# Limiting absorption principle and absence of eigenvalues for massless
Klein–Gordon operators on perturbations of the Minkowski spacetime

Haruya Mizutani  Department of Mathematics, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan<EMAIL_ADDRESS>
###### Abstract.

We prove a uniform weighted resolvent estimate for the massless Klein–Gordon
operator on a curved spacetime which is sufficiently close to the Minkowski
spacetime. This particularly implies the existence and Hölder continuity of
the limiting resolvents at all energies, as well as the absolute continuity,
of the massless Klein–Gordon operator. The proof is based on a simple version
of Mourre’s commutator method and does not rely on microlocal analysis. We
also prove the absence of eigenvalues via the Virial theorem under an
ellipticity condition on the commutator of the massless Klein–Gordon operator
against the generator of a wick-rotated dilation, which is weaker than the
smallness condition for the metric perturbation.

##  1\. Introduction and main results

###  1.1. Introduction

Let $P$ be the variable coefficient massless Klein–Gordon operator (also
called the wave operator) on $L^{2}({\mathbb{R}}^{1+n},dx)$ of the form

$P=DGD^{T}=\sum_{j,k=0}^{n}D_{j}g^{jk}(x)D_{k},\quad
x=(t,y)\in{\mathbb{R}}^{1+n},$

where $y=(y_{1},...,y_{n})\in{\mathbb{R}}^{n}$, $D=(D_{0},D_{1},...,D_{n})$
with $D_{0}=-i\partial_{t}$, $D_{j}=-i\partial_{y_{j}}$ for $1\leq j\leq n$
and $D^{T}$ is the transpose of $D$. The coefficient matrix
$G(x)=(g^{jk}(x))_{j,k=0}^{n}\in M_{1+n}({\mathbb{R}})$ is assumed to be a
(smooth) Lorentzian dual metric on ${\mathbb{R}}^{1+n}$, that is, $G$ is
symmetric and has the signature $(-1,1,...,1)$. Throughout the paper, we
always assume

$\displaystyle|\partial_{x}^{\alpha}G(x)|\leq
C_{\alpha}{\langle}x{\rangle}^{-|\alpha|}$ ( 1.1)

uniformly in ${\mathbb{R}}^{1+n}$ for all $\alpha\in{\mathbb{Z}}_{+}^{1+n}$,
and

$P$ is essentially self-adjoint on the Schwartz class
$\mathcal{S}({\mathbb{R}}^{1+n})$, ( 1.2)

where ${\langle}x{\rangle}=\sqrt{1+|x|^{2}}$,
${\mathbb{Z}}_{+}={\mathbb{N}}\cup\\{0\\}$ ans $|G(x)|$ denotes a (fixed)
matrix norm of $G(x)$. We again denote by $P$ the unique self-adjoint
extension of $P|_{\mathcal{S}({\mathbb{R}}^{1+n})}$.

######  Remark 1.1 (Essential self-adjointness).

We record a sufficient condition to ensure that $P$ is essentially self-
adjoint. Let $G_{0}=\mathop{\mathrm{diag}}\nolimits(-1,1,...,1)$ be the exact
Minkowski dual metric. $G$ is said to be asymptotically Minkowski if
$G(x)-G_{0}$ belong to the symbol class $S^{-\mu}$ with some $\mu>0$, namely
for any $\alpha\in{\mathbb{Z}}_{+}^{1+n}$ there exists $C_{\alpha}>0$ such
that

$\displaystyle|\partial_{x}^{\alpha}(G(x)-G_{0})|\leq
C_{\alpha}{\langle}x{\rangle}^{-\mu-|\alpha|},\quad x\in{\mathbb{R}}^{1+n}.$ (
1.3)

Let $p(x,\xi)=g^{jk}(x)\xi_{j}\xi_{k}$ with $(x,\xi)\in
T^{*}{\mathbb{R}}^{1+n}$ be the principal symbol of $P$ and

$(z(s,x,\xi),\eta(s,x,\xi))=\exp sH_{p}(x,\xi)$

the Hamilton flow generated by $p$, namely the solution to

$\partial_{s}z=(\partial_{\xi}p)(z,\eta),\quad\partial_{s}\eta=-(\partial_{x}p)(z,\eta),\quad(z,\eta)|_{s=0}=(x,\xi).$

Then $G$ is said to be null non-trapping if for any $(x,\xi)\in
T^{*}{\mathbb{R}}^{1+n}$ satisfying $\xi\neq 0$ and $p(x,\xi)=0$,

$|z(s,x,\xi)|\to\infty\quad\text{as}\quad|s|\to\infty.$

It has been proved by Vasy [V] (see also Nakamura–Taira [NT1]) that $P$ is
essentially self-adjoint on $\mathcal{S}({\mathbb{R}}^{1+n})$ if $G$ is
asymptotically Minkowski and null non-trapping. We also refer to [NT2, NT3,
WZ] on the essential self-adjointness of $P$ for $G$ being not necessarily
asymptotically Minkowski.

In the present note, we study the limiting absorption principle (LAP) and some
spectral properties of $P$. The main result is the uniform weighted resolvent
estimate

$\sup_{z\in{\mathbb{C}}\setminus{\mathbb{R}}}\|{\langle}x{\rangle}^{-1}(P-z)^{-1}{\langle}x{\rangle}^{-1}\|<\infty$

for $G$ being a sufficiently small perturbation of $G_{0}$, where
$\|\cdot\|=\|\cdot\|_{\mathbb{B}(L^{2}({\mathbb{R}}^{1+n}))}$ denotes the
operator norm on $L^{2}({\mathbb{R}}^{1+n})$. One of consequences of this
uniform bound is the existence and Hölder continuity of the limiting
resolvents

${\langle}x{\rangle}^{-\rho}(P-\lambda\mp
i0)^{-1}{\langle}x{\rangle}^{-\rho}=\lim_{{\varepsilon}\searrow
0}{\langle}x{\rangle}^{-\rho}(P-\lambda\mp
i{\varepsilon})^{-1}{\langle}x{\rangle}^{-\rho},\quad\rho>1,$

with respect to $\lambda\in{\mathbb{R}}$ (including the zero energy case
$\lambda=0$, in other words, the massless case). In particular, $P$ is shown
to be purely absolutely continuous so that $P$ has no eigenvalues. As a second
result, we also prove that $P$ still has no eigenvalues even if $G$ is not a
small perturbation of $G_{0}$, as long as the commutator of $G$ against the
generator of a wick-rotated dilation satisfies a suitable uniform ellipticity
condition. For both results, $G$ does not have to be asymptotically Minkowski
as long as $P$ is essentially self-adjoint.

