# Time dependent Quantum Perturbations uniform in the semiclassical regime

François Golse  CMLS, École polytechnique, CNRS, Université Paris-Saclay ,
91128 Palaiseau Cedex, France<EMAIL_ADDRESS>and  Thierry
Paul  Laboratoire J.-L. Lions, Sorbonne Université & CNRS, boîte courrier 187,
75252 Paris Cedex 05, France<EMAIL_ADDRESS>
###### Abstract.

We present a time dependent quantum perturbation result, uniform in the Planck
constant, for perturbations of potentials whose gradients are Lipschitz
continuous by potentials whose gradients are only bounded a.e.. Though this
low regularity of the full potential is not enough to provide the existence of
the classical underlying dynamics, at variance with the quantum one, our
result shows that the classical limit of the perturbed quantum dynamics
remains in a tubular neighbourhood of the classical unperturbed one of size of
order of the square root of the size of the perturbation. We treat both
Schrödinger and von Neumann-Heisenberg equations.

in memory of Arthur Wightman

###### Contents

  1. 1 Introduction
  2. 2 Main result
  3. 3 Applications to the classical limit
  4. 4 The case of mixed states
  5. 5 Proof of Theorems 2.1 and 4.1, and Proposition 3.2
  6. 6 Proof of Corollaries 2.2 and 4.2
  7. 7 Proof of Proposition 2.4

##  1\. Introduction

Perturbation theory has a very special status in Quantum Mechanics. On one
side, it is responsible to most of its more spectacular success, from atomic
to nuclear physics. On the other side, it has a very peculiar epistemological
status: it was while he was working with Max Born [B25] on the Bohr-Sommerfeld
quantization of celestial perturbations series, as explicitly stated by
Poincaré in his famous “Mémoires” [P1892], that Heisenberg went to the idea of
replacing the commutative algebra of convolution — corresponding to multiple
multiplications of Fourier series appearing in computations on action-angle
variables — by the famous noncommutative algebra of matrices [H25].

After quantum mechanics was truly settled, perturbation theory took a
completely different form, in the paradigm of functional analysis “à la Kato”
and appeared then mostly in the framework of the so-called Rayleigh-
Schrödinger series. A kind of paradox is that it took a long time to link back
the Rayleigh-Schrödinger series to the “original” formalism of quantization
of, say, Birkhoff series [B28], though, in the mean time, the latter continued
to be extensively used for applied purpose e.g. in heavy chemical
computations.

It seems that Arthur Wightman proposed to several PhD students to work on this
problem. One of the difficulty is that, starting with the second term of the
Rayleigh-Schrödinger expansion,

$E^{2}_{i}=\sum\limits_{k}\frac{\langle\psi^{0}_{i},V\psi^{0}_{k}\rangle\langle\psi^{0}_{k},V\psi^{0}_{i}\rangle}{E^{0}_{i}-E^{0}_{k}},$

there appears formally poles at zero in the Planck constant, for example when
the unperturbed Hamiltonian is the harmonic oscillator with unperturbed
eigenvalues $E^{0}_{i}=(i+~{}\tfrac{1}{2})\hbar$. Although this poles
disappear at the classical limit $\hbar\to 0$ because the sum
$\sum\limits_{k}\frac{\langle\psi^{0}_{i},V\psi^{0}_{k}\rangle\langle\psi^{0}_{k},V\psi^{0}_{i}\rangle}{i-k}$
vanishes in this limit for parity reasons, controlling all the terms of the
series remained for years a task considered as unachievable.

To our knowledge, the first proof on the convergence term by term of the
Rayleigh-Schrödinger expansion to the quantized Birkhoff one, for
perturbations of non-resonant harmonic oscillators, was given in [G87], by
implementing the perturbation procedure in the so-called Bargman
representation (see also [D91] for an implementation in the framework of the
Lie method). The reader interested in this subject can also consult [P16,
P162] for a proof (also for general non harmonic unperturbed Hamiltonians) in
a generalization of Écalle’s mould theory and [NPST18] for a link between
Rayleigh-Schrödinger expansion and Hopf algebras.

When one considers time dependent perturbation theory, i.e. comparison between
two quantum evolution associated to two “close” Hamiltonians $H,H^{\prime}$,
the situation is more difficult. The simple Duhamel formula

$e^{-i\frac{tH}{\hbar}}-e^{-i\frac{tH^{\prime}}{\hbar}}=\frac{1}{i\hbar}\int_{0}^{t}e^{-i\frac{(t-s)H}{\hbar}}(H^{\prime}-H)e^{-i\frac{sH^{\prime}}{\hbar}}ds$

shows clearly that a pole at zero in the Planck constant is again involved.
But to our knowledge, no combinatorics or normal form can help to remove it in
general and one is usually reduced to the trivial estimate

$\|e^{-i\frac{tH}{\hbar}}-e^{-i\frac{tH^{\prime}}{\hbar}}\|\leq
t\frac{\|H^{\prime}-H\|}{\hbar}$

valid for, e.g. any Schatten norm, the operator, Hilbert-Schmidt or trace norm
for example.

In the present paper, we will get rid of this pole in $\hbar$ phenomenon by
estimating the difference between two quantum evolutions (in a weak topology
consisting in tracing against a set of test observables) in two forms:

\- one linear in the norm of the difference of the Hamiltonians plus a term
vanishing with $\hbar$

\- the other proportional to the norm of the difference of the Hamiltonians to
the power $1/3$ and independent of $\hbar$.

The proofs of our results, Sections 5, 6 and 7, will be using the framework of
the von Neumann-Heisenberg equation for density operators $D$,

$\partial_{t}D=\tfrac{1}{i\hbar}[D,H],$

but our results, Theorem 2.1 and Corollary 2.2, will be first presented for
pure states, Section 2, that is when $D=|\psi\rangle\langle\psi|$, in which
case it reduces to the usual Schrödinger equation (modulo a global phase of
the wave function)

$i\hbar\partial_{t}\psi=H\psi.$

The mixed states situation will be treated in Section 4, Theorem 4.1.

Our results will need very low regularity of the perturbed potential, namely
the boundness of its gradient, and of the unperturbed one, Lipschitz
continuity of its gradient. In this situation, the classical underlying
dynamics is well posed for the unperturbed Hamiltonian, but not for the
perturbed one. To our knowledge, the classical limit for pure state in this
perturbed situation is unknown. We show, in Section 3 Theorem 3.1, that the
limit as $\hbar\to 0$ of the Wigner function of the wave function at time $t$
is close to the one of the initial state pushed forward by the unperturbed
classical flow.

##  2\. Main result

For $\lambda,\mu\in[0,1]$, let us consider the quantum Hamiltonian

$\mathcal{H}^{\lambda,\mu}_{0}=\mathcal{H}_{0}:=-\tfrac{1}{2}{\hbar}^{2}{\Delta}_{x}+\tfrac{\lambda}{2}|x|^{2}+\mu
V$

on $\mathfrak{H}:=L^{2}(\mathbf{R}^{d})$. Here $V\equiv V(x)\in\mathbf{R}$
such that $V\in C^{1,1}(\mathbf{R}^{d})$. For any other real potential $U\in
W^{1,\infty}(\mathbf{R}^{d})$, we define, for ${\epsilon}\in[0,1]$

$\mathcal{H}^{\lambda,\mu}_{\epsilon}=\mathcal{H}_{\epsilon}:=-\tfrac{1}{2}{\hbar}^{2}{\Delta}_{x}+\tfrac{\lambda}{2}|x|^{2}+\mu
V+{\epsilon}U.$

Henceforth we denote

$\displaystyle\mathcal{H}$ $\displaystyle:=$
$\displaystyle\mathcal{H}_{0}^{1,0}=-\tfrac{1}{2}{\hbar}^{2}{\Delta}_{x}+\tfrac{1}{2}|x|^{2}\mbox{
\hfill(harmonic oscillator)}$ $\displaystyle\mathcal{D}(\mathfrak{H})$
$\displaystyle:=$ $\displaystyle\\{R\in\mathcal{L}^{1}(\mathfrak{H})\text{
s.t. }R=R^{*}\geq 0\text{ and
}\operatorname{trace}_{\mathfrak{H}}(R)=1\\}\,\mbox{ (density operators)},$
$\displaystyle\mathcal{D}_{2}(\mathfrak{H})$ $\displaystyle:=$
$\displaystyle\\{R\in\mathcal{D}(\mathfrak{H})\text{ s.t.
}\operatorname{trace}_{\mathfrak{H}}(R^{1/2}{\mathcal{H}}R^{1/2})<\infty\\}\,\mbox{
(finite second moments)}.$

For $\psi\in\mathfrak{H}$, we define

(1)
$\Delta(\psi):=\sqrt{\left(\psi,(x-(\psi,x\psi))\psi)^{2}+(-i\hbar\nabla_{x}-(\psi,-i\hbar\nabla_{x}\psi))^{2}\psi\right)}$

Note that the Heisenberg inequalities

$\sqrt{(\psi,(x_{k}-(\psi,x_{k}\psi))^{2},\psi)}\sqrt{(\psi,(-i\hbar\nabla_{x_{k}}-(\psi,-i\hbar\nabla_{x_{k}}\psi))^{2}\psi)}\geq\hbar/2,\
k=1,\dots,d,$

imply that

$\Delta(\psi)\geq\sqrt{2d\hbar}.$

On $\mathcal{D}(\mathfrak{H})$ we define the following distance

(2) $\mbox{\bf
d}(R,S):=\sup_{\begin{subarray}{c}\max\limits_{\begin{subarray}{c}|{\alpha}|,|{\beta}|\leq
2[\frac{d}{4}]+3\end{subarray}}\|\mathcal{D}^{\alpha}_{-i\hbar\nabla}\mathcal{D}^{\beta}_{x}F\|_{1}\leq
1\end{subarray}}|\operatorname{trace}{(F(R-S))}|\,,$

where $\mathcal{D}_{A}=\tfrac{1}{i\hbar}[A,\cdot]$ for each (possibly
unbounded) self-adjoint operator $A$ on $\mathfrak{H}$ and $\|.\|_{1}$ is the
trace norm on $\mathcal{D}(\mathfrak{H})$. The fact that d is a distance has
been proved in [GJP20, Appendix A].

