar))
\\
\sum_{k=0}^N \hbar^{k} \OpW( a_{\varepsilon,k}) \sum_{j=0}^N
\hbar^{j} \OpW(b_{\varepsilon,z,j})
+ \sum_{j=0}^N \hbar^{N+1+j} R_N(\varepsilon,\hbar)
\OpW(b_{\varepsilon,z,j}).
\end{aligned}
\end{equation}

If we consider the first part then this corresponds to a composition
of two strongly $\hbar$-$\varepsilon$-admissible operators. As we
want to apply Theorem <ref> we need to ensure
$N$ satisfies the inequality

\begin{equation*}
\delta(N +2d + 2-\tau) +\tau > 2d+1.
\end{equation*}

As this is the condition that ensures a positive power in front of
the error term. If we assume $N$ satisfies the inequality. Then by
Theorem <ref> we have

\begin{equation*}
\begin{aligned}
\MoveEqLeft \sum_{k=0}^N \hbar^{k} \OpW( a_{\varepsilon,k}) \sum_{j=0}^N
\hbar^{j} \OpW(b_{\varepsilon,z,j})
\\
&= \sum_{l=0}^N \hbar^l
\OpW(c_{\varepsilon,l})
+ \hbar^{N+1}
\OpW(\mathcal{R}_\varepsilon(a_\varepsilon(\hbar),
\end{aligned}
\end{equation*}

\begin{equation*}
c_{\varepsilon,l}(x,p) = \sum_{\abs{\alpha}+\abs{\beta}+k+j=l} \frac{1}{\alpha!\beta!}\Big(\frac{1}{2} \Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}} (\partial_p^\alpha D_x^\beta a_{\varepsilon,k}) (\partial_p^\beta D_x^\alpha b_{\varepsilon,z,j})(x,p).
\end{equation*}

The error term satisfies for every multi indices $\alpha$, $\beta$
in $\N^d$, that there exists a constant $C(\alpha,\beta,N)$ independent of
$a_\varepsilon$ and $B_{\varepsilon,z,N}$ and an integer $M$ such

\begin{equation*}
\begin{aligned}
\hbar^{N+1} |\partial_x^\alpha \partial_p^\beta \mathcal{R}_\varepsilon
(a_\varepsilon(\hbar), b_\varepsilon(\hbar);x,p;\hbar) |
& \leq
C(\alpha,\beta,N) \hbar^{\delta(\tau-N-2d-2)_{-} + \tau -2d-1}
\varepsilon^{-\abs{\alpha}}
\\
&\times \sum_{ j+k \leq 2N}
\mathcal{G}^{\alpha,\beta}_{M,\tau-j-k}(a_{\varepsilon,k},(a_{\varepsilon,0}-\zeta_1),b_{\varepsilon,z,j},(a_{\varepsilon,0}-\zeta_1)^{-1}),
\end{aligned}
\end{equation*}

where $\mathcal{G}^{\alpha,\beta}_{M,\tau-j-k}$ are as defined in
Theorem <ref>. By
Lemma <ref> we have for all
$j+k\leq 2N$

\begin{equation*}
\begin{aligned}
\MoveEqLeft \mathcal{G}^{\alpha,\beta}_{M,\tau-j-k}(a_{\varepsilon,k},(a_{\varepsilon,0}-\zeta_1),b_{\varepsilon,z,j},(a_{\varepsilon,0}-\zeta_1)^{-1})
\\
&= \sup_{\substack{\abs{\gamma_1 + \gamma_2}+\abs{\eta_1 +
\eta_2}\leq M \\ (x,\xi) \in \R^{2d}}} \varepsilon^{(\tau -j
- k -M)_{-}+\abs{\alpha}}
\abs{\partial_x^\alpha\partial_\xi^\beta
( \partial_{x}^{\gamma_1} \partial_{\xi}^{\eta_1}a_{\varepsilon,k}(x,\xi) \partial_{x}^{\gamma_2} \partial_{\xi}^{\eta_2}b_{\varepsilon,z,j}(x,\xi))}
\\
&\leq C_{\alpha,\beta,M}\; \sup_{\mathclap{\substack{\abs{\gamma_1 +
\gamma_2}+\abs{\eta_1 + \eta_2}\leq M \\ (x,\xi) \in
\R^{2d}}}} \; \varepsilon^{(\tau -j - k
-M)_{-}-(\tau-k-\gamma_1)_{-}-(\tau-j-\gamma_2)_{-}}\left( \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{2j+M+\abs{\alpha}+\abs{\beta}}
\\
&\leq C_{\alpha,\beta,M} \left(
\frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{2j+M+\abs{\alpha}+\abs{\beta}}.
\end{aligned}
\end{equation*}