Before stating the precise statement, we shall discuss some known literatures,
as well as the motivation of this note. For the free massive Klein–Gordon
operator $P_{0}+m^{2}=\partial_{t}^{2}-\Delta_{y}+m^{2}$ with $m>0$, the
limiting resolvent $(P_{0}+m^{2}+i0)^{-1}$ (resp. $(P_{0}+m^{2}-i0)^{-1}$) is
called the Feynman propagator (resp. anti-Feynman propagator) and known to
play a fundamental role in Quantum Field Theory (QFT)111Precisely speaking,
the Schwartz kernel $(P_{0}+m^{2}+i0)^{-1}(x,y)$ is often called the Feynman
propagator in physics instead of the resolvent $(P_{0}+m^{2}+i0)^{-1}$ itself.
. In fact, it is possible to prove the existence of the limits
${\langle}x{\rangle}^{-1}(P_{0}\pm i0)^{-1}{\langle}x{\rangle}^{-1}$ even for
the free massless operator $P_{0}$ if $n\geq 2$ (see e.g. [KRS]). Note that
$(P_{0}-z)^{-1}$ can be computed explicitly via the Fourier transform, while
it is almost impossible in general to write down the explicit formula of
$(P-z)^{-1}$ in curved spacetimes. For the case of curved spacetimes,
Duistermaat–Hörmander [DH] have introduced the notion (and proved its unique
existence) of the Feynman parametrices, that is approximate inverses of $P$
(modulo smoothing operators) whose Schwartz kernels satisfy a suitable
structural condition characterized by the wave front set (see [GW1, Section 1]
for details). Since then, there are several results on mathematical proofs of
the existence and its properties of the Feynman propagators for various
spacetimes (see [DS1, DS2, DS3, GHV, GW1, GW2, IS, V, T1]). In particular, the
Feynman and anti-Feynman propagators in the sprit of [DH] on the null-
nontrapping asymptotically Minkowski spacetime with $\mu>1$ have been
constructed by Gérard–Wrochna [GW2] for the massive operator $P+m^{2}$ with
$m>0$. Precisely, they showed the existence of the actual inverses
$(P+m^{2})_{F}^{-1}$ and $(P+m^{2})_{\overline{F}}^{-1}$, called the Feynman
and anti-Feynman inverses, of $P+m^{2}$ with Schwartz kernels satisfying the
same structural condition as in [DH]. For the massless operator $P$, the
Feynman inverses $P_{F,\overline{F}}^{-1}$ have been constructed by Gell-
Redman–Haber–Vasy [GHV] for sufficiently small perturbations of the Minkowski
spacetime in the setting of Lorentzian scattering spaces (which are
generalizations of asymptotically Minkowski spacetimes). The existence of the
limiting resolvents $(P_{0}+m^{2}\pm i0)^{-1}$ with $m>0$ between appropriate
weighted Sobolev spaces have been established by Dereziński–Siemssen [DS1,
DS2, DS3] on the static spacetimes coupled by electromagnetic potentials, by
Vasy [V] on Lorentzian scattering spaces, and by [T1] on the asymptotically
Minkowski spacetime.

While the constructions of $(P+m^{2})_{F/\overline{F}}^{-1}$ and $(P+m^{2}\pm
i0)^{-1}$ are very different from each other, Taira [T1] showed that
$(P+m^{2})_{\overline{F}}^{-1}$ constructed by [GW2] coincides with
$(P+m^{2}-i0)^{-1}$ on ${\mathcal{S}}({\mathbb{R}}^{1+n})$. This result
motivates the following questions:

  * •

Existence of $(P\pm i0)^{-1}$ and the continuity $(P+m^{2}\pm i0)^{-1}\to(P\pm
i0)^{-1}$ as $m\to 0$ ?

  * •

Do $(P\pm i0)^{-1}$ coincide with the Feynman propagators constructed by [GHV]
?

It is worth noting that a combination of Taira’s result [T1] and these
questions would imply that the Feynman propagators by [GHV] are the massless
limit $m\to 0$ of the ones by [GW2].

Our main result answers the above first question for the setting of small
perturbations and could be a starting point of tackling the second one. At the
level of technical issues, one advantage of our argument is to use only
operator theoretical techniques, especially Mourre’s theory, and a few basic
properties of pseudo-differential operators. In particular, we do not rely on
advanced microlocal analysis, making the proofs much simpler than the
aforementioned papers.

###  1.2. Main results

Now we state the results. Let ${\mathbb{C}}^{\pm}=\\{z\in{\mathbb{C}}\
|\pm\mathop{\mathrm{Im}}\nolimits z>0\\}$. For any $f\in
L^{2}({\mathbb{R}}^{1+n})$ and bounded operator
$A\in\mathbb{B}(L^{2}({\mathbb{R}}^{1+n}))$, we use the notation

$\|f\|:=\|f\|_{L^{2}({\mathbb{R}}^{1+n})},\quad\|A\|:=\|A\|_{\mathbb{B}(L^{2}({\mathbb{R}}^{1+n}))}.$

######  Theorem 1.2.

Let $n\geq 2$ and $G$ satisfy 1.1 and 1.2. If the quantity

$\sup_{|\alpha|\leq
n+3}\|{\langle}x{\rangle}^{|\alpha|}\partial_{x}^{\alpha}(G-G_{0})\|_{L^{\infty}({\mathbb{R}}^{1+n})}$

is small enough, then

$\displaystyle\sup_{z\in{\mathbb{C}}^{\pm}}\|{\langle}x{\rangle}^{-1}(P-z)^{-1}{\langle}x{\rangle}^{-1}\|<\infty.$
( 1.4)

Moreover, for any $\rho>1$, the maps ${\mathbb{C}}^{\pm}\ni
z\mapsto{\langle}x{\rangle}^{-\rho}(P-z)^{-1}{\langle}x{\rangle}^{-\rho}\in\mathbb{B}(L^{2}({\mathbb{R}}^{1+n}))$
are Hölder continuous. In particular, the following limits exist and are
Hölder continuous for all $\lambda\in{\mathbb{R}}$:

${\langle}x{\rangle}^{-\rho}(P-\lambda\mp
i0)^{-1}{\langle}x{\rangle}^{-\rho}:=\lim_{{\varepsilon}\searrow
0}{\langle}x{\rangle}^{-\rho}(P-\lambda\mp
i{\varepsilon})^{-1}{\langle}x{\rangle}^{-\rho}\in\mathbb{B}(L^{2}({\mathbb{R}}^{1+n})).$

The following is an immediate consequence of this theorem:

######  Corollary 1.3.

Under the condition in Theorem 1.2, $P$ is purely absolutely continuous, that
is $P$ has neither eigenvalues nor singular continuous spectrum.