Abusing the notation, for $\phi,\psi\in\mathfrak{H}$, we set for
$\psi,\varphi\in\mathfrak{H}$

$\mbox{\bf d}(\psi,\varphi):=\mbox{\bf
d}(|\psi\rangle\langle\psi|,|\varphi\rangle\langle\varphi|)$

Consider the family of Schrödinger equations, for ${\epsilon}\in[0,1]$,

(3)
$i{\hbar}{\partial}_{t}\psi_{\epsilon}(t)=\mathcal{H}_{\epsilon}\psi_{\epsilon}(t),\,\qquad\psi_{\epsilon}(0)=\psi^{in}_{\epsilon}\in
H^{2}(\mathbf{R}^{d})\,.$

######  Theorem 2.1.

Let $\psi^{in}_{\epsilon}$ satisfy the following hypothesis:

(4) $\Delta(\psi^{in})=O(\sqrt{\hbar}).$

Then, for every $t$,

$\mbox{\bf d}(\psi_{0}(t),\psi_{\epsilon}(t))^{2}\leq C(t)\epsilon+D(t)\hbar,$

where $C(t),D(t)$, given by (22),(23), satisfy

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\frac{e^{|t|(1-\lambda+\mu\operatorname{Lip}(\nabla
V))}-1}{1-\lambda+\mu\operatorname{Lip}(\nabla
V)}C_{(\psi_{\epsilon}^{in},\mathcal{H}\psi_{\epsilon}^{in}),\|V\|_{\infty},\|U\|_{\infty},\|\nabla
U\|_{\infty}}<\infty$ $\displaystyle D(t)$ $\displaystyle=$ $\displaystyle
e^{|t|(1-\lambda+\mu\operatorname{Lip}(\nabla V))}D_{d}<\infty$

The following result gives an upper bound independent of $\hbar$.

######  Corollary 2.2.

Under the same assumptions as in Theorem 2.1,

$\mbox{\bf d}(\psi_{0}(t),\psi_{\epsilon}(t))\leq E(t)\epsilon^{\frac{1}{3}},$

with

$E(t)=\min{(\sqrt{C(t)+D(t}),2|t|\|U\|^{2}_{\infty})}.$

The function $z\mapsto\frac{e^{z}-1}{z}$ is extended by continuity at $z=0$,
so that, when $\lambda=1,\ \operatorname{Lip}(\nabla V)=0$ (perturbation of
the harmonic oscillator), $C(t)$ increases linearly in time and $D(t)$ is
independent of time and $C(t),E(t)$ increase linearly in time.

######  Remark 2.3.

Other choices than the hypothesis (i) are possible, that we didn’t mention for
sake of clarity of the main statements. For example

(i)’

$\Delta(\psi^{in})=O(\sqrt{\epsilon})$:

in this case the statement of both Theorem 2.1 and Corollary 2.2 remain the
same with a slight chance of the constants $C(t),D(t)$.

(i)”

$\Delta(\psi^{in})=O(\hbar^{\alpha}),\ 0\leq\alpha<1/2$:

in this case the statement of both Theorem 2.1 and Corollary 2.2 become

$\displaystyle\mbox{\bf d}(\psi_{0}(t),\psi_{\epsilon}(t))^{2}$
$\displaystyle\leq$ $\displaystyle C(t)\epsilon+D^{\prime}(t)\hbar^{\alpha}$
$\displaystyle\mbox{\bf d}(\psi_{0}(t),\psi_{\epsilon}(t))$
$\displaystyle\leq$ $\displaystyle
E^{\prime}(t)\epsilon^{\frac{\alpha}{\alpha+2}}$

for constants $C(t),D(t),E(t)$ easily computable from the proofs of Section 5.

Let us finish this section by some topological remarks, inspired by [GP18,
Section 4]. The distance d defines a weak topology, very different a priori of
the usual strong topologies associated to Hilbert spaces in quantum mechanics.
Nevertheless, it seems to us better adapted to the semiclassical approximation
for the following reason.

Let us consider two coherent states pinned up at two points
$z_{1}=(p_{1},q_{1}),z_{2}=(p_{2},q_{2})$ of the phase-space
$T^{*}\mathbf{R}^{d}$:
$\psi_{z_{j}}(x)=(\pi\hbar)^{-d/4}e^{-i\tfrac{p_{j}.x}{{\hbar}}}e^{-\tfrac{(x-q_{j})^{2}}{2\hbar}},\
j=1,2$.

An easy computation shows that

$\|\psi_{z_{1}}-\psi_{z_{2}}\|^{2}_{L^{2}(\mathbf{R}^{d})}=\||\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|-|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}||_{Hilbert-
Schmidt}^{2}=1-e^{-\frac{|z_{1}-z_{2}|^{2}}{2\hbar}}$

so that, as $\hbar\to 0$,

$\displaystyle\|\psi_{z_{1}}-\psi_{z_{2}}\|^{2}_{L^{2}(\mathbf{R}^{d})}=\||\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|-|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}||_{Hilbert-
Schmidt}^{2}$ $\displaystyle=$ $\displaystyle 0\ \ \mbox{ if }z_{1}=z_{2}$
$\displaystyle\to$ $\displaystyle 1\ \ \ \forall\ z_{1}\neq z_{2}.$

In other words, the Lebesgue or Schatten norms behave for small values of
$\hbar$ as the discrete topology, the one which only discriminates points.

On the contrary, d is much more sensitive to the localization on phase space
as shows our next result, proven in Section 7 below.

######  Proposition 2.4.

For any bounded convex domain $\Omega\subset\mathbf{R}^{2d}$, there exists
$C_{\Omega}>0$ such that, for any $z_{1},z_{2}\in\Omega$,

$C_{\Omega}|z_{1}-z_{2}|-\hbar\leq 2^{d}\mbox{\bf
d}(\psi_{z_{1}},\psi_{z_{2}})\leq
2^{d}\sqrt{|z_{1}-z_{2}|^{2}+2d\hbar}+C_{d}\hbar,$

where $C_{d}$ is defined in Lemma 5.3 Section 5 below.

##  3\. Applications to the classical limit

The estimates provided by the results of the two preceding sections do not
require ${\nabla}U$ to be continuous — in other words, the classical dynamics
underlying the quantum dynamics generated by $\mathcal{H}_{\epsilon},\
{\epsilon}>0$, fails to satisfy the assumptions of the Cauchy-Peano-Arzelà
Theorem.

Let us recall that one way to look at the transition from quantum to classical
dynamics as $\hbar\to 0$ is to associate to a quantum (pure or mixed) state,
namely a positive trace one operator $R^{\hbar}$ on $\mathfrak{H}$ (density
operator) with integral kernel $r^{\hbar}(x,x^{\prime})$, e.g. a pure state
$R^{\hbar}=|\psi^{\hbar}\rangle\langle\psi^{\hbar}|$, for any vector
$\psi^{\hbar}$ in $\mathfrak{H}$ the so-called Wigner transform defined on
phase-space by (with a slight abuse of notation again)

(5) $\displaystyle W_{\hbar}[R^{\hbar}](x,\xi)$ $\displaystyle:=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbf{R}^{d}}e^{-i\xi\cdot
y}r^{\hbar}(x+\tfrac{1}{2}{\hbar}y,x-\tfrac{1}{2}{\hbar}y)dy\,$ (6)
$\displaystyle W_{\hbar}[\psi^{\hbar}](x,\xi)$ $\displaystyle:=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbf{R}^{d}}e^{-i\xi\cdot
y}{\psi^{\hbar}(x+\tfrac{1}{2}{\hbar}y)}\overline{\psi_{\epsilon}(x-\tfrac{1}{2}{\hbar}y)}dy\,.$

An easy computation shows that $W_{\hbar}[\psi^{\hbar}]$ is linked to
$\psi^{\hbar}$ by the two following marginal properties

(7) $\displaystyle\int_{\mathbf{R}^{d}}W_{\hbar}[\psi^{\hbar}](x,\xi)d\xi$
$\displaystyle=$ $\displaystyle|\psi^{\hbar}(x)|^{2}$ (8)
$\displaystyle\int_{\mathbf{R}^{d}}W_{\hbar}[\psi^{\hbar}](x,\xi)dx$
$\displaystyle=$ $\displaystyle|\widehat{\psi}^{\hbar}(p)|^{2},\
\widehat{\psi}^{\hbar}(p):=\int_{\mathbf{R}^{d}}e^{-ip\cdot
x/\hbar}\psi^{\hbar}(x)\tfrac{dx}{(2\pi\hbar)^{d/2}}$

It has been proved, see e.g. [LP93], that, under the tightness conditions

(9)
$\displaystyle\lim_{R\to+\infty}\sup_{{\hbar}\in(0,1)}\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}r^{\hbar}(x,x)\;dx$ $\displaystyle=$ $\displaystyle 0,$ (10)
$\displaystyle\lim_{R\to+\infty}\sup_{{\hbar}\in(0,1)}\frac{1}{(2\pi{\hbar})^{d}}\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}\mathcal{F}r^{\hbar}\Bigl{(}\frac{p}{{\hbar}},\frac{p}{{\hbar}}\Bigr{)}\;dp$
$\displaystyle=$ $\displaystyle 0,$

where $B_{R}^{(d)}$ is the ball of radius $R$ in $\mathbf{R}^{d}$ and
$\mathcal{F}$ is the Fourier transform on $\mathbf{R}^{2d}$, the family of
Wigner functions $W_{\hbar}[R^{\hbar}]$ converges weakly, in particular in
$\mathcal{S}^{\prime}(\mathbf{R}^{2d})$, after extraction of a subsequence of
values of $\hbar$, to $W_{0}\in\mathcal{P}(\mathbf{R}^{2d})$, the space of
probability measures on $\mathbf{R}^{2d}$. The measure $W_{0}$ is called the
Wigner measure of the family $R_{\hbar}$.