Now by Theorem <ref>
there exists a number $M_d$ such

\begin{equation*}
\hbar^{N+1}\norm{\OpW(\mathcal{R}_\varepsilon(a_\varepsilon(\hbar), B_\varepsilon,z,N(\hbar);\hbar))}_{\mathcal{L}(L^2(\R^d))} \leq C \hbar^{\kappa(N)} \left( \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{M_d}.
\end{equation*}

If we now consider the symbols $c_{\varepsilon,l}(x,p)$ for
$0\leq l \leq N$. For $l=0$ we have

\begin{equation*}
c_{\varepsilon,0}(x,p) = (a_{\varepsilon,0}(x,p)-z) b_{\varepsilon,z,0}(x,p) =1,
\end{equation*}

by definition of $b_{\varepsilon,z,0}(x,p) $. Now for
$1\leq l \leq N$ we have

\begin{equation*}
\begin{aligned}
c_{\varepsilon,l} ={}& \sum_{\abs{\alpha}+\abs{\beta}+k+j=l}
\frac{1}{\alpha!\beta!}\Big(\frac{1}{2}
\Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}}
(\partial_p^\alpha D_x^\beta a_{\varepsilon,k}) (\partial_p^\beta
D_x^\alpha b_{\varepsilon,z,j})
\\
={}& \sum_{\substack{\abs{\alpha}+\abs{\beta}+k+j=l\\0\leq j\leq
l-1}} \frac{1}{\alpha!\beta!}\Big(\frac{1}{2}
\Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}}
(\partial_p^\alpha D_x^\beta a_{\varepsilon,k}) (\partial_p^\beta
D_x^\alpha b_{\varepsilon,z,j}) +(a_{\varepsilon,0}-z)
\\
={}& \sum_{\substack{\abs{\alpha}+\abs{\beta}+k+j=l\\0\leq j\leq
l-1}} \frac{1}{\alpha!\beta!}\Big(\frac{1}{2}
\Big)^{\abs{\alpha}}\Big(-\frac{1}{2} \Big)^{\abs{\beta}}
(\partial_p^\alpha D_x^\beta a_{\varepsilon,k}) (\partial_p^\beta
D_x^\alpha b_{\varepsilon,z,j})
\\
&-\sum_{\substack{\abs{\alpha} + \abs{\beta} + k +j = l\\
0\leq j \leq l-1 }} \frac{1}{\alpha ! \beta !}
\frac{1}{2^{\abs{\alpha}}} \frac{1}{(-2)^{\abs{\beta}}}
(\partial_p^{\alpha} D_x^{\beta} a_{\varepsilon,k})
(\partial_p^{\beta} D_x^{\alpha} b_{\varepsilon,z,j})
\\
\end{aligned}
\end{equation*}

by definition of $b_{\varepsilon,z,l}$. These two equalities implies that

\begin{equation}\label{B.res_con_2}
\sum_{k,j=0}^N \hbar^{k+j} \OpW( a_{\varepsilon,k})  \OpW(b_{\varepsilon,z,j})  =  I + \hbar^{N+1}\OpW(\mathcal{R}_\varepsilon(a_\varepsilon(\hbar), B_\varepsilon,z,N(\hbar);\hbar)).
\end{equation}