In fact, we can prove at least the absence of eigenvalues for a larger class
of metrics as follows. Define the matrix coefficient
$G_{11}(x)=(g_{1}^{jk}(x))\in M_{1+n}({\mathbb{R}})$ by

$\displaystyle
G_{11}=(g_{1}^{jk})=\begin{pmatrix}-2g^{00}+t\partial_{t}g^{00}-y\cdot\partial_{y}g^{00}&t\partial_{t}g^{0m}-y\cdot\partial_{y}g^{0m}\\\
t\partial_{t}g^{\ell 0}-y\cdot\partial_{y}g^{\ell 0}&2g^{\ell
m}+t\partial_{t}g^{\ell m}-y\cdot\partial_{y}g^{\ell
m}\end{pmatrix}_{\ell,m=1,...,n}.$ ( 1.5)

######  Theorem 1.4.

Suppose $n\geq 2$, $G$ satisfies 1.1 and 1.2. If $G_{11}$ is uniformly
elliptic, that is

$\displaystyle\xi
G_{11}(x)\xi^{T}=\sum_{j,k=0}^{n}g_{1}^{jk}(x)\xi_{j}\xi_{k}\geq
C_{0}|\xi|^{2}$ ( 1.6)

with some $C_{0}>0$, uniformly in $x,\xi\in{\mathbb{R}}^{1+n}$, then $P$ has
no eigenvalues.

######  Remark 1.5.

The LAP for $P$ away from the zero energy $\lambda=0$ was already established
by Taira [T1] for the asymptotically Minkowski spacetime with $\mu>0$.
Precisely, it was shown that

  * •

for each interval $I\Subset{\mathbb{R}}\setminus\\{0\\}$, $P$ has at most
finitely many eigenvalues in $I$;

  * •

for any $I^{\prime}\Subset I\setminus\sigma_{\mathrm{p}}(P)$, 1.4 holds with
${\mathbb{C}}^{\pm}$ and ${\langle}x{\rangle}^{-1}$ replaced by
$I^{\prime}_{\pm}=\\{z\in{\mathbb{C}}\ |\ \mathop{\mathrm{Re}}\nolimits z\in
I^{\prime},\ \pm\mathop{\mathrm{Im}}\nolimits z>0\\}$ and
${\langle}x{\rangle}^{-\rho}$ for any $\rho>1/2$. Moreover,
${\langle}x{\rangle}^{-\rho}(P-\lambda\mp i0)^{-1}{\langle}x{\rangle}^{-\rho}$
exist and are Hölder continuous for $\lambda\in I^{\prime}$.

Compared with this result, interesting points of our theorems are including
the zero energy case $\lambda=0$, as well as showing the absence of
eigenvalues. Moreover, in Theorems 1.2 and 1.4, $G$ does not have to be
asymptotically Minkowski as long as $P$ is essentially self-adjoint.

######  Remark 1.6.

It was also proved in [T1] that $P$ has no negative eigenvalues if $G$ is
asymptotically Minkowski with $\mu>1$, null non-trapping and there exists a
time-function $\tilde{t}$ with $\tilde{t}-t\in
C_{0}^{\infty}({\mathbb{R}}^{1+n})$. However, the absence of non-negative
eigenvalues of $P$ for general asymptotically Minkowski metrics is still open
even for the short-range case $\mu>1$.

######  Remark 1.7.

The weight ${\langle}x{\rangle}^{-1}$ in 1.4 is optimal in the sense that
one cannot replace it by ${\langle}x{\rangle}^{-\rho}$ with $\rho<1$ even for
the free case $P=P_{0}$ (or even if $P$ is replaced by the Laplacian
$\Delta_{{\mathbb{R}}^{1+n}}$). This is related with the scaling invariant
structure of $(P_{0}\pm i0)^{-1}$ and can be seen by applying the scaling
transformation $x\mapsto\delta x$ and taking the limit $\delta\to\infty$ in
1.4.

###  1.3. Outline of the proof

The proof of Theorem 1.2 follows an argument by [BoMi], which is based on
Mourre’s theory [Mo] and can be regarded as a simple version of the method of
weakly conjugate operators [BM]. An important step in Mourre’s theory is
choosing an appropriate conjugate operator $A$ to obtain a suitable positivity
of the commutator $[P,iA]$ and a relative boundedness of the double commutator
$[[P,iA],iA]$ with respect to $[P,iA]$. One of natural choices of conjugate
operators in the present setting could be the operator

$\displaystyle B=\frac{-tD_{t}-D_{t}t+y\cdot D_{y}+D_{y}\cdot y}{2}.$

Indeed, one has $[P,iB]=DG_{11}D^{T}$ on $C_{0}^{\infty}$, which is comparable
to $-\Delta_{{\mathbb{R}}^{1+n}}$ if $G-G_{0}$ is small enough. However, it is
known that $D(P)\notin H^{1}({\mathbb{R}}^{1+n})$ due to the lack of the
ellipticity of $P$ (see [T2]). In particular, $D(P)\not\subset D([P,iB])$ and
$[P,iB]$ does not make sense on $D(P)$ even for the form sense. Moreover,
$[[P,iB],iB]$ is not bounded from below, so how to choose its (canonical)
closed extension is already problematic. Indeed, it seems to be unclear
whether $[[P,iB],iB]$ is essentially self-adjoint even if $P$ is assumed to be
essentially self-adjoint. To overcome these difficulties, instead of $B$, we
will work with the operator

$A=QBQ,\quad
Q=(r-\Delta_{{\mathbb{R}}^{1+n}})^{-1/2}:L^{2}({\mathbb{R}}^{1+n})\to
H^{1}({\mathbb{R}}^{1+n}),$

where $r\geq 1$ is a constant. Then the commutators $[P,iA]$ and $[[P,iA],iA]$
are bounded on $L^{2}({\mathbb{R}}^{1+n})$. Moreover, if
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G-G_{0}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{1}$
is small enough then a strict Mourre inequality of the form

$[P,iA]\gtrsim Q(-\Delta_{{\mathbb{R}}^{1+n}})Q$

holds in the form sense on $L^{2}({\mathbb{R}}^{1+n})$. With these properties
of commutators at hand, we apply Mourre’s differential inequality technique to
the operator

$F_{\varepsilon}(z)=Q|D|(A+i\kappa)^{-1}(P-i{\varepsilon}[P,iA]-z)^{-1}(A-i\kappa)^{-1}|D|Q,\quad{\varepsilon}\mathop{\mathrm{Im}}\nolimits
z>0,\quad\kappa>r,$

obtaining the uniform bound 1.4. The remaining results of Theorem 1.2 is
obtained by estimating the difference
$K_{{\varepsilon},\rho}(z)-K_{{\varepsilon},\rho}(z^{\prime})$ with

$K_{{\varepsilon},\rho}(z)={\langle}x{\rangle}^{-\rho}(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{-1}{\langle}x{\rangle}^{-\rho}.$

The choice of $r\geq 1$ will not play any important role in the proof of
Theorem 1.2 and one can take $r=1$. For proving Theorem 1.4, we study the
behavior of $[P,iA]$ as $r\to\infty$ more carefully, showing that for any
$u\in D(P)$ there exists $r\geq 1$ such that ${\langle}[P,iA]u,u{\rangle}>0$
under the condition 1.6. This positivity, together with the Virial theorem,
yields the absence of eigenvalues of $P$.