As it is quite standard, we will omit to mention the extraction of
subsequences, together with the explicit dependence of states in the Planck
constant, and we will just write, when it does not create any confusion,

$\lim_{\hbar\to 0}W_{\hbar}[R]=W_{0}.$

Note that when $R=|\psi\rangle\langle\psi|$ is a pure state, (9)-(10) reads

(11)
$\displaystyle\lim_{R\to+\infty}\sup_{{\hbar}\in(0,1)}\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}|\psi(x)|^{2}\;dx$ $\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle\lim_{R\to+\infty}\sup_{{\hbar}\in(0,1)}\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}|\widehat{\psi}(p)|^{2}\;dp$ $\displaystyle=$ $\displaystyle 0$

Considering the quantum Hamiltonian $\mathcal{H}_{\epsilon}$, the expected
underlying classical dynamics is the one driven by the Liouville equation

(13) $\partial_{t}\rho=\\{\tfrac{1}{2}{(p^{2}+\lambda q^{2})}+\mu
V(q)+{\epsilon}U,\rho\\},\ \rho^{t}|_{t=0}=\rho^{in}$

where $\\{.,.\\}$ is the Poisson bracket on the symplectic manifold
$T^{*}\mathbf{R}^{d}\sim\mathbf{R}^{2d}$.

When ${\epsilon}=0$, the Hamiltonian vector field of Hamiltonian
$\tfrac{p^{2}+\lambda q^{2}}{2}+\mu V(q)$ is Lipschitz continuous. Moreover,
it was proven, [LP93], that $R_{\hbar}(t)$ is tight for any $t\in\mathbf{R}$
and $W_{0}^{t}:=\lim\limits_{\hbar\to 0}W_{\hbar}[R_{\hbar}(t)]=\rho^{t}$
solving (13) with $\rho^{in}=W_{0}$.

When ${\epsilon}>0$, the Liouville equation (13) exits the Cauchy-Peano-Arzelà
category: the associated Hamiltonian vector field might fail to have a
characteristic through every point of the phase-space. Nevertheless, as shown
in [AFFGP10, Theorem 6.1], (13) is still well posed in
$L^{\infty}_{+}([0,T];L^{1}(\mathbf{R}^{2d})\cap
L^{\infty}(\mathbf{R}^{2d}))$, and it was proven in [FLP13] (after [AFFGP10]),
that the Wigner function $W_{\hbar}[R_{\hbar}(t)]$ of the solution of the von
Neumann equation

(14)
$i{\hbar}{\partial}_{t}R_{\epsilon}(t)=[\mathcal{H}_{\epsilon},R_{\epsilon}(t)]\,,\qquad
R_{\epsilon}(0)=R^{in}_{\epsilon}$

tends weakly to the solution of (13), under certain conditions on
$R_{\epsilon}^{in}$.

Unfortunately, these conditions exclude definitively pure states, as, for
example, one of them impose that $\|R_{\epsilon}^{in}\|=O(\hbar^{d})$ and, to
our knowledge, nothing is known concerning the dynamics of the (possible)
limit of $W_{\hbar}[\psi_{\epsilon}(t)]$ as $\hbar\to 0$ where
$\psi_{\epsilon}(t)$ solves the Schrödinger equation (14).

Our next result will show that such a limit remains $\sqrt{\epsilon}$-close to
the push-forward of the Wigner measure of the initial condition by the flow of
the unperturbed classical Hamiltonian.

######  Theorem 3.1.

Let $R_{\epsilon}(t)$ be the solution of the von Neumann equation (14) with
$R_{\epsilon}^{in}$ satisfying

$\Delta(R_{\epsilon}^{in})=O(\sqrt{\hbar}).$

Let $R_{\epsilon}^{in}$ be tight, in the sense that it satisfies (9)-(10), so
that $W_{\hbar}[R_{\epsilon}^{in}]\to W_{0}^{in}$, its Wigner measure, as
$\hbar\to 0$.

Then, for any $t\in\mathbf{R}$, the family $R_{\epsilon}(t)$ is tight, so that

$W_{\hbar}[R_{\epsilon}(t)]\to W_{0}(t)\in\mathcal{P}(\mathbf{R}^{2d})\mbox{
as }\hbar\to 0.$

Moreover,

  1. (1)

$W_{0}(t)$ is $\sqrt{\epsilon}$-close to $\Phi^{t}\\#W_{0}^{in}$, where
$\Phi^{t}$ is the flow of Hamiltonian

$\tfrac{p^{2}+(1-\lambda)q^{2}}{2}+\mu V(q),$

in the sense that

$\sup_{\begin{subarray}{c}\int_{\mathbf{R}^{d}}\sup\limits_{q\in\mathbf{R}^{d}}|\mathcal{F}_{p}f(q,z)|dz\leq
1\\\ \max\limits_{\begin{subarray}{c}|{\alpha}|+|{\beta}|\\\ \leq
2[d/4]+3\end{subarray}}\|{\partial}^{\alpha}_{q}{\partial}^{\beta}_{p}f\|_{L^{\infty}(\mathbf{R}^{2d})}\leq
1\end{subarray}}|\int_{\mathbf{R}^{2d}}f(q,p)\left(W_{0}(t)-\Phi^{t}\\#W_{0}^{in}\right)(q,p)dqdp|\leq
2^{-d}\sqrt{C(t)}\sqrt{\epsilon},$

where $\mathcal{F}_{p}f(q,z):=\int_{\mathbf{R}^{d}}e^{-ip.z}f(q,p)dp$ and
$C(t)$ is as in Theorem 2.1 after replacing
$(\psi_{\epsilon}^{in},\mathcal{H}\psi_{\epsilon}^{in})$ by
$\int_{\mathbf{R}^{2d}}(\tfrac{p^{2}+q^{2}}{2})W_{0}(dpdq)$.

  2. (2)

In particular, $W_{0}(t)$ is weakly $\sqrt{\epsilon}$-close to
$\Phi^{t}\\#W_{0}^{in}$ in the sense of distribution as, for all test
functions $\varphi\in\mathcal{S}(\mathbf{R}^{2d})$,

$|\int_{\mathbf{R}^{2d}}\varphi(q,p)\left(W_{0}(t)-\Phi^{t}\\#W_{0}^{in}\right)(dp,dq)|\leq
C\varphi(t)\sqrt{\epsilon}$

with

$C_{\varphi}(t)=\max{\left(\int_{\mathbf{R}^{d}}\sup\limits_{q\in\mathbf{R}^{d}}|\mathcal{F}_{p}\varphi(q,z)|dz,\max\limits_{|{\alpha}|+|{\beta}|\leq
2[d/4]+3}\|{\partial}^{\alpha}_{q}{\partial}^{\beta}_{p}\varphi\|_{L^{\infty}(\mathbf{R}^{2d})}\right)}2^{-d}\sqrt{C(t)}.$

  3. (3)

Finally,

$\operatorname{dist_{MK,2}}(W_{0}(t),\Phi^{t}\\#W_{0}^{in})\leq\sqrt{2^{-2d-1}C(t)}\sqrt{\epsilon},$

where $\operatorname{dist_{MK,2}}$ is the Monge-Kantorovitch-Wasserstein
distance of order two, whose definition is recalled, for example, in [GP20b,
Section 1].

To our knowledge, no existence result is known for the Liouville equation
associated to a vector field whose components are in
$L^{\infty}(\mathbf{R}^{2d})$ and not a priori continuous and for general
measure initial data - the only known to us result being the existence and
uniqueness in $L^{1}(\mathbf{R}^{2d})\cap L^{\infty}(\mathbf{R}^{2d})$ result
recalled before, excluding concentration to trajectories of the ODE associated
to the vector field (note that well posedness for this ODE has been proved in
[CJ10] under the extra hypothesis for the vector field to be in
$H^{\frac{3}{4}}$).

Our result includes the case of concentrating initial data, by taking for
example a coherent state for $\psi^{in}$ which provides a Dirac mass as Wigner
measure:

if
$\psi(x)=(\pi\hbar)^{-d/4}e^{-i\tfrac{p.x}{{\hbar}}}e^{-\tfrac{(x-q)^{2}}{2\hbar}},\
(p,q)\in T^{*}\mathbf{R}^{d}$, one shows easily that.