This was the first part of equation (<ref>). If we now
consider the second part of (<ref>):

\begin{equation*}
\sum_{j=0}^N \hbar^{N+1+j}  R_N(\varepsilon,\hbar) \OpW(b_{\varepsilon,z,j}).
\end{equation*}

By Theorem <ref> and
Lemma <ref> there exist constants
$M_d$ and $C$ such that

\begin{equation*}
\hbar^j \norm{\OpW(b_{\varepsilon,z,j})}_{\mathcal{L}(L^2(\R^d))} \leq C \left( \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{2j+M_d}
\end{equation*}

for all $j$ in $\{0,\dots,N\}$. Hence by assumption we have

\begin{equation*}
\sum_{j=0}^N \hbar^{N+1+j}  \norm{R_N(\varepsilon,\hbar) \OpW(b_{\varepsilon,z,j})}_{\mathcal{L}(L^2(\R^d))} \leq C \hbar^{\kappa(N)}  \left( \frac{\abs{z-\zeta_1}}{\dist(z,[\zeta_1,\infty))}\right)^{q(N)}.
\end{equation*}

Now by combining this with (<ref>) and
(<ref>) we get

\begin{equation*}
(A_\varepsilon(h) - z ) \OpW B_{\varepsilon,z,N} = I + h^{N+1} \Delta_{z,N+1}(h)
\end{equation*}

\begin{equation*}
\hbar^{N+1}\norm{ \Delta_{z,N+1}(h)}_{\mathcal{L}(L^2(\R^d))} \leq C \hbar^{\kappa(N)} \left( \frac{\abs{z}}{\dist(z,[\zeta_1,\infty))} \right)^{q(N)},
\end{equation*}

where $\kappa$ is a positive strictly increasing function and $q(N)$
is a positive integer depending on $N$. Which is the desired form and this ends the proof.

§.§ Functional calculus for rough pseudo-differential operators

We are now almost ready to construct/prove a functional calculus for
operators satisfying Assumption <ref>. First we
need to settle some terminology and recall a theorem.

For a smooth function $f:\R\rightarrow\R$ we define the almost
analytical extension $\tilde{f}:\C\rightarrow\C$ of $f$ by

\begin{equation*}
\tilde{f}(x+iy) = \left( \sum_{r=0}^n f^{(r)}(x)\frac{(iy)^r}{r!} \right) \sigma(x,y),
\end{equation*}

where $n \geq1$ and

\begin{equation*}
\sigma(x,y) = \omega\left(\tfrac{y}{\lambda(x)}\right),
\end{equation*}

for some smooth function $\omega$, defined on $\R$ such that
$\omega(t)=1$ for $\abs{t}\leq1$ and $\omega(t)=0$ for
$\abs{t}\geq2$. Moreover we will use the notation

\begin{equation*}
\begin{aligned}
\bar{\partial} \tilde{f}(x+iy)& \coloneqq \frac{1}{2} \left(
\frac{\partial \tilde{f}}{\partial x} + i \frac{\partial
\tilde{f}}{\partial y} \right)
\\
& = \frac{1}{2} \left( \sum_{r=0}^n f^{(r)}(x)\frac{(iy)^r}{r!}
\right) ( \sigma_x(x,y) + i \sigma_y(x,y)) + \frac{1}{2}
f^{(n+1)}(x)\frac{(iy)^n}{n!} \sigma(x,y),
\end{aligned}
\end{equation*}

where $\sigma_x$ and $\sigma_y$ are the partial derivatives of
$\sigma$ with respect to $x$ and $y$ respectively.

The above choice is one way to define an almost analytic extension
and it is not unique. Once an $n$ has been fixed the extension has
the property that

\begin{equation*}
|\bar{\partial} \tilde{f}(x+iy)| =\mathcal{O}( \abs{y}^n)
\end{equation*}

as $y\rightarrow0$. Hence when making calculation a choice has to be
made concerning how fast $|\bar{\partial} \tilde{f}|$ vanishes when
approaching the real axis. If $f$ is a $C_0^\infty(\R)$ function one
can find an almost analytic extension $\tilde{f}$ in
$C_0^\infty(\C)$ such $f(x)=\tilde{f}(x)$ for $x$ in $\R$ and

\begin{equation*}
|\bar{\partial} \tilde{f}(x+iy)| \leq C_N  \abs{y}^N, \quad\text{for all } N\geq0.
\end{equation*}

without chancing the extension. This type of extension could be
based on a Fourier transform hence it may not work for a general
smooth function. For details see <cit.> or <cit.>.