###  1.4. Notation

Throughout the paper, the following notation will be used frequently:

  * •

For a matrix-valued function $M(x)$, we denotes
$\|M\|_{L^{p}}=\||M(\cdot)|\|_{L^{p}}$, where $|M(x)|$ denotes a (fixed)
matrix norm of $M(x)$.

  * •

For $k\in{\mathbb{Z}}_{+}$,
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{k}:=\sup_{k\leq|\alpha|\leq
n+3}\|{\langle}x{\rangle}^{|\alpha|}\partial_{x}^{\alpha}G\|_{L^{\infty}({\mathbb{R}}^{1+n})}.$

  * •

${\langle}u,v{\rangle}=\int_{{\mathbb{R}}^{1+n}}u(x)\overline{v(x)}dx$.

  * •

$H^{s}=H^{s}({\mathbb{R}}^{1+n})$ denotes the $L^{2}$-Sobolev space of order
$s$.

  * •

${\mathcal{S}}={\mathcal{S}}({\mathbb{R}}^{1+n})$ denotes the Schwartz class.

  * •

Let $\mathbb{B}(L^{2})$ be the space of bounded operators on $L^{2}$.

  * •

$S^{m,\ell}=S({\langle}x{\rangle}^{\ell}{\langle}\xi{\rangle}^{m},{\langle}x{\rangle}^{-2}dx^{2}+{\langle}\xi{\rangle}^{-2}d\xi^{2})$
denotes Hörmander’s symbol class, namely $a(x,\xi)$ belongs to $S^{m,\ell}$ if
$a\in C^{\infty}({\mathbb{R}}^{2(n+1)})$ and for any
$\alpha,\beta\in{\mathbb{Z}}_{+}^{1+n}$

$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq
C_{\alpha\beta}{\langle}x{\rangle}^{\ell-|\alpha|}{\langle}\xi{\rangle}^{m-|\beta|},\quad
x,\xi\in{\mathbb{R}}^{1+n}.$

Let $a(x,D)$ denote the pseudodifferential operator with symbol $a(x,\xi)$:

$a(x,D)f(x)=(2\pi)^{-n}\iint e^{i(x-y)\cdot\xi}a(x,\xi)f(y)dyd\xi.$

  * •

$\mathop{\mathrm{Op}}\nolimits S^{m,\ell}$ denotes the space of
pseudodifferential operators with symbol class $S^{m,\ell}$. If
$A_{j}\in\mathop{\mathrm{Op}}\nolimits S^{m_{j},\ell_{j}}$ for $j=1,2$ then
$A_{1}A_{2}\in\mathop{\mathrm{Op}}\nolimits
S^{m_{1}+m_{2},\ell_{1}+\ell_{2}}$. Moreover, the Calderón-Vaillancourt
theorem states that $\mathop{\mathrm{Op}}\nolimits
S^{0,0}\subset\mathbb{B}(L^{2})$ and

$\|a(x,D)\|\lesssim\sum_{|\alpha|,|\beta|\leq
n+3}\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a\|_{L^{\infty}({\mathbb{R}}^{1+n})}.$

We refer to Martinez [Ma] for more details on pseudodifferential operators.

##  2\. Preliminaries

In this section, we introduce our conjugate operator $A$ and prepare some
basic properties of $P,A$, especially the formula of the commutator $[P,iA]$.

For short we let $E$ be the identity matrix on ${\mathbb{R}}^{1+n}$ and
$\widetilde{M}:=MG_{0}$ for a given $k\times n$ matrix $M$. For example,
$\widetilde{D}=DG_{0}=(-D_{0},D_{1},...,D_{n})$. Let $B$ be the generator of
the dilation group

$\displaystyle e^{i\lambda
B}f(x)=e^{(n-1)\lambda/2}f(e^{-\lambda}t,e^{\lambda}y),\quad\lambda\in{\mathbb{R}}.$
( 2.1)

Note that

$\displaystyle
B=\frac{1}{2}(x\widetilde{D}^{T}+\widetilde{D}x^{T})=\frac{1}{2}(-tD_{t}-D_{t}t+y\cdot
D_{y}+D_{y}\cdot y)$ ( 2.2)

on ${\mathcal{S}}$. Then we define
$Q=(r-\Delta)^{-1/2}\in\mathop{\mathrm{Op}}\nolimits S^{-1,0}$ with $r>0$ and

$A=QBQ.$

Since $B$ is essentially self-adjoint on ${\mathcal{S}}$ by Nelson’s
commutator theorem and $Q:{\mathcal{S}}\to{\mathcal{S}}$ is symmetric and
bounded on $L^{2}$, $A$ is also essentially self-adjoint on ${\mathcal{S}}$.
We denote a unique self-adjoint extension of $A|_{{\mathcal{S}}}$ by the same
symbol $A$. Note that $A\in\mathop{\mathrm{Op}}\nolimits S^{-1,1}$.

Recall that $P$ is said to be of $C^{k}(A)$ if the map ${\mathbb{R}}\ni
t\mapsto e^{itA}(P-z)^{-1}e^{-itA}\in\mathbb{B}(L^{2}({\mathbb{R}}^{1+n})$ is
of $C^{k}$ for some (and hence all) $z\in{\mathbb{C}}\setminus{\mathbb{R}}$.
Define the multiple commutators $\mathrm{ad}_{A}^{k}P$ by

$\mathrm{ad}_{A}^{0}P:=P,\quad\mathrm{ad}_{A}^{k}P:=[\mathrm{ad}_{A}^{k-1}P,iA].$

Then $P\in C^{k}(A)$ is if and only if $\mathrm{ad}_{A}^{\ell}P$ defined
originally on $\mathcal{S}({\mathbb{R}}^{1+n})$ extend to bounded self-adjoint
operators from $D(P)^{*}$ to $D(P)$ for all $\ell\leq k$. In particular, we
know $P\in C^{2}(A)$ if $[P,iA]$ and $[[P,iA],iA]$ extend to bounded self-
adjoint operators on $L^{2}({\mathbb{R}}^{1+n})$.

######  Lemma 2.1.

The commutators $[P,iA]$ and $[[P,iA],iA]$, defined initially on
${\mathcal{S}}$, extend to bounded self-adjoint operators on
$L^{2}({\mathbb{R}}^{1+n})$ and hence $P\in C^{2}(A)$. Moreover, we have

$\displaystyle[P,iA]=QDG_{1}D^{T}Q,\quad[[P,iA],iA]=QDG_{2}D^{T}Q,$ ( 2.3)

where $G_{1}=G_{11}+G_{12}$ with $G_{11}$ given by 1.5,

$G_{12}=[Q^{-1},G]QiB+iBQ[Q^{-1},G]+[Q^{-1},G]QG_{0}+G_{0}Q[G,Q^{-1}],$

and each entries of $G_{12}$ and $G_{2}$ are bounded self-adjoint operators on
$L^{2}$.