$W_{\hbar}[|\psi\rangle\langle\psi|](x,\xi)=(\pi\hbar)^{-d}e^{-\frac{(x-q)^{2}+(\xi-p)^{2}}{{\hbar}}}\longrightarrow\delta_{(q,p)}\mbox{
as }\hbar\to 0.$

The meaning of the Theorem 3.1 can be summarized by the following diagram: one
can “regularizes” the Liouville equation associated to an $L^{\infty}$
perturbation of a Cauchy-Lipschitz vector field with a Wigner measure for
initial data by the associated Heisenberg-von Neumann equation, one remains
close to the unperturbed solution.

| “Schrödinger” |
---|---|---
$R_{\hbar}^{in}$ | $\stackrel{{\scriptstyle 0\to t}}{{\longrightarrow}}$ | $R_{\hbar}(t)$
${\hbar\to 0}$ $\downarrow$ |  |  $\mathbf{\downarrow}$ ${\hbar\to 0}$
$\begin{array}[]{rrl}W_{0}^{in}&:=&\lim\limits_{\hbar\to 0}W_{\hbar}[R_{\hbar}^{in}]\end{array}$ | $\stackrel{{\scriptstyle 0\to t}}{{\longrightarrow}}$ | $\begin{array}[]{rrl}W_{0}(t)&:=&\lim\limits_{\hbar\to 0}W_{\hbar}[R_{\hbar}(t)]\\\ &=&\Phi^{t}_{{\epsilon}=0}\\#W_{0}^{in}+O(\sqrt{\epsilon})\end{array}$
| “Liouville” |

Semiclassical regularization of rough Liouville equation ${\epsilon}$-close to
Cauchy-Lipschitz

###### Proof.

The propagation of tightness is proved as follows.

Let $\chi\in C^{\infty}(\mathbf{R}^{d})$, $0\leq\chi\leq 1$ such that
$\chi(x)=0$ if $|x|<1/2$ and $\chi(x)=1$ if $|x|>1$, and define
$\chi_{R}(x):=\chi(x/R)$. Obviously

$\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}|\psi_{\epsilon}(t)(x)|^{2}\;dx\leq\int_{\mathbf{R}^{d}}\chi_{R}(x)|\psi_{\epsilon}(t)(x)|^{2}\;dx$

Moreover, for some $C>0$,
$\|\nabla\chi_{R}\|_{\infty},\|\Delta\chi_{R}\|_{\infty}\leq C/R^{2}$, and

$-\frac{i}{{\hbar}}[\chi_{R},\mathcal{H}_{\epsilon}]=-\frac{i}{{\hbar}}[\chi_{R},-\tfrac{\hbar^{2}}{2}\Delta]=-i\hbar(\tfrac{1}{2}\Delta\chi_{R}-i\nabla\chi_{R}\cdot\nabla).$

Therefore

$\displaystyle\partial_{t}\int_{\mathbf{R}^{d}}\chi_{R}(x)|\psi_{\epsilon}(t)(x)|^{2}\;dx$
$\displaystyle=$
$\displaystyle-i\hbar\int_{\mathbf{R}^{d}}\bar{\psi}_{\epsilon}(t)(x)((\tfrac{1}{2}\Delta\chi_{R}-i\nabla\chi_{R}.\nabla)\psi_{\epsilon}(t))(x)dx$
$\displaystyle=$
$\displaystyle\int_{\mathbf{R}^{d}}(-i\hbar\tfrac{1}{2}\Delta\chi_{R}(x)|\psi_{\epsilon}(t)|^{2}$
$\displaystyle+\bar{\psi}_{\epsilon}(t)(x)\nabla\chi_{R}(x)\cdot(-i\hbar\nabla\psi_{\epsilon}(t)(x))dx,$

so that

$\displaystyle\partial_{t}\int_{\mathbf{R}^{d}}\chi_{R}(x)|\psi_{\epsilon}(t)(x)|^{2}\;dx$
$\displaystyle\leq$
$\displaystyle\hbar\frac{C}{2R^{2}}+\frac{C}{R}\|-i\hbar\nabla\psi_{\epsilon}(t)\|_{L^{2}(\mathbf{R}^{d})}$
$\displaystyle=$
$\displaystyle\hbar\frac{C}{2R^{2}}+2\frac{C}{R}(\psi_{\epsilon}(t),\mathcal{H}_{0}^{0,0}\psi_{\epsilon}(t))_{L^{2}(\mathbf{R}^{d})}$
$\displaystyle\leq$
$\displaystyle\hbar\frac{C}{2R^{2}}+2\frac{C}{R}\left((\psi_{\epsilon}(t),\mathcal{H}_{\epsilon}^{\lambda,\mu}\psi_{\epsilon}(t))_{L^{2}(\mathbf{R}^{d})}+\mu\|V\|_{\infty}+{\epsilon}\|U\|_{\infty}\right)$
$\displaystyle=$
$\displaystyle\hbar\frac{C}{2R^{2}}+2\frac{C}{R}\left((\psi_{\epsilon}^{in},\mathcal{H}_{\epsilon}^{\lambda,\mu}\psi_{\epsilon}^{in})_{L^{2}(\mathbf{R}^{d})}+\mu\|V\|_{\infty}+{\epsilon}\|U\|_{\infty}\right)$

and finally, for $t\in[0,T]$

$\displaystyle\int_{\mathbf{R}^{d}}\chi_{R}(x)|\psi_{\epsilon}(t)(x)|^{2}\;dx$
$\displaystyle\leq$
$\displaystyle\int_{\mathbf{R}^{d}}\chi_{R}(x)|\psi_{\epsilon}^{in}(x)|^{2}\;dx$
$\displaystyle+\left(\hbar\frac{C}{2R^{2}}+2\frac{C}{R}\left((\psi_{\epsilon}^{in},\mathcal{H}_{\epsilon}^{\lambda,\mu}\psi_{\epsilon}^{in})_{L^{2}(\mathbf{R}^{d})}+\mu\|V\|_{\infty}+{\epsilon}\|U\|_{\infty}\right)\right)T.$

Therefore $\psi_{\epsilon}(t)$ satisfies (12) as soon as
$\psi_{\epsilon}^{in}$ does.

Finally, let us remark that

(15) $\displaystyle\int_{\mathbf{R}^{d}\setminus
B_{R}^{(d)}}|\widehat{\psi}(p)|^{2}\;dp$ $\displaystyle\leq$
$\displaystyle\frac{1}{R^{2}}\int_{\mathbf{R}^{d}}p^{2}|\widehat{\psi}(p)|^{2}\;dp$
$\displaystyle=$
$\displaystyle\frac{2}{R^{2}}(\psi_{\epsilon}(t),\mathcal{H}_{0}^{0,0}\psi_{\epsilon}(t))_{L^{2}(\mathbf{R}^{d})}$

and one concludes the same way.

The rest of the Theorem is proved as follows.

  1. (1)

One knows from [LP93] that the convergence of Wigner functions to Wigner
measure as $\hbar\to 0$ takes place in the dual of the set of test functions
$f$ on $\mathbf{R}^{2d}$ satisfying

(16)
$\int_{\mathbf{R}^{d}}\sup\limits_{q\in\mathbf{R}^{d}}|\mathcal{F}_{p}f(q,z)|dz<\infty.$

Since $V\in C^{1,1}$ one knows that, for such a test function,

$\lim\limits_{\hbar\to
0}\int_{\mathbf{R}^{2d}}f(x,\xi)W_{\hbar}[\psi_{0}(t)](x,\xi)dxd\xi=\int_{\mathbf{R}^{2d}}f(x,\xi)\Phi^{t}\\#W_{0}(x,\xi)dxd\xi.$

On the other hand, we have the slight variant of Theorem 2.1, proven also in
Section 5.

######  Proposition 3.2.

Let $\delta$ be defined by (19) below. Then

$\delta(W_{\hbar}[\psi_{0}(t)],W_{\hbar}[\psi_{\epsilon}(t)])\leq
2^{-d}\sqrt{C(t)\epsilon+D(t)\hbar},$

where $C(t),D(t)$ are the constants defined in Theorem 2.1.

Proposition 3.2 tells us that, for any $f$ satisfying

$\max\limits_{|{\alpha}|+|{\beta}|\leq
2[d/4]+3}\|{\partial}^{\alpha}_{q}{\partial}^{\beta}_{p}f\|_{L^{\infty}(\mathbf{R}^{2d})}\leq
1,$
$|\int_{\mathbf{R}^{2d}}f(q,p)(W_{\hbar}[\psi_{\epsilon}(t)]-W_{\hbar}[\psi_{0}(t)])(dpdq)|\leq
2^{-d}\sqrt{C(t){\epsilon}+D(t)\hbar}$

Hence for any $f$ satisfying

$\int_{\mathbf{R}^{d}}\sup\limits_{q\in\mathbf{R}^{d}}|\mathcal{F}_{p}f(q,z)|dz\leq
1,\ \max\limits_{|{\alpha}|+|{\beta}|\leq
2[d/4]+3}\|{\partial}^{\alpha}_{q}{\partial}^{\beta}_{p}f\|_{L^{\infty}(\mathbf{R}^{2d})}\leq
1,$

we have

$\displaystyle|\int_{\mathbf{R}^{2d}}f(q,p)(W_{\hbar}[\psi_{\epsilon}(t)]-\Phi^{t}\\#W_{0}(x,\xi)])(dpdq)|$
$\displaystyle\leq
2^{-d}\sqrt{C(t){\epsilon}+D(t)\hbar}+|\int_{\mathbf{R}^{2d}}f(q,p)(W_{\hbar}[\psi_{0}(t)]-\Phi^{t}\\#W_{0}(x,\xi))(dpdq)|$

and we conclude by taking first the supremum on the functions $f$ and then the
limit $\hbar\to 0$ on both sides.