The type of functions for which we can construct a functional calculus
is introduced in the next definition:

For $\rho$ in $\R$ we define the set $S^\rho$ to be the set of
smooth functions $f:\R\rightarrow\R$ such that

\begin{equation*}
|f^{(r)}(x)| \coloneqq \Big|\frac{d^r f}{dx^r}(x) \Big| \leq c_r \lambda(x)^{\rho-r}
\end{equation*}

for some $c_r< \infty$, all $x$ in $\R$ and all integers $r\geq0$. Moreover we define
$\mathcal{W}$ by

\begin{equation*}
\mathcal{W} \coloneqq \bigcup_{\rho<0} S^\rho.
\end{equation*}

We can now recall the form of the spectral theorem which we will use:

Let $H$ be a self-adjoint operator acting on a Hilbert space
$\mathscr{H}$ and $f$ a function from $\mathcal{W}$. Moreover let
$\tilde{f}$ be an almost analytic extension of $f$ with $n$
terms. Then the bounded operator $f(H)$ is given by the equation

\begin{equation*}
f(H) =- \frac{1}{\pi} \int_\C   \bar{\partial }\tilde{f}(z) (z-H)^{-1} \, L(dz),
\end{equation*}

where $L(dz)=dxdy$ is the Lebesgue measure on $\C$. The formula
holds for all numbers $n\geq1$.

A proof of the above theorem can be found in e.g. <cit.> or <cit.>. We are now ready to state and prove the functional calculus for a
certain class of rough pseudo-differential operators.

Let $A_\varepsilon(\hbar)$, for $\hbar$ in $(0,\hbar_0]$, be a
$\hbar$-$\varepsilon$-admissible operator of regularity
$\tau \geq 1$ and with symbol

\begin{equation*}
a_\varepsilon(\hbar) = \sum_{j\geq 0} \hbar^j a_{\varepsilon,j}.
\end{equation*}

Suppose that $A_\varepsilon(h)$ satisfies
Assumption <ref>. Then for any function $f$
from $\mathcal{W}$, $f(A_\varepsilon(h))$ is a
$\hbar$-$\varepsilon$-admissible operator of regularity $\tau$ with
respect to a constant tempered weight
function. $f(A_\varepsilon(h))$ has the symbol

\begin{equation*}
a_{\varepsilon}^f(\hbar) = \sum_{j\geq 0} \hbar^j a_{\varepsilon,j}^f,
\end{equation*}

\begin{equation}\label{B.func_cal_sym}
\begin{aligned}
a_{\varepsilon,0}^f &= f(a_{\varepsilon,0}),
\\
a_{\varepsilon,j}^f &= \sum_{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon,j,k} f^{(k)}(a_{\varepsilon,0}) \quad\quad\text{for $j\geq1$},
\end{aligned}
\end{equation}

the symbols $d_{\varepsilon,j,k}$ are the polynomials from
Lemma <ref>. Especially we have

\begin{equation*}
a_{\varepsilon,1}^f =
\end{equation*}

The proof is an application of Theorem <ref> and
the fact that the resolvent is a $\hbar$-$\varepsilon$-admissible
operator as well.