###### Proof.

Let $f\in C^{1}({\mathbb{R}}^{1+n})$ be of polynomial growth. By 2.1, we
have

$\displaystyle e^{-i\lambda B}f(x)e^{i\lambda B}$
$\displaystyle=f(e^{\lambda}t,e^{-\lambda}y),\quad e^{-i\lambda
B}f(D)e^{i\lambda B}=f(e^{-\lambda}D_{t},e^{\lambda}D_{y}).$ ( 2.4)

These formulas, together with the fact that $Q$ commutes with $D$ and
$\widetilde{D}$, imply

$\displaystyle[f(D),iA]$
$\displaystyle=Q[f(D),iB]Q=Q\frac{d}{d\lambda}\left(e^{-i\lambda
B}f(D)e^{i\lambda B}\right)\Big{|}_{\lambda=0}Q=Qf_{1}(D)Q,$ ( 2.5)
$\displaystyle[G,iA]$ $\displaystyle=Q[G,iB]Q+[G,Q]iBQ-QiB[Q,G]$
$\displaystyle=Q\left(-x\widetilde{D}^{T}G+[Q^{-1},G]QiB+iBQ[Q^{-1},G]\right)Q$
( 2.6)

on ${\mathcal{S}}$, where
$f_{1}(\xi)=(\xi\widetilde{\partial}_{\xi}^{T}f)(\xi)=(-\xi_{0}\partial_{0}+\xi_{1}\partial_{1}+\cdots+\xi_{n}\partial_{n})f$.
In particular,

$\displaystyle[D,iA]=Q[D,iB]Q=Q\widetilde{D}Q.$ ( 2.7)

Moreover, $[G,iA]\in\mathop{\mathrm{Op}}\nolimits S^{0,0}$ by the symbolic
calculus and 1.1. Since $G$ is symmetric, we have

$\displaystyle
DG\widetilde{D}^{T}+\widetilde{D}GD^{T}=D(GG_{0}+G_{0}G)D^{T}=DHD^{T},$ ( 2.8)

where

$\displaystyle H=\begin{pmatrix}-2g^{00}&0\\\ 0&2g^{\ell
m}\end{pmatrix}_{\ell,m=1,...,n}.$ ( 2.9)

Since $G_{11}=H-x\widetilde{D}^{T}G$, we find by 2.5–2.8 that

$[P,iB]=DG[D,iB]^{T}+D[G,iB]D^{T}+[D,iB]GD^{T}=DG_{11}D^{T}$

and

$\displaystyle[P,iA]$ $\displaystyle=DG[D,iA]^{T}+D[G,iA]D^{T}+[D,iA]GD^{T}$
$\displaystyle=DGQ\widetilde{D}^{T}Q+Q\widetilde{D}QGD^{T}+D[G,iA]D^{T}$
$\displaystyle=Q(DG\widetilde{D}^{T}+\widetilde{D}GD^{T})Q+D[G,iA]D^{T}+D[G,Q]\widetilde{D}^{T}Q+Q\widetilde{D}[Q,G]D^{T}$
$\displaystyle=QD\left(G_{11}+Q^{-1}[G,Q]G_{0}+G_{0}[Q,G]Q^{-1}+Q^{-1}[G,Q]iB+iB[G,Q]Q^{-1}\right)D^{T}Q$
$\displaystyle=QDG_{1}D^{T}Q$

on ${\mathcal{S}}$. Note that $G_{1}\in\mathbb{B}(L^{2})$ since $G_{11}\in
L^{\infty}({\mathbb{R}}^{1+n})$ and

$\displaystyle Q^{-1}[G,Q]G_{0}\in\mathop{\mathrm{Op}}\nolimits
S^{1-2,-1}=\mathop{\mathrm{Op}}\nolimits
S^{-1,-1},\quad[Q^{-1},G]QB\in\mathop{\mathrm{Op}}\nolimits
S^{0-1+1,-1+1}=\mathop{\mathrm{Op}}\nolimits S^{0,0}.$

Moreover, since $D^{T}Q$ and $QD$ are also bounded on $L^{2}$, $[P,iA]$
extends to a bounded self-adjoint operator on $L^{2}$ and 2.3 holds on
$L^{2}$ by density argument. For $[[P,iA],iA]$, we similarly have

$\displaystyle[[P,iA],iA]=QD\left(G_{1}Q\widetilde{Q}G_{0}-G_{0}\widetilde{Q}QG_{1}+[G_{1},iA]\right)D^{T}Q,$

where each entries of
$G_{2}:=G_{1}Q\widetilde{Q}G_{0}-G_{0}\widetilde{Q}QG_{1}+[G_{1},iA]$ belong
to $\mathop{\mathrm{Op}}\nolimits S^{0,-1}\subset\mathbb{B}(L^{2})$ and are
symmetric on $L^{2}$. Hence $[[P,iA],iA]$ also extends to a bounded self-
adjoint operator on $L^{2}$. ∎

######  Remark 2.2.

It is known that, due to the lack of the ellipticity of $P$, $D(P)\not\subset
H^{1}({\mathbb{R}}^{1+n})$ even in the exact Minkowski spacetime (see [T2]).
Note that $H^{1}({\mathbb{R}}^{1+n})$ is the form domain of $DG_{11}D^{T}$.
Therefore, $[P,iB]$ does not make sense in general even in the form sense on
$H^{1}$. Moreover, it is easy to see that the double commutator $[[P,iB],iB]$
is not bounded from below on ${\mathcal{S}}$. Thus, how to define a canonical
closure of $[[P,iB],iB]$ is already problematic in general. This is why we use
$A$ as our conjugate operator instead of $B$.

######  Lemma 2.3.

${\|G_{12}u\|}\lesssim\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{1}{\|Q{\langle}D{\rangle}u\|}$
for $u\in L^{2}$ uniformly in $r\geq 1$.

###### Proof.

We prove
$\|[Q^{-1},G]QiBu\|\lesssim\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{1}{\|Q{\langle}D{\rangle}u\|}$
only, the proof for the other parts of $G_{12}$ being analogous. Using the
formula $iB=x\widetilde{D}^{T}+(n-1)/2$, we rewrite $[Q^{-1},G]QiB$ as

$\displaystyle[Q^{-1},G]QiB=[Q^{-1},G]\left(xQ\widetilde{D}^{T}+Q[x,Q^{-1}]Q\widetilde{D}^{T}+\frac{n-1}{2}Q\right).$

Since $|\partial_{\xi}^{\beta}(r+|\xi|^{2})^{1/2}|\leq C_{\beta}$ for
$|\beta|\geq 1$ uniformly in $r\geq 1$, we have
$[Q^{-1},G]{\langle}x{\rangle},[x,Q^{-1}]\in\mathbb{B}(L^{2})$ with uniform
operator norm bounds in $r\geq 1$. Moreover, the Calderón-Vaillancourt theorem
shows
${\|[Q^{-1},G]{\langle}x{\rangle}\|}\lesssim\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{1}$.
Thus the result follows. ∎

In what follows, we set for short $S(\xi)=(r+|\xi|^{2})^{-1/2}|\xi|$, and

$\displaystyle M_{1}:=[P,iA],\quad M_{2}:=[[P,iA],iA],\quad S:=S(D)=Q|D|.$ (
2.10)

######  Lemma 2.4.

Let $n\geq 2$. Then ${\|{\langle}x{\rangle}^{-1}u\|}\lesssim{\|Su\|}$ for
$u\in L^{2}$.