  2. (2)

The proof is obvious by homogeneity.

  3. (3)

Since functions in the Schwartz class satisfy (16), one knows that, as
$\hbar\to 0$, $W_{\hbar}[\psi_{\epsilon}(t)]\to W_{0}(t)$ and
$W_{\hbar}[\psi_{0}(t)]\to\Phi^{t}\\#W_{0}^{in}$ both in
$\mathcal{S}^{\prime}(\mathbf{R}^{2d})$. Therefore, by [GMP16, Theorem 2.3.
(2)], one knows that

$\operatorname{dist_{MK,2}}(W_{0}(t),\Phi^{t}\\#W_{0}^{in})\leq\varliminf_{\hbar\to
0}MK_{\hbar}(|\psi_{\epsilon}(t)\rangle\langle\psi_{\epsilon}(t)|,|\psi_{0}(t)\rangle\langle\psi_{0}(t)|)\,.$

By Theorem 5.2 with
$R_{0}^{in}=R_{\epsilon}^{in}=|\psi_{0}^{in}\rangle\langle\psi_{0}^{in}|=|\psi_{\epsilon}^{in}\rangle\langle\psi_{\epsilon}^{in}|$,
we have that

$MK_{\hbar}(|\psi_{\epsilon}(t)\rangle\langle\psi_{\epsilon}(t)|,|\psi_{0}(t)\rangle\langle\psi_{0}(t)|)\leq\sqrt{\gamma(t){\epsilon}},$

where $\gamma(t)$ is defined in (20).

We conclude by noticing that, as $\hbar\to 0$, $2^{2d+1}\gamma(t)\to C(t)$
defined in item $(1)$.

∎

##  4\. The case of mixed states

Consider the family of von Neumann equations, for ${\epsilon}\in[0,1]$,

(17)
$i{\hbar}{\partial}_{t}R_{\epsilon}(t)=[\mathcal{H}_{\epsilon},R_{\epsilon}(t)]\,,\qquad
R_{\epsilon}(0)=R^{in}_{\epsilon}\in\mathcal{D}(\mathfrak{H}).\,$

Let $R^{in}_{\epsilon}=R^{in}_{0}=R\in\mathcal{D}_{2}(\mathfrak{H})$ satisfy
one of the five following hypothesis:

  1. (i)

$\Delta(R^{in})=O(\sqrt{\hbar})$ where the standard deviation $\Delta(R)$ is
defined in Lemma 5.4 below;

  2. (ii)

$\sqrt{R^{in}}$ satisfies, for some $C>0$,

$\sup_{\begin{subarray}{c}|\beta_{1}|,\dots,|\beta_{d}|\leq
7\end{subarray}}|\prod_{m=1}^{d}D_{(x,\xi)}^{\beta_{m}}W_{\hbar}[\sqrt{R^{in}}](x,\xi)|\leq\frac{C(2\pi\hbar)^{-\frac{d}{2}}}{((\xi^{2}+x^{2})^{2}+d)^{\frac{10}{4}+3\epsilon}}\hfil\
\ \ \forall(x,\xi)\in\mathbf{R}^{2d},$

where $W_{\hbar}[\sqrt{R^{in}}]$ is the Wigner transform of $R^{in}$;

  3. (iii)

$\sqrt{R^{in}}$ satisfies, for $C>0$ and all $j\in\mathbf{N}^{d}$,

    1. (a)

$|(H_{i},\sqrt{R^{in}}H_{j})|\leq
C(2\pi\hbar)^{\frac{d}{2}}{\prod\limits_{1\leq l\leq d}|\hbar
j_{l}+\frac{1}{2}|^{-\frac{3}{4}-\epsilon}(|i_{l}-j_{l}|+1)^{-2-\epsilon}}$,

    2. (b)

$\sup\limits_{O\in\Omega_{1}}{|(H_{i},\tfrac{1}{i\hbar}[O,\sqrt{R^{in}}]H_{j})|}\leq
C{(2\pi\hbar)^{\frac{d}{2}}}{\prod\limits_{1\leq l\leq d}|\hbar
j_{l}+\frac{1}{2}|^{-\frac{1}{2}-\epsilon}(|i_{l}-j_{l}|+1)^{-1-\epsilon}}$,

where $\Omega_{1}=\\{y_{j},\pm\hbar\partial_{y_{j}}\ on\
L^{2}(\mathbf{R}^{d},dy),\ j=1,\dots,d\\}$ and the $H_{j}s$ are the
semiclassical Hermite functions;

  4. (iv)

$R^{in}$ is a Töplitz operator;

  5. (v)

there exist a Töplitz operator $T_{F}$ such that
$T_{F}^{-\frac{1}{2}}\sqrt{R^{in}}\ ,\sqrt{R^{in}}T_{F}^{-\frac{1}{2}}$ and
$T_{F}^{-\frac{1}{2}}\sqrt{R^{in}}\frac{1}{i\hbar}[O,\sqrt{R^{in}}T_{F}^{-\frac{1}{2}}],\
O\in\Omega_{1}$ are bounded on $L^{2}(\mathbf{R}^{d})$.

######  Theorem 4.1.

For every $t\geq 0$,

$\mbox{\bf d}(R_{0}(t),R_{\epsilon}(t))^{2}\leq C(t)\epsilon+D(t)\hbar,$

where $C(t),D(t)$ are given by (22)-(23), and are of the form

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\frac{e^{|t|(1-\lambda+\mu\operatorname{Lip}(\nabla
V))}-1}{1-\lambda+\mu\operatorname{Lip}(\nabla
V)}C_{\|\mathcal{H}R^{in}\|_{1},\|V\|_{\infty},\|U\|_{\infty},\|\nabla
U\|_{\infty}}<\infty,$ $\displaystyle D(t)$ $\displaystyle=$ $\displaystyle
e^{|t|(1-\lambda+\mu\operatorname{Lip}(\nabla V))}D_{d}<\infty.$

######  Corollary 4.2.

For every $t\geq 0$,

$\mbox{\bf d}(R_{0}(t),R_{\epsilon}(t))\leq E(t)\epsilon^{\frac{1}{3}}.$

In Theorem 4.1 and Corollary 4.2, the constants $C(t),D(t),E(t)$ are the same
as in Theorem 2.1 and Corollary 2.2 after replacing
$(\psi^{in},\mathcal{H}\psi^{in})$ by $\|\mathcal{H}R^{in}\|_{1}$ in $C(t)$.

##  5\. Proof of Theorems 2.1 and 4.1, and Proposition 3.2

For all $R,S\in\mathcal{D}(\mathfrak{H})$, we denote by $\mathcal{C}(R,S)$ the
set of couplings of $R$ and $S$, i.e.

$\mathcal{C}(R,S):=\\{Q\in\mathcal{D}(\mathfrak{H}\otimes\mathfrak{H})\text{
s.t. }\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}((A\otimes
I+I\otimes B)Q)=\operatorname{trace}_{\mathfrak{H}}(AR+BS)\\}\,.$

We recall the definition of the pseudo-distance $MK_{\hbar}$ (see Definition
2.2 in [GMP16]).

######  Definition 5.1.

For each $R,S\in\mathcal{D}_{2}(\mathfrak{H})$,

$MK_{\hbar}(R,S):=\inf_{Q\in\mathcal{C}(R,S)}\sqrt{\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q^{1/2}CQ^{1/2})}\,,$

where

$C:=\sum_{j=1}^{d}((q_{j}\otimes I-I\otimes
q_{j})^{2}-{\hbar}^{2}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}})^{2})\,.$

Theorem 2.1 and Proposition 3.2 are a consequence of the following inequality,
which controls the continuous dependence of the solution to the von Neumann
equation in terms of the initial data and on the potential.

######  Theorem 5.2.

Let $R_{\epsilon}^{in}\in\mathcal{D}_{2}(\mathfrak{H})$ and $R_{\epsilon}(t)$
be the solution of (17), ${\epsilon}\in[0,1]$. Then, for each
$t\in\mathbf{R},{\epsilon}\in[0,1]$, one has

$\displaystyle MK_{\hbar}(R_{0}(t),R_{\epsilon}(t))^{2}\leq
e^{|t|\Lambda({\nabla}V)}MK_{\hbar}(R_{0}^{in},R_{\epsilon}^{in})^{2}$
$\displaystyle+\epsilon\frac{e^{|t|\Lambda({\nabla}V)}-1}{\Lambda({\nabla}V)}\|{\nabla}U\|_{L^{\infty}}\sqrt{\operatorname{trace}((R_{0}^{in})^{1/2}{\mathcal{H}}(R_{0}^{in})^{1/2})+2\mu\|V\|_{L^{\infty}}}$
$\displaystyle+\epsilon\frac{e^{|t|\Lambda({\nabla}V)}-1}{\Lambda({\nabla}V)}\|{\nabla}U\|_{L^{\infty}}\sqrt{\operatorname{trace}((R_{\epsilon}^{in})^{1/2}{\mathcal{H}}(R_{\epsilon}^{in})^{1/2})+2(\mu\|V\|_{L^{\infty}}+\epsilon\|U\|_{L^{\infty}})}$
$\displaystyle\,,$

where

$\Lambda({\nabla}V)=1-\lambda+\mu\operatorname{Lip}({\nabla}V).$

Note that, as mentioned before, when $\lambda=1$,
$\Lambda({\nabla}V)=\mu\operatorname{Lip}({\nabla}V)$ and in the inequality
above, the function

$z\mapsto\frac{e^{z}-1}{z}$

is extended by continuity at $z=0$.