By Theorem <ref> the operator
$A_\varepsilon(\hbar)$ is essentially self-adjoint for sufficiently small $\hbar$. Hence
Theorem <ref> gives us

\begin{equation*}
f(A_\varepsilon(\hbar)) =- \frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) (z-A_\varepsilon(\hbar))^{-1} \, L(dz),
\end{equation*}

where $\tilde{f}$ is an almost analytic extension of $f$. For the
almost analytic extension of $f$ we will need a suffiently large
number of terms which we assume to have chosen from the
start. Theorem <ref> gives that the resolvent
is a $\hbar$-$\varepsilon$-admissible operator and the explicit form
of it as well. Hence

\begin{equation*}
\begin{aligned}
\MoveEqLeft  f(A_\varepsilon(\hbar))
\\
={}& \frac{1}{\pi} \int_\C \bar{\partial
}\tilde{f}(z) \sum_{j=0}^M \hbar^j \OpW(b_{\varepsilon,z,j}) \,
- \frac{1}{\pi} \int_\C \bar{\partial }\tilde{f}(z)
h^{N+1} (z-A_\varepsilon(\hbar))^{-1}\Delta_{z,N+1}(h) \, L(dz),
\end{aligned}
\end{equation*}

where the symbols $b_{\varepsilon,z,j}$ and the operator
$\Delta_{z,N+1}(h)$ are as defined in
Theorem <ref>. If we start by considering the
error term we have by Theorem <ref> the

\begin{equation*}
\begin{aligned}
\norm{(z-A_\varepsilon(\hbar))^{-1}\Delta_{z,N+1}(h)}_{\mathcal{L}(L^2(\R^d))}
\leq{}& C \hbar^{\kappa(N)} \frac{1}{\abs{\im(z)}} \left(
\frac{\abs{z}}{\dist(z,[\zeta_1,\infty))} \right)^{q(N)}
\\
\leq{}& C \hbar^{\kappa(N)}
\frac{\abs{z}^{q(N)}}{\abs{\im(z)}^{q(N)+1} },
\end{aligned}
\end{equation*}

for $N$ sufficiently large and where $q(N)$ is some integer
dependent of the number $N$. We have

\begin{equation*}
|\bar{\partial}\tilde{f}(z)| \leq c_1 \sum_{r=0}^n |\tilde{f}^{(r)}(\re(z))| \lambda(\re(z))^{r-1} \boldsymbol{1}_U(z) + c_2 |\tilde{f}^{(n+1)}(\re(z))| |\im(z)|^n \boldsymbol{1}_V(z),
\end{equation*}

\begin{align*}
U &=\{z \in \C\, |\, \lambda(\re(z)) <
\abs{\im(z)}<2\lambda(\re(z))\}, \shortintertext{and} V &=\{z \in
\C \, | \, 0 < \abs{\im(z)}<2\lambda(\re(z))\}.
\end{align*}

This estimate follows directly from the definition of
$\tilde{f}$. By combining these estimates and the definition of the
class of functions $\mathcal{W}$ we have

\begin{equation*}
\Big\lVert\frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) h^{N+1} (z-A_\varepsilon(\hbar))^{-1}\Delta_{z,N+1}(h) \, L(dz)\Big\rVert_{\mathcal{L}(L^2(\R^d))} \leq C \hbar^{\kappa(N)}.
\end{equation*}

What remains to prove the following equality

\begin{equation}\label{B.sum_symbol_func_proof}
\sum_{j=0}^M \hbar^j \frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) \OpW(b_{\varepsilon,z,j}) \, L(dz) =   \sum_{j=0}^M \hbar^j \OpW(a_{\varepsilon,j}^f),
\end{equation}

where the symbols $a_{\varepsilon,j}^f$ are as defined in the
statement. We will only consider one of the terms as the rest is
treated analogously. Hence we need to establish the equality

\begin{equation*}
\frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) \OpW(b_{\varepsilon,z,j}) \, L(dz) =   \OpW(a_{\varepsilon,j}^f).
\end{equation*}