###### Proof.

It follows from Hardy’s inequality ${\||x|^{-1}u\|}\lesssim{\||D|u\|}$ for
$1+n\geq 3$ that

$\displaystyle{\|{\langle}x{\rangle}^{-1}u\|}$
$\displaystyle\leq{\|{\langle}x{\rangle}^{-1}\mathds{1}_{|\xi|\leq
1}(D)u\|}+{\|{\langle}x{\rangle}^{-1}\mathds{1}_{|\xi|\geq 1}(D)u\|}$
$\displaystyle\lesssim{\||D|\mathds{1}_{|\xi|\leq
1}(D)u\|}+{\|\mathds{1}_{|\xi|\geq 1}(D)u\|}\lesssim{\|Su\|},$

where we have used the bounds $S(\xi)\lesssim|\xi|$ if $|\xi|\leq 1$ and
$S(\xi)\lesssim 1$ if $|\xi|\geq 1$ in the last line. ∎

######  Lemma 2.5.

Let $\kappa>r$. Then ${\|S(A+i\kappa)^{-1}u\|}\lesssim{\|Su\|}$ for $u\in
L^{2}$.

###### Proof.

We may assume $u\in D(A)$ without loss of generality since $D(A)$ is dense in
$L^{2}$ and $S(A+i\kappa)^{-1}$ and $S$ are bounded on $L^{2}$. We rewrite
$S(A+i\kappa)^{-1}$ as

$\displaystyle
S(A+i\kappa)^{-1}u=\frac{1}{i}\int_{0}^{\infty}e^{-\kappa\lambda}Se^{i\lambda
A}ud\lambda.$

To prove the lemma, it is enough to show $\|Se^{i\lambda A}u\|\leq
e^{r\lambda}\|Su\|$. To this end, we observe

$[S^{2},iA]=Q[S^{2},iB]Q=2rQ[Q^{2},iB]Q=2rQ^{2}DG_{0}D^{T}Q^{2}$

on ${\mathcal{S}}$. Thus $[S^{2},iA]$ extends a bounded operator on $L^{2}$
and $|{\langle}[S^{2},iA]v,v{\rangle}|\leq 2r{\|Sv\|}^{2}$ for any $v\in
D(A)$. In particular, $S^{2}\in C^{1}(A)$. Then we obtain, for $\lambda\geq
0$,

$\displaystyle{\|Se^{i\lambda A}u\|}^{2}$
$\displaystyle={\langle}S^{2}u,u{\rangle}+\int_{0}^{\lambda}\frac{d}{ds}{\langle}S^{2}e^{isA}u,e^{isA}u{\rangle}ds$
$\displaystyle={\|Su\|}^{2}+\int_{0}^{\lambda}{\langle}[S^{2},iA]e^{isA}u,e^{isA}u{\rangle}ds\leq{\|Su\|}^{2}+2r\int_{0}^{\lambda}{\|Se^{isA}u\|}^{2}ds.$

Gronwall’s inequality then yields ${\|Se^{i\lambda A}u\|}^{2}\leq
e^{2r\lambda}{\|Su\|}^{2}$. ∎

##  3\. Proof of Theorem 1.2

The proof of Theorem 1.2 is based on a simple version (see [BM, BoMi]) of
Mourre’s theory [Mo]. At first, we observe that since

$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}H-2E\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{0}\leq
2\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G-G_{0}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{0},\quad\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{1}\leq
2\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G-G_{0}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{0},$

where $H$ is given by 2.9 and the norm
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\cdot\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{k}$
is given in Subsection 1.4, Lemmas 2.1 and 2.3 imply the following strict
Mourre inequality

$\displaystyle M_{1}=QDG_{1}D^{T}Q\gtrsim-Q\Delta Q=S^{2}\geq 0$ ( 3.1)

if
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G-G_{0}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{0}$
small enough. In this case, $M_{1}$ and $S^{2}$ is equivalent in the form
sense:

$C^{-1}\|Su\|^{2}\leq{\langle}M_{1}u,u{\rangle}\leq C\|Su\|^{2}$

with some $C>0$ independent of $u\in L^{2}$.

The following theorem is a crucial step in the proof of Theorem 1.2.

######  Theorem 3.1.

Let $n\geq 1$, $\kappa>r$ and
$\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}G-G_{0}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{0}$
be small enough. Then

$\displaystyle\sup_{z\in{\mathbb{C}}\setminus{\mathbb{R}}}{\|S(A+i\kappa)^{-1}(P-z)^{-1}(A-i\kappa)^{-1}S\|}<\infty.$
( 3.2)

###### Proof.

Let
$({\varepsilon},z)\in\overline{{\mathbb{R}}^{\pm}}\times{\mathbb{C}}^{\pm}$
and set

$\displaystyle R_{\varepsilon}(z):=(P-i{\varepsilon}M_{1}-z)^{-1},\quad
W=S(A+i\kappa)^{-1},\quad F_{{\varepsilon}}(z):=WR_{\varepsilon}(z)W^{*}.$

The proof is decomposed into several steps.

We first check that $F_{{\varepsilon}}(z)$ is well-defined. The Mourre
inequality 3.1 implies

$\displaystyle\mp\mathop{\mathrm{Im}}\nolimits(P-z-i{\varepsilon}M_{1})=\pm\mathop{\mathrm{Im}}\nolimits(P-z-{\varepsilon}M_{1})^{*}=\pm\mathop{\mathrm{Im}}\nolimits
z\pm{\varepsilon}M_{1}\geq\pm\mathop{\mathrm{Im}}\nolimits z,$ ( 3.3)

so ${\|(P-z-i{\varepsilon}M_{1})u\|}\geq|\mathop{\mathrm{Im}}\nolimits
z|{\|u\|}$ and
${\|(P-z-i{\varepsilon}M_{1})^{*}u\|}\geq|\mathop{\mathrm{Im}}\nolimits
z|{\|u\|}$ for $u\in D(P)$. Since $D(P)$ is dense in $L^{2}$, by the closed
graph theorem, $R_{\varepsilon}(z):L^{2}\to D(P)$ is well-defined and
satisfies ${\|R_{\varepsilon}(z)\|}\leq|\mathop{\mathrm{Im}}\nolimits
z|^{-1}$. In particular,
${\|F_{{\varepsilon}}(z)\|}\lesssim|\mathop{\mathrm{Im}}\nolimits z|^{-1}$
since $W\in\mathbb{B}(L^{2})$.