###### Proof.

In order to lighten the formulas we will use the following notations

(18) $V_{1}=\mu V,\ V_{2}=\mu V+{\epsilon}U,$

so that

$\mathcal{H}_{0}=\mathcal{H}^{\lambda,0}_{0}+V_{1}\mbox{ and
}\mathcal{H}_{\epsilon}=\mathcal{H}^{\lambda,0}_{0}+V_{2}.$

Let $Q^{in}\in\mathcal{C}(R_{0}^{in},R_{\epsilon}^{in})$, and let $Q$ be the
solution of the von Neumann equation

$i{\hbar}{\partial}_{t}Q=[({\mathcal{H}^{\lambda,0}_{0}}+V_{1})\otimes
I+I\otimes({\mathcal{H}^{\lambda,0}_{0}}+V_{2}),Q]\,,\quad
Q{\big{|}}_{t=0}=Q^{in}\,.$

Then

$Q(t)\in\mathcal{C}(R_{0}(t),R_{\epsilon}(t))\,,\quad\text{ for each }t\geq 0$

(see for instance Lemma 5.1 in [GMP16]).

Next we compute

$\displaystyle\frac{d}{dt}\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})$
$\displaystyle=\frac{i}{{\hbar}}\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}[({\mathcal{H}^{\lambda,0}_{0}}+V_{1})\otimes
I+I\otimes({\mathcal{H}^{\lambda,0}_{0}}+V_{2}),C]Q(t)^{1/2})$
$\displaystyle\,.$

One finds that

$\frac{i}{{\hbar}}[-\tfrac{1}{2}{\hbar}^{2}({\Delta}\otimes
I+I\otimes{\Delta}),C]=\sum_{j=1}^{d}(q_{j}\otimes I-I\otimes
q_{j})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))\,,$

while

$\frac{i}{{\hbar}}[(V_{1}\otimes I+I\otimes
V_{2}),C]=-\sum_{j=1}^{d}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{2})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))\,,$

with the notation

$A\vee B:=AB+BA\,.$

In particular

$\frac{i}{{\hbar}}[\tfrac{1}{2}(|q|^{2}\otimes
I+I\otimes|q|^{2}),C]=-\sum_{j=1}^{d}(q_{j}\otimes I-I\otimes
q_{j})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))\,.$

See [GMP16] on p. 190. Hence

$\displaystyle\frac{d}{dt}\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})$
$\displaystyle-(1-\lambda)\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}\left(Q(t)^{1/2}\sum_{j=1}^{d}(q_{j}\otimes
I-I\otimes q_{j})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))Q(t)^{1/2}\right)$
$\displaystyle=\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}\left(Q(t)^{1/2}\sum_{j=1}^{d}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{2})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))Q(t)^{1/2}\right)$
$\displaystyle=\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}\left(Q(t)^{1/2}\sum_{j=1}^{d}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{1})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))Q(t)^{1/2}\right)$
$\displaystyle+\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}\left(Q(t)^{1/2}\sum_{j=1}^{d}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))Q(t)^{1/2}\right)$
$\displaystyle=:\tau_{1}+\tau_{2}$ $\displaystyle\,.$

At this point, we recall the elementary operator inequality

$A^{*}B+B^{*}A\leq A^{*}A+B^{*}B\,.$

Therefore,

$\sum_{j=1}^{d}(q_{j}\otimes I-I\otimes
q_{j})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))\leq C$

and, for each $\ell>\operatorname{Lip}({\nabla}V_{1})^{1/2}$, one has

$\displaystyle\sum_{j=1}^{d}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{1})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))$
$\displaystyle=\sum_{j=1}^{d}\frac{1}{\ell}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{1})\vee\ell(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))$
$\displaystyle\leq\sum_{j=1}^{d}\left(\frac{\operatorname{Lip}({\nabla}V_{1})^{2}}{\ell^{2}}(q_{j}\otimes
I-I\otimes q_{j})^{2}+\ell^{2}(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))^{2}\right)$ $\displaystyle\,.$

Letting $\ell\to\operatorname{Lip}({\nabla}V)^{1/2}$ shows that

$\sum_{j=1}^{d}({\partial}_{q_{j}}V_{1}\otimes
I-I\otimes{\partial}_{q_{j}}V_{1})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))\leq\operatorname{Lip}({\nabla}V_{1})C\,,$

so that

$\tau_{1}\leq\operatorname{Lip}({\nabla}V_{1})\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})\,.$

For the term $\tau_{2}$, we simply use the Cauchy-Schwarz inequality:

$\displaystyle\tau_{2}=\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}\left(Q(t)^{1/2}\sum_{j=1}^{d}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2})\vee(-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}}))Q(t)^{1/2}\right)$
$\displaystyle\leq\sum_{j=1}^{d}\|Q(t)^{1/2}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2}))\|_{2}||-i{\hbar}({\partial}_{q_{j}}\otimes
I-I\otimes{\partial}_{q_{j}})Q(t)^{1/2}||_{2}$
$\displaystyle\leq\sum_{j=1}^{d}\|Q(t)^{1/2}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2}))\|_{2}||(-i{\hbar}{\partial}_{q_{j}}\otimes
I)Q(t)^{1/2}||_{2}$
$\displaystyle+\sum_{j=1}^{d}\|Q(t)^{1/2}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2}))\|_{2}+||(I\otimes(-i{\hbar}{\partial}_{q_{j}}))Q(t)^{1/2}||_{2}$
$\displaystyle=\tau_{21}+\tau_{22}$ $\displaystyle\,.$

Now

$\displaystyle\tau_{21}\leq\left(\sum_{j=1}^{d}\|Q(t)^{1/2}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2}))\|^{2}_{2}\right)^{1/2}\left(\sum_{j=1}^{d}\|(-i{\hbar}{\partial}_{q_{j}}\otimes
I)Q(t)^{1/2}\|^{2}_{2}\right)^{1/2}$
$\displaystyle=\left(\sum_{j=1}^{d}\operatorname{trace}\left(Q(t)^{1/2}(I\otimes{\partial}_{q_{j}}(V_{1}-V_{2}))^{2}Q(t)^{1/2}\right)\right)^{1/2}$
$\displaystyle\times\left(\operatorname{trace}\left(Q(t)^{1/2}(-{\hbar}^{2}{\Delta}\otimes
I)Q(t)^{1/2}\right)\right)^{1/2}$

so that

$\tau_{21}\leq\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\operatorname{trace}(R_{0}(t)^{1/2}{\mathcal{H}^{\lambda,0}_{0}}R_{0}(t)^{1/2})^{1/2}\,,$

and likewise

$\tau_{22}\leq\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\operatorname{trace}(R_{\epsilon}(t)^{1/2}{\mathcal{H}^{\lambda,0}_{0}}R_{\epsilon}(t)^{1/2})^{1/2}\,.$

Summarizing, we have proved that

$\displaystyle\frac{d}{dt}\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})\leq\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})$
$\displaystyle+\operatorname{Lip}({\nabla}V_{1})\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})$
$\displaystyle+\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}(\operatorname{trace}(R_{0}(t)^{1/2}{\mathcal{H}^{\lambda,0}_{0}}R_{0}(t)^{1/2})^{1/2}$
$\displaystyle+\operatorname{trace}(R_{\epsilon}(t)^{1/2}{\mathcal{H}^{\lambda,0}_{0}}R_{\epsilon}(t)^{1/2})^{1/2})$
$\displaystyle\,.$

On the other hand, since

$i{\hbar}{\partial}_{t}R_{j}(t)=[{\mathcal{H}^{\lambda,0}_{0}}+V_{j},R_{j}(t)]\,,$

one has

$\operatorname{trace}(R_{j}(t)^{1/2}({\mathcal{H}^{\lambda,0}_{0}}+V_{j})R_{j}(t)^{1/2})=\operatorname{trace}((R_{j}^{in})^{1/2}({\mathcal{H}^{\lambda,0}_{0}}+V_{j})(R_{j}^{in})^{1/2})$

so that

$\operatorname{trace}(R_{j}(t)^{1/2}{\mathcal{H}^{\lambda,0}_{0}}R_{j}(t)^{1/2})\leq\operatorname{trace}((R_{j}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{j}^{in})^{1/2})+2\|V_{j}\|_{L^{\infty}(\mathbf{R}^{d})}\,.$

Hence

$\displaystyle\frac{d}{dt}\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})\leq(1-\lambda+\operatorname{Lip}({\nabla}V_{1}))\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})$
$\displaystyle+\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{0}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{0}^{in})^{1/2})+2\|V_{1}\|_{L^{\infty}(\mathbf{R}^{d})}}$
$\displaystyle+\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{\epsilon}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{\epsilon}^{in})^{1/2})+2\|V_{2}\|_{L^{\infty}(\mathbf{R}^{d})}}$
$\displaystyle\,.$

By Gronwall’s inequality, choosing $Q^{in}$ to be an optimal coupling of
$R_{0}^{in}$ and $R_{\epsilon}^{in}$, one finds that, denoting
$\Lambda({\nabla}V_{1})=1-\lambda+\operatorname{Lip}({\nabla}V_{1})$,