As both operators are bounded we need only establish the equality
weekly for a dense subset of $L^2(\R^d)$. Hence let $\varphi$ and
$\psi$ be two functions from $C_0^\infty(\R^d)$ and a $j$ be
given. We have

\begin{equation}\label{B.func_cal_1}
\langle \frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) \OpW(b_{\varepsilon,z,j}) \, L(dz) \varphi , \psi \rangle =  \frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) \langle \OpW(b_{\varepsilon,z,j}) \varphi , \psi \rangle \, L(dz) ,
\end{equation}

where we have

\begin{equation}\label{B.func_cal_2}
\begin{aligned}
\langle \OpW&(b_{\varepsilon,z,j}) \varphi , \psi \rangle
\\
={}& \frac{1}{(2\pi\hbar)^d}\int_{\R^{3d}}
e^{i\hbar^{-1}\langle x-y,p\rangle}
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx
\\
={}& \lim_{\sigma\rightarrow\infty} \frac{1}{(2\pi\hbar)^d}
\int_{\R^{3d}} e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx,
\end{aligned}
\end{equation}

where the function $g$ is a positiv Schwartz function bounded by $1$
and identical $1$ in a neighbourhood of $0$. I the above we have set
$g_{\sigma}(x,y,p)=g(\tfrac{x}{\sigma},\tfrac{y}{\sigma},\tfrac{p}{\sigma})$. The
next step in the proof is to apply dominated convergence to move the
limit outside the integral over $z$.

We let $\chi$ be in $C_0^\infty(\R^d)$ such that $\chi(p)=1$ for
$\abs{p}\leq1$ and $\chi(p)=0$ for $\abs{p}\geq2$. With this
function we have

\begin{equation}\label{B.func_cal_3}
\begin{aligned}
\MoveEqLeft \frac{1}{(2\pi\hbar)^d}\int_{\R^{3d}}
e^{i\hbar^{-1}\langle x-y,p\rangle} g_\sigma(x,y,p)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx
\\
&\phantom{\frac{1}{2}}{}= \frac{1}{(2\pi\hbar)^d}\big[
\int_{\R^{3d}}e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p) \chi(p)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx
\\
&\phantom{\frac{1}{2}=}{}+ \int_{\R^{3d}}
e^{i\hbar^{-1}\langle x-y,p\rangle} g_\sigma(x,y,p)(1-\chi(p))
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx\big].
\end{aligned}
\end{equation}

By Lemma <ref> we have

\begin{equation}\label{B.func_cal_4}
\begin{aligned}
\MoveEqLeft \Big|\int_{\R^{3d}}e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p) \chi(p)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx \Big|
\\
&\leq C_j \varepsilon^{-(\tau-j)_{-}} \left(
\frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j},
\end{aligned}
\end{equation}

where the $\zeta_1$ is the number from
Assumption <ref>. The factor
$\varepsilon^{-(\tau-j)_{-}}$ is not an issue as the operator we
consider has $\hbar^j$ in front. We have just omitted to write this
factor. This bound is clearly independent of $\sigma$. We now need
to bound the term with $1-\chi(p)$. Here we use that on the support
of $1-\chi(p)$ we have $\abs{p}>1$. Hence the operator

\begin{equation*}
M= \frac{(-i\hbar)^{2d}}{ \abs{p}^{2d}}(\sum_{k=1}^d \partial_{y_k}^2)^d = \frac{(-i\hbar)^{2d}}{ \abs{p}^{2d}} \sum_{\abs{\alpha}=d} \partial_{y}^{2\alpha},
\end{equation*}

is well defined when acting on functions supported in
$\supp(1-\chi)$. We then obtain that

\begin{equation*}
\begin{aligned}
\MoveEqLeft \int_{\R^{3d}} e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p) (1-\chi(p))
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx
\\
&= \int_{\R^{3d}} (M e^{i\hbar^{-1}\langle
x-y,p\rangle}) g_\sigma(x,y,p) (1-\chi(p))
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx
\\
&= \int_{\R^{3d}} e^{i\hbar^{-1}\langle
x-y,p\rangle}(1-\chi(p)) M^t(g_\sigma(x,y,p)
\varphi(y))\overline{\psi}(x) \, dydpdx,
\end{aligned}
\end{equation*}

where we have used thet $ M e^{i\hbar^{-1}\langle x-y,p\rangle} = e^{i\hbar^{-1}\langle x-y,p\rangle}$. Just considering the action of $M^t$ inside the integral we have by Leibniz's formula