Next we derive the following formula for ${\varepsilon},z$ satisfying
${\varepsilon}\mathop{\mathrm{Im}}\nolimits z>0$:

$\displaystyle\partial_{\varepsilon}F_{{\varepsilon}}(z)=2\kappa
iF_{{\varepsilon}}(z)-SR_{\varepsilon}(z)W^{*}+WR_{\varepsilon}(z)S-{\varepsilon}WR_{\varepsilon}(z)M_{2}R_{\varepsilon}(z)W^{*}.$
( 3.4)

A direct calculation implies
$R_{\varepsilon}(z)-R_{{\varepsilon}^{\prime}}(z)=i({\varepsilon}-{\varepsilon}^{\prime})R_{\varepsilon}(z)M_{1}R_{{\varepsilon}^{\prime}}(z)$
and

${\|R_{\varepsilon}(z)-R_{{\varepsilon}^{\prime}}(z)\|}\lesssim|{\varepsilon}-{\varepsilon}^{\prime}||\mathop{\mathrm{Im}}\nolimits
z|^{-2}.$

Hence, for each $z\in{\mathbb{C}}^{\pm}$, the map
${\mathbb{R}}^{\pm}\ni{\varepsilon}\mapsto R_{\varepsilon}(z)\in L^{2}$ is
continuously differentiable, respectively, and
$\partial_{\varepsilon}R_{\varepsilon}(z)=iR_{\varepsilon}(z)M_{1}R_{{\varepsilon}}(z)$.
In what follows, we simply write $R=R_{\varepsilon}(z)$,
$F=F_{{\varepsilon}}(z)$ and $F^{\prime}=\partial_{\varepsilon}F$ for short.
We compute

$\displaystyle F^{\prime}=WR^{\prime}W^{*}=iWRM_{1}RW^{*}.$ ( 3.5)

Here we observe $R_{\varepsilon}(z)W^{*}:{\mathcal{S}}\to D(A)\cap D(P)$.
Indeed, $W^{*}:{\mathcal{S}}\to D(A)$ and $R_{\varepsilon}(z):D(A)\to D(A)\cap
D(P)$ since $P-i{\varepsilon}M_{1}-z\in C^{1}(A)$. With this property at hand,
we compute

$\displaystyle\frac{d}{d\lambda}\left(We^{-i\lambda A}Re^{i\lambda
A}W^{*}\right)\Big{|}_{\lambda=0}$ $\displaystyle=W[R,iA]W^{*}$
$\displaystyle=WR(-M_{1}+i{\varepsilon}M_{2})RW^{*}$
$\displaystyle=iF^{\prime}+i{\varepsilon}WRM_{2}RW^{*}$ ( 3.6)

on ${\mathcal{S}}$. Moreover, we have

$\displaystyle\frac{d}{d\lambda}e^{i\lambda A}W^{*}\Big{|}_{\lambda=0}$
$\displaystyle=iA(A-i\kappa)^{-1}S=iS-\kappa W^{*}$

which, together with its adjoint formula, implies

$\displaystyle\frac{d}{d\lambda}\left(We^{-i\lambda A}Re^{i\lambda
A}W^{*}\right)\Big{|}_{\lambda=0}$ $\displaystyle=(-iS-\kappa
W)RW^{*}+WR(iS-\kappa W^{*})$ $\displaystyle=-2\kappa F-iSRW^{*}+iWRS$ ( 3.7)

3.4 now follows from 3.5–3.7 and density argument.

We now prove 3.2 for $({\varepsilon},z)\in(0,1]\times{\mathbb{C}}^{+}$.
Lemma 2.5 and 3.1 imply

$\displaystyle{\|Fu\|}^{2}$
$\displaystyle\lesssim{\|SRW^{*}u\|}^{2}={\langle}S^{2}RW^{*}u,RW^{*}u{\rangle}\lesssim{\langle}M_{1}RW^{*}u,RW^{*}u{\rangle}$
$\displaystyle\leq{\varepsilon}^{-1}\left({\varepsilon}{\langle}M_{1}RW^{*}u,RW^{*}u{\rangle}+\mathop{\mathrm{Im}}\nolimits
z{\langle}RW^{*}u,RW^{*}u{\rangle}\right)$
$\displaystyle=-{\varepsilon}^{-1}\mathop{\mathrm{Im}}\nolimits{\langle}(H-i{\varepsilon}M_{1}-z)RW^{*}u,RW^{*}u{\rangle}$
$\displaystyle=-{\varepsilon}^{-1}\mathop{\mathrm{Im}}\nolimits{\langle}u,Fu{\rangle}\leq{\varepsilon}^{-1}{\|Fu\|}{\|u\|}$

for $u\in L^{2}$ and $\rho\geq 1$. Thus we have for
$({\varepsilon},z)\in{\mathbb{R}}^{+}\times{\mathbb{C}}^{+}$,

$\displaystyle{\|F\|}\lesssim|{\varepsilon}|^{-1},\quad{\|SRW^{*}\|}\lesssim|{\varepsilon}|^{-1/2}{\|F\|}^{1/2}.$
( 3.8)

Taking the complex conjugate implies
${\|S\overline{R}W^{*}\|}\lesssim|{\varepsilon}|^{-1/2}{\|F\|}^{1/2}$, where
$\overline{R_{\varepsilon}(z)}:=R_{-{\varepsilon}}(\overline{z})$. Since
$\overline{R_{\varepsilon}(z)^{*}\overline{u}}=R_{\varepsilon}(z)u$, where
$\overline{u}$ denotes the complex conjugate of $u$, we also obtain

$\displaystyle{\|WRS\|}={\|SR^{*}W^{*}\|}={\|S\overline{R}W^{*}\|}\lesssim|{\varepsilon}|^{-1/2}{\|F\|}^{1/2}.$
( 3.9)

Moreover, we know by Lemma 2.1, 3.1, 3.8 and 3.9 that

$\displaystyle|{\langle}WRM_{2}RW^{*}u,v{\rangle}|\lesssim{\|SRW^{*}u\|}{\|SR^{*}W^{*}v\|}|\lesssim{\varepsilon}^{-1}{\|F\|}{\|u\|}{\|v\|}.$

Taking the supremum over $u,v\in L^{2}$ then implies

$\displaystyle{\varepsilon}{\|WRM_{2}RW^{*}\|}\lesssim{\|F\|}.$ ( 3.10)

Putting 3.4, 3.8–3.10 all together, we obtain

$\displaystyle{\|F_{{\varepsilon}}^{\prime}(z)\|}\lesssim{\|F_{{\varepsilon}}(z)\|}+{\varepsilon}^{-1/2}{\|F_{{\varepsilon}}(z)\|}^{1/2}$
( 3.11)

with an implicit constant independent of
$({\varepsilon},z)\in{\mathbb{R}}^{+}\times{\mathbb{C}}^{+}$. Using Gronwall’s
inequality (see [Mo]), we conclude