$\displaystyle
MK_{\hbar}(R_{0}(t),R_{\epsilon}(t))^{2}\leq\operatorname{trace}_{\mathfrak{H}\otimes\mathfrak{H}}(Q(t)^{1/2}CQ(t)^{1/2})\leq
e^{t\Lambda({\nabla}V_{1})}MK_{\hbar}(R_{0}^{in},R_{\epsilon}^{in})^{2}$
$\displaystyle+\frac{e^{t\Lambda({\nabla}V_{1})}-1}{\Lambda({\nabla}V_{1})}\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{0}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{0}^{in})^{1/2})+2\|V_{1}\|_{L^{\infty}(\mathbf{R}^{d})}}$
$\displaystyle+\frac{e^{t\Lambda({\nabla}V_{1})}-1}{\Lambda({\nabla}V_{1})}\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{\epsilon}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}{\mathcal{H}^{\lambda,0}_{0}}(R_{\epsilon}^{in})^{1/2})+2\|V_{2}\|_{L^{\infty}(\mathbf{R}^{d})}}$
$\displaystyle\,,$

which is the desired inequality by coming back to $V,U$ through (18) and using

$\|\mu V+{\epsilon}U\|\leq\mu\|V\|+{\epsilon}\|U\|.$

∎

###### Proof of Theorem 2.1 and Proposition 3.2.

Observe that the estimate above is uniform in ${\hbar}$ — more precisely, the
moduli of continuity in the initial data and in the potential are independent
of ${\hbar}$. Of course, the pseudo-distance $MK_{\hbar}$ itself is not
independent of ${\hbar}$. For $R,S\in\mathcal{D}(\mathfrak{H})$ let us define

(19) $\displaystyle\delta(W_{\hbar}[R],W_{\hbar}[S]):=$
$\displaystyle\sup_{\begin{subarray}{c}\max\limits_{|{\alpha}|+|{\beta}|\leq
2[d/4]+3}\|{\partial}^{\alpha}_{q}{\partial}^{\beta}_{p}f\|_{L^{\infty}(\mathbf{R}^{2d})}\leq
1\end{subarray}}|\int_{\mathbf{R}^{2d}}f(q,p)(W_{\hbar}[R],W_{\hbar}[S])*q,p)dqdp|.$

######  Lemma 5.3.

For any $R,S\in\mathcal{D}_{2}(\mathfrak{H})$,

$\mbox{\bf d}(R,S)\leq 2^{d}\delta(W_{\hbar}[R],W_{\hbar}[S])\leq
2^{d}(MK_{\hbar}(R,S)+C_{d}\hbar)\ \mbox{ with }\
C_{d}=(1+\frac{\gamma_{d}}{\sqrt{\pi}})2d,$

where
$\gamma_{d}\leq\frac{d^{3/4}(192e^{-\frac{1}{4}}\pi^{-\frac{5}{4}})^{d}}{4e^{\frac{1}{4}}}(d^{d})^{11/4}$
is the constant appearing in the Caldaron-Vaillancourt theorem (see Appendix C
in [GP20a]).

###### Proof.

The proof consists in applying Theorem A.7 in [GJP20] and Theorem 2.3 (2) in
[GMP16].∎

Using Lemma 5.3 and Theorem 5.2 we get that

$\displaystyle(2^{-d}\mbox{\bf d}(R_{0}(t),R_{\epsilon}(t))-C_{d}\hbar)^{2}$
$\displaystyle\leq(\delta(W_{\hbar}[R_{0}(t)],W_{\hbar}[R_{\epsilon}(t)])-C_{d}\hbar)^{2}$
$\displaystyle\leq MK_{\hbar}(R_{0}(t),R_{\epsilon}(t))^{2}$
$\displaystyle\leq
e^{t\Lambda(\epsilon{\nabla}V_{1})}MK_{\hbar}(R_{0}^{in},R_{\epsilon}^{in})^{2}$
$\displaystyle+\epsilon\frac{e^{t\Lambda(\epsilon{\nabla}V_{1})}-1}{\Lambda({\nabla}V_{1})}\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{0}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{0}^{in})^{1/2})+2\epsilon\|V_{1}\|_{L^{\infty}(\mathbf{R}^{d})}}$
$\displaystyle+\epsilon\frac{e^{t\Lambda(\epsilon{\nabla}V_{1})}-1}{\Lambda({\nabla}V_{1})}\|{\nabla}(V_{1}-V_{2})\|_{L^{\infty}(\mathbf{R}^{d})}\sqrt{\operatorname{trace}((R_{\epsilon}^{in})^{1/2}{\mathcal{H}^{\lambda,0}_{0}}(R_{\epsilon}^{in})^{1/2})+2\epsilon\|V_{2}\|_{L^{\infty}(\mathbf{R}^{d})}}$
(20)
$\displaystyle:=e^{t\Lambda({\nabla}V_{1})}MK_{\hbar}(R_{0}^{in},R_{\epsilon}^{in})^{2}+\gamma(t)\epsilon.$

Therefore Theorem 2.1 and Proposition 3.2 are proven as soon as

(21) $MK_{\hbar}(R^{in},R^{in})^{2}\leq D^{\prime}\hbar$

since then

$\displaystyle\mbox{\bf d}(R_{0}(t),R_{\epsilon}(t))$ $\displaystyle\leq$
$\displaystyle 2^{d}\delta(W_{\hbar}[R_{0}(t)],W_{\hbar}[R_{\epsilon}(t)])$
$\displaystyle\leq$
$\displaystyle\sqrt{2^{2d}e^{|t|\mu\operatorname{Lip}{(\nabla
V)}}D^{\prime}\hbar+2^{2d}\gamma(t){\epsilon}}+2^{d}C_{d}\hbar$
$\displaystyle\leq$ $\displaystyle\sqrt{C(t){\epsilon}+D(t)\hbar}$

with, since $\hbar\leq 1$,

(22) $\displaystyle C(t)$ $\displaystyle=$ $\displaystyle 2^{2d+1}\gamma(t)$
(23) $\displaystyle D(t)$ $\displaystyle=$ $\displaystyle
e^{|t|\mu\operatorname{Lip}{(\nabla V)}}2^{2d+1}(D^{\prime}(t)+C_{d}^{2})$

######  Lemma 5.4.

For $R\in\mathcal{D}_{2}(\mathfrak{H})$ let

$\Delta(R):=\sqrt{\operatorname{trace}{\big{(}R^{\frac{1}{2}}(((x-\operatorname{trace}{(R^{\frac{1}{2}}xR^{\frac{1}{2}})})^{2}+(-i\nabla_{x}-\operatorname{trace}{(R^{\frac{1}{2}}(-i\nabla_{x})R^{\frac{1}{2}})})^{2})R^{\frac{1}{2}}\big{)}}}.$

Note that when $R$ is a pure state i.e. $R=|\psi\rangle\langle\psi|$,
$\Delta(R)$ is, modulo a slight abuse of notation, the same as in the
definition (1).

Then

$MK_{\hbar}(R,R)\leq\sqrt{2}\Delta(R).$

Moreover, for any $\psi\in\mathfrak{H}$,

$MK_{\hbar}(|\psi\rangle\langle\psi|,|\psi\rangle\langle\psi|)=\sqrt{2}\Delta(|\psi).$

###### Proof.

The proof consists in remarking that $R\otimes R$ is indeed a coupling between
$R$ and itself. Therefore

$\displaystyle MK_{\hbar}(R)^{2}$ $\displaystyle\leq$
$\displaystyle\operatorname{trace}{\big{(}(R^{\frac{1}{2}}\otimes
R^{\frac{1}{2}})C(R^{\frac{1}{2}}\otimes R^{\frac{1}{2}})\big{)}}$
$\displaystyle=$
$\displaystyle\operatorname{trace}{\big{(}(R^{\frac{1}{2}}\otimes
R^{\frac{1}{2}})(x^{2}\otimes I+I\otimes x^{2}-2x\otimes
x)(R^{\frac{1}{2}}\otimes R^{\frac{1}{2}})\big{)}}$ $\displaystyle+\mbox{ same
with }x\leftrightarrow-i\hbar\nabla_{x}$ $\displaystyle=$
$\displaystyle\operatorname{trace}{\big{(}R^{\frac{1}{2}}x^{2}R^{\frac{1}{2}}+R^{\frac{1}{2}}x^{2}R^{\frac{1}{2}}-2R^{\frac{1}{2}}xR^{\frac{1}{2}}.R^{\frac{1}{2}}xR^{\frac{1}{2}}\big{)}}$
$\displaystyle+\mbox{ same with }x\leftrightarrow-i\hbar\nabla_{x}$
$\displaystyle=$ $\displaystyle
2\operatorname{trace}{\big{(}R^{\frac{1}{2}}x^{2}R^{\frac{1}{2}}-(R^{\frac{1}{2}}xR^{\frac{1}{2}})^{2}\big{)}}$
$\displaystyle+\mbox{ same with }x\leftrightarrow-i\hbar\nabla_{x}$
$\displaystyle=$ $\displaystyle 2\Delta(R).$

The equality is proven the same way, after Lemma 2.1 (ii) in [GP20b] which
stipulates that the only coupling between $|\psi\rangle\langle\psi|$ and
itself is $|\psi\rangle\langle\psi|\otimes|\psi\rangle\langle\psi|$. ∎

This proves Theorem 2.1, and Theorem 4.1 when $R^{in}$ satisfies hypothesis
(i).