\begin{equation*}
\begin{aligned}
|M^tg_\sigma(x,y,p) b_{\varepsilon,z,j}(\tfrac{x+y}{2},p)
\varphi(y) |
&= \frac{\hbar^{2d}}{ \abs{p}^{2d}} \Big| \sum_{\abs{\alpha}=d}
\sum_{\beta\leq
2\alpha} \partial_{y}^{2\alpha-\beta}(g_\sigma(x,y,p)
\varphi(y)) \partial_{y}^{\beta}
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \Big|
\\
&\leq  C_j \frac{\hbar^{2d}}{ \abs{p}^{2d}}
\boldsymbol{1}_{\supp(\varphi)}(y) \sum_{\abs{\alpha}=d}
\sum_{\beta\leq 2\alpha} \varepsilon^{-(\tau-j-\abs{\beta})_{-}}
\left( \frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j +
\abs{\beta}}
\\
&\leq C \frac{\hbar^{2d}}{ \abs{p}^{2d}}
\varepsilon^{-(\tau-j-2d)_{-}} \boldsymbol{1}_{\supp(\varphi)}(y)
\left( 1+\frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j + 2d},
\end{aligned}
\end{equation*}

where we again have used
Lemma <ref>. This imply the estimate

\begin{equation*}
\begin{aligned}
\MoveEqLeft\Big|\int_{\R^{3d}} e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p) (1-\chi(p))
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx \Big|
\\
&\leq C_j \varepsilon^{-(\tau-j)_{-}} \left(
1+\frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j + 2d}.
\end{aligned}
\end{equation*}

If we combine this estimate with (<ref>) and
(<ref>) we have

\begin{equation*}
\begin{aligned}
\MoveEqLeft\Big| \frac{1}{(2\pi\hbar)^d} \int_{\R^{3d}}
e^{i\hbar^{-1}\langle x-y,p\rangle} g_\sigma(x,y,p)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \varphi(y)
\overline{\psi}(x) \, dydpdx \Big|
\\
&\leq C \varepsilon^{-(\tau-j)_{-}} \left(
1+\frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j + 2d}.
\end{aligned}
\end{equation*}

As above we have

\begin{equation*}
\Big| \int_\C \bar{\partial }\tilde{f}(z) \left(
1+\frac{\abs{z-\zeta_1}}{\abs{\im(z)}}\right)^{2j + 2d} \,
L(dz)\Big| < \infty.
\end{equation*}

Hence we can apply dominated convergence and by an analogous
argument we can also apply Fubini's Theorem. This gives

\begin{equation}\label{B.func_cal_5}
\begin{aligned}
\MoveEqLeft  \frac{1}{\pi} \int_\C \bar{\partial }\tilde{f}(z) \langle
\OpW(b_{\varepsilon,z,j}) \varphi , \psi \rangle \, L(dz)
\\
={}& \lim_{\sigma\rightarrow\infty} \frac{1}{(2\pi\hbar)^d}
\int_{\R^{3d}} e^{i\hbar^{-1}\langle
x-y,p\rangle} g_\sigma(x,y,p)
\frac{1}{\pi} \int_\C \bar{\partial }\tilde{f}(z)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \, L(dz) \varphi(y)
\overline{\psi}(x) \, dydpdx
\end{aligned}
\end{equation}

If we only consider the integral over $z$ then we have by a Cauchy
formula and Lemma <ref> that

\begin{equation*}
\begin{aligned}
\frac{1}{\pi} \int_\C \bar{\partial }\tilde{f}(z)
b_{\varepsilon,z,j}(\tfrac{x+y}{2},p) \, L(dz)
&= \frac{1}{\pi}
\int_\C \bar{\partial }\tilde{f}(z) \sum_{k=1}^{2j-1}
d_{\varepsilon, j ,k}(\tfrac{x+y}{2},p)
b_{\varepsilon,z,0}^{k+1}(\tfrac{x+y}{2},p) \, L(dz)
\\
&= \sum_{k=1}^{2j-1} d_{\varepsilon, j ,k}(\tfrac{x+y}{2},p)
\frac{1}{\pi} \int_\C \bar{\partial }\tilde{f}(z)
\Big(\frac{1}{a_{\varepsilon,0}(\tfrac{x+y}{2},p)-z}\Big)^{k+1}\,
\\
&= \sum_{k=1}^{2j-1} \frac{(-1)^k}{k!} d_{\varepsilon, j ,k}(\tfrac{x+y}{2},p)
f^{(k)}( a_{\varepsilon,0}(\tfrac{x+y}{2},p)) =
\end{aligned}
\end{equation*}

which is the desired form of $a_{\varepsilon,j}^f$ given in
(<ref>). Now combing this with (<ref>),
(<ref>) and (<ref>) we arrive at

\begin{equation*}
\langle \frac{1}{\pi} \int_\C  \bar{\partial }\tilde{f}(z) \OpW(b_{\varepsilon,z,j}) \, L(dz) \varphi , \psi \rangle = \langle \OpW(a_{\varepsilon,j}^f) \varphi , \psi \rangle.
\end{equation*}

The remaning $j$' can be treated analogously and hence we obtain the
equality (<ref>). This ends the proof.

§.§ Applications of functional calculus

With the functional calculus we can now prove some useful theorems and lemmas. One of them is an asymptotic expansion of
certain traces.

Let $A_\varepsilon(\hbar)$, for $\hbar$ in $(0,\hbar_0]$, be a
$\hbar$-$\varepsilon$-admissible operator of regularity
$\tau \geq 1$ and symbol

\begin{equation*}
a_\varepsilon(\hbar) = \sum_{j\geq 0} \hbar^j a_{\varepsilon,j}.
\end{equation*}

Suppose that $A_\varepsilon(\hbar)$ satisfies
Assumption <ref>. Let $E_1<E_2$ be two real
numbers and suppose there exists an $\eta>0$ such
$a_{\varepsilon,0}^{-1}([E_1 - \eta, E_2+\eta])$ is compact. Then we have

\begin{equation}
\spec(A_\varepsilon(\hbar))\cap[E_1.E_2] \subseteq \spec_{pp}(A_\varepsilon(\hbar)),
\end{equation}

for $\hbar$ sufficiently small, where $\spec_{pp}(A_\varepsilon(\hbar))$ is the pure point spectrum of $A_\varepsilon(\hbar)$.

Let $f$ and $g$ be in
$C_0^\infty((E_1-\eta,E_2+\eta))$ such $g(t)=1$ for
$t\in[E_1,E_2]$ and $f(t)=1$ for $t$ in $\supp(g)$. By Theorem <ref> we have

\begin{equation}\label{B.punkt_spec_1}
f(A_\varepsilon(\hbar)) = A_{\varepsilon,f,N}(\hbar) + \hbar^{N+1}
\end{equation}

where the terms $A_{\varepsilon,f,N}(\hbar)$ consists of the first $N$ terms in the
expansion in $\hbar$ of $ f(A_\varepsilon(\hbar))$. We get by (<ref>) and the definition of $g$ and $f$ that

\begin{equation*}
g(A_\varepsilon(\hbar)) (I - \hbar^{N+1}
R_{N+1,f}(\varepsilon,\hbar))=  g(A_\varepsilon(\hbar))A_{\varepsilon,f,N}(\hbar).
\end{equation*}

Hence for $\hbar$ sufficiently small we have

\begin{equation*}
g(A_\varepsilon(\hbar)) =  g(A_\varepsilon(\hbar))A_{\varepsilon,f,N}(\hbar) (I - \hbar^{N+1}
\end{equation*}

thereby we have the inequality

\begin{equation}\label{B.punkt_spec_2}
\norm{ g(A_\varepsilon(\hbar))}_{\Tr} \leq c \norm{g}_{\infty} \norm{A_{\varepsilon,f,N}(\hbar)}_{\Tr} \leq C \hbar^{-d},