If the initial data $R^{in}$ is a Töplitz operator, specifically if

$R^{in}=\operatorname{OP}^{T}_{\hbar}[(2\pi{\hbar})^{d}\mu^{in}]\,,$

with the notation of [GMP16] one can go further and apply Theorem 2.3 (1) in
[GMP16]:

$MK_{\hbar}(R^{in},R^{in})\leq 2d\hbar,$

so that (21) is again satisfied and Theorem 4.1 is proven when $R^{in}$
satisfies the hypothesis (iv).

The proof in the case of the hypothesis (ii), (iii) and (v) follows directly
the first inequality of Theorem 8.1 in [GP20b] with $R=S=R^{in}$, together
with item $(I)$ (through the Corollary of Theorem 3.1 in [GP20a]) and item
$(II)$ of Theorem 4.1 in [GP20a], with $\mu(\hbar)=\mu^{\prime}(\hbar)=C,\
\nu(\hbar)=\nu^{\prime}(\hbar)=C\sqrt{\hbar}\mbox{ and }\tau(\hbar)=\hbar$,
respectively.

Indeed, [GP20b, Theorem 8.1, (iii) first inequality] stipulates that, for all
density matrix $R$,

$MK_{\hbar}(R^{in},R^{in})\leq
2E_{\hbar}(\widetilde{W}_{\hbar}[{R^{in}}],R^{in})$

where $E_{\hbar}$ is a semiquantum pseudometric whose knowledge of the
definition [GP17, Definition 2.2] is not strictly necessary for our purpose
here since Theorem 4.1 in [GP20a]) shows that, when
$\mu(\hbar)=\mu^{\prime}(\hbar)=C,\
\nu(\hbar)=\nu^{\prime}(\hbar)=C\sqrt{\hbar}\mbox{ and }\tau(\hbar)=\hbar$,

$E_{\hbar}(\widetilde{W}_{\hbar}[{R^{in}}],R^{in})=O(\sqrt{\hbar}).$

Hence

$MK_{\hbar}(R^{in},R^{in})^{2}\leq D^{\prime}\hbar$

for some constant $D^{\prime}$ explicitely recoverable from [GP20a, Theorem
4.1].

This completes the proof of Theorems 2.1 and 4.1.∎

##  6\. Proof of Corollaries 2.2 and 4.2

Let us first derive the easy standard following estimate.

######  Proposition 6.1.

For every $t\in\mathbf{R}$,

$\|R_{0}(t)-R_{\epsilon}(t)\|_{1}\leq 2t\frac{\epsilon}{\hbar}\|U\|_{\infty}$

###### Proof.

The solution of (17) is explicitly given by

$R_{\epsilon}(t)=e^{-i\frac{t\mathcal{H}_{\epsilon}}{{\hbar}}}R^{in}e^{i\frac{t\mathcal{H}_{\epsilon}}{{\hbar}}}$

Therefore, one easily shows that

$R_{0}(t)-R_{\epsilon}(t)=\frac{1}{i\hbar}\int_{0}^{t}e^{-i\frac{(t-s)\mathcal{H}_{\epsilon}}{{\hbar}}}[{\epsilon}U,R_{0}(t)]e^{i\frac{(t-s)\mathcal{H}_{\epsilon}}{{\hbar}}}.$

and the result follows from

$\|[{\epsilon}U,R_{0}(t)]\|_{1}\leq 2{\epsilon}\|U\|\|R_{0}(t)\|_{1}.$

∎

Corollaries 2.2, 4.2 will follow by interpolation between Theorem 5.2 and
Proposition 6.1, through the following inequality.

######  Lemma 6.2.

For any $R,S\in\mathcal{D}_{2}(\mathfrak{H})$,

$\mbox{\bf d}(R,S)\leq 2^{d}\|R-S\|_{1}$

###### Proof.

This is Theorem A7 in [GJP20], item $(2)$ ∎

Therefore, by Theorem 2.1 and Lemma 6.2 we get, for each
$\hbar,\epsilon\in[0,1]$,

$\mbox{\bf
d}(R_{1}(t),R_{2}(t))\leq\min{(\sqrt{C(t)\epsilon+D(t)\hbar},2|t|\frac{\epsilon}{{\hbar}}\|U\|_{\infty})}.$

Obviously,
$\min{(\sqrt{C(t)\epsilon+D(t)\hbar},2|t|\frac{\epsilon}{{\hbar}}\|U\|_{\infty})}\leq\min{(\sqrt{C(t)+D(t)},2|t|\|U\|_{\infty})}{\epsilon}^{\frac{1}{3}}$
for $\epsilon,\hbar\leq 1$, since, when $\hbar\leq\epsilon^{\frac{2}{3}}$,
$\hbar,{\epsilon}\leq{\epsilon}^{\frac{2}{3}}$, and, when
$\hbar\geq\epsilon^{\frac{2}{3}}$,
$\frac{\epsilon}{{\hbar}}\leq\epsilon^{\frac{1}{3}}$. The Corollary is proved.

##  7\. Proof of Proposition 2.4

The upper bound is given simply by Lemma 5.3 and the following inequality,
proved in [GP18, Section 4]

$MK_{\hbar}(|\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|,|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}|)^{2}\leq|z_{1}-z_{2}|^{2}+2d\hbar.$

For the lower bound, we will pick a test operator in the form of a Töplitz
operator with symbol $f\geq 0$:

$F:=\operatorname{OP}^{T}_{\hbar}[(2\pi{\hbar})^{d}f].$

One easily verifies that
$\operatorname{trace}{F}=\int_{\mathbf{R}^{2d}}f(q,p)dqdp$.

Moreover, see [GJP20, Appendix B],

$\displaystyle\tfrac{1}{i\hbar}[F,-i\hbar\partial_{x_{j}}]$ $\displaystyle=$
$\displaystyle\operatorname{OP}^{T}_{\hbar}[(2\pi{\hbar})^{d}\partial_{p_{j}}f],\
j=1,\dots,d,$ $\displaystyle\tfrac{1}{i\hbar}[F,x_{j}]$ $\displaystyle=$
$\displaystyle\operatorname{OP}^{T}_{\hbar}[(2\pi{\hbar})^{d}(-\partial_{q_{j}}f)],\
j=1,\dots,d,$ $\displaystyle\|F\|_{1}$ $\displaystyle\leq$
$\displaystyle\|f\|_{L^{1}(\mathbf{R}^{2d})}.$

Therefore, it is easy to construct functions $f$ such that
$F:=\operatorname{OP}^{T}_{\hbar}[(2\pi{\hbar})^{d}f]$ satisfies the
constraints of the maximization problem in the definition of d.

Moreover, denoting $z=(q,p)\in\mathbf{R}^{2d}$,

$\displaystyle\mbox{\bf
d}(|\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|,|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}|)$
$\displaystyle\geq$
$\displaystyle|\operatorname{trace}{\left(F(|\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|-|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}|)\right)}|$
$\displaystyle=$
$\displaystyle|\langle\psi_{z_{1}}|F|\psi_{z_{1}}\rangle-\langle\psi_{z_{2}}|F|\psi_{z_{2}}\rangle|$
$\displaystyle=$
$\displaystyle|\int_{\mathbf{R}^{2d}}f(z)(|\langle\psi_{z}|\psi_{z_{1}}\rangle|^{2}-|\langle\psi_{z}|\psi_{z_{2}}\rangle|^{2})\tfrac{dqdp}{(2\pi\hbar)^{d}}|$
$\displaystyle=$
$\displaystyle|\int_{\mathbf{R}^{2d}}f(z)(e^{-\frac{|z-z_{1}|^{2}}{2\hbar}}-e^{-\frac{|z-z_{2}|^{2}}{2\hbar}})\tfrac{dqdp}{(2\pi\hbar)^{d}}|$
$\displaystyle\geq$ $\displaystyle|f(z_{1})-f(z_{2})|$ $\displaystyle-$
$\displaystyle\tfrac{2}{e}{\hbar}\max_{\|{\alpha}|,|{\beta}|\leq
2}\int_{\mathbf{R}^{2d}}|\partial_{q}^{\alpha}\partial_{p}^{\beta}f(q,p)|dqdp.$

Let us suppose now that $f\in\mathcal{S}(\mathbf{R}^{2d})$ and $f$ is convex
in a convex domain containing $z_{1},z_{2}$. Then, one can certainly rescale,
translate and rotate $f$ such that

  * •

$f(z_{1})-f(z_{2})\geq\nabla f(z_{2}).(z_{1}-z_{2})$ convexity

  * •

$|f(z_{1})-f(z_{2})|\geq C|z_{1}-z_{2}|,\ C>0$ rotation and translation

  * •

$\max\limits_{\begin{subarray}{c}|{\alpha}|,|{\beta}|\leq
2[\frac{d}{4}]+3\end{subarray}}\|\mathcal{D}^{\alpha}_{-i\hbar\nabla}\mathcal{D}^{\beta}_{x}F\|_{1}\leq\max\limits_{\|{\alpha}|,|{\beta}|\leq
2}\|\partial_{q}^{\alpha}\partial_{p}^{\beta}f\|_{L^{1}(\mathbf{R}^{2d})}\leq
1$ rescaling.

Hence

$C|z_{1}-z_{2}|-\hbar\leq\mbox{\bf
d}(|\psi_{z_{1}}\rangle\langle\psi_{z_{1}}|,|\psi_{z_{2}}\rangle\langle\psi_{z_{2}}|).$

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