$\displaystyle\sup_{{\varepsilon}\in(0,1],z\in{\mathbb{C}}^{+}}{\|F_{{\varepsilon}}(z)\|}<\infty.$
( 3.12)

Letting ${\varepsilon}\searrow 0$, we obtain the desired uniform bound 3.2
with $z\in{\mathbb{C}}\setminus{\mathbb{R}}$ replaced by
$z\in{\mathbb{C}}^{+}$. Taking the complex conjugate, we also have 3.2 for
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$. ∎

We are now ready to complete the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

Let $n\geq 2$ and $\kappa>r$. We first show 1.4. To this end, we shall prove

$\displaystyle{\|{\langle}x{\rangle}^{-1}(A+i\kappa)u\|}\lesssim{\|Su\|},\quad
u\in D(A).$ ( 3.13)

Note that 3.13 implies ${\langle}x{\rangle}^{-1}(A+i\kappa)S^{-1}$ and its
adjoint extend to bounded operators on $L^{2}$. Since
$Q{\langle}x{\rangle}^{-1}x,[{\langle}x{\rangle}^{-\rho},Q]x\in\mathbb{B}(L^{2})$
by the symbolic calculus, if we write

$\displaystyle{\langle}x{\rangle}^{-1}iA$
$\displaystyle={\langle}x{\rangle}^{-1}Q\left(xD^{T}+\frac{n-1}{2}\right)Q$
$\displaystyle=Q{\langle}x{\rangle}^{-1}xD^{T}Q+[{\langle}x{\rangle}^{-1},Q]xD^{T}Q+\frac{n-1}{2}{\langle}x{\rangle}^{-1}Q^{2},$
( 3.14)

then 3.13 follows from Lemma 2.4 and the fact
$Q{\langle}x{\rangle}^{-1}x,[{\langle}x{\rangle}^{-1},Q]x\in\mathbb{B}(L^{2})$.
Now 1.4 follows from 3.13 and Theorem 3.1 since
${\langle}x{\rangle}^{-1}(P-z)^{-1}{\langle}x{\rangle}^{-1}$ is decomposed as

$\displaystyle{\langle}x{\rangle}^{-1}(A+i\kappa)S^{-1}\cdot
S(A+i\kappa)^{-1}(P-z)^{-1}(A-i\kappa)^{-1}S\cdot
S^{-1}(A-i\kappa){\langle}x{\rangle}^{-1}.$

We next prove the Hölder continuity of the maps ${\mathbb{C}}^{\pm}\ni
z\mapsto{\langle}x{\rangle}^{-\rho}(P-z)^{-1}{\langle}x{\rangle}^{-\rho}$.
Define

$K_{{\varepsilon},\rho}(z):={\langle}x{\rangle}^{-\rho}(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{-1}{\langle}x{\rangle}^{-\rho}$

for $\rho\geq 1$, which is well-defined for
$({\varepsilon},z)\in{\mathbb{R}}^{+}\times{\mathbb{C}}^{+}$ since

$-\mathop{\mathrm{Im}}\nolimits(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)\geq{\varepsilon}{\langle}x{\rangle}^{-2}+\mathop{\mathrm{Im}}\nolimits
z.$

We also set
$K_{0,\rho}(z):={\langle}x{\rangle}^{-\rho}(P-z)^{-1}{\langle}x{\rangle}^{-\rho}$.
By the resolvent identity, we have

$\displaystyle
K_{{\varepsilon},1}(z)-K_{0,1}(z)=i{\varepsilon}K_{0,1}(z)K_{{\varepsilon},1}(z).$

Since ${\|K_{0,1}(z)\|}\lesssim 1$ uniformly in $z\in{\mathbb{C}}^{+}$ by
1.4, we obtain ${\|K_{{\varepsilon},1}(z)\|}\lesssim
1+{\varepsilon}{\|K_{{\varepsilon},1}(z)\|}$ uniformly in
$({\varepsilon},z)\in{\mathbb{R}}^{+}\times{\mathbb{C}}^{+}$. Taking
${\varepsilon}_{0}$ small enough (independently of $z$), we arrive at

$\displaystyle\sup_{{\varepsilon}\in[0,{\varepsilon}_{0}],z\in{\mathbb{C}}^{+}}{\|K_{{\varepsilon},\rho}(z)\|}\leq\sup_{{\varepsilon}\in[0,{\varepsilon}_{0}],z\in{\mathbb{C}}^{+}}{\|K_{{\varepsilon},1}(z)\|}<\infty,$
( 3.15)
$\displaystyle\sup_{z\in{\mathbb{C}}^{+}}{\|K_{{\varepsilon},\rho}(z)-K_{0,\rho}(z)\|}\lesssim{\varepsilon}.$
( 3.16)

On the other hand, since

$\displaystyle 2{\|(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)u\|}$
$\displaystyle\geq{\|(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)u-(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{*}u\|}$
$\displaystyle=2{\|{\varepsilon}{\langle}x{\rangle}^{-2}u+(\mathop{\mathrm{Im}}\nolimits
z)u\|}\geq 2{\varepsilon}^{2}{\|{\langle}x{\rangle}^{-2}u\|}$

for $u\in D(P)$, we know by density argument that

${\|{\langle}x{\rangle}^{-2}(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{-1}\|}\leq{\varepsilon}^{-1}.$

Taking the complex conjugate and then the adjoint also implies

${\|(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{-1}{\langle}x{\rangle}^{-2}\|}\leq{\varepsilon}^{-1}.$

Since
$K_{{\varepsilon},2}(z)-K_{{\varepsilon},2}(z^{\prime})=(z^{\prime}-z){\langle}x{\rangle}^{-2}(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z)^{-1}(P-i{\varepsilon}{\langle}x{\rangle}^{-2}-z^{\prime})^{-1}{\langle}x{\rangle}^{-2}$,
we have

$\displaystyle{\|K_{{\varepsilon},2}(z)-K_{{\varepsilon},2}(z^{\prime})\|}$
$\displaystyle\leq|z-z^{\prime}|{\varepsilon}^{-2}.$ ( 3.17)

We now optimize the right hand sides of 3.16 and 3.17 by taking
${\varepsilon}=|z-z^{\prime}|^{1/3}$, to find that the map
${\mathbb{C}}^{+}\ni z\mapsto K_{{\varepsilon},2}(z)\in\mathbb{B}(L^{2})$ is
Hölder continuous of order $1/3$, so is the map ${\mathbb{C}}^{+}\ni
z\mapsto{\langle}x{\rangle}^{-2}(P-z)^{-1}{\langle}x{\rangle}^{-2}$ by 3.16
and the triangle inequality. Hence, we have

${\|K_{0,2}(z)-K_{0,2}(z^{\prime})\|}\lesssim|z-z^{\prime}|^{1/3},\quad
z,z^{\prime}\in{\mathbb{C}}^{+}.$

By interpolating this and the uniform bound 3.15 with ${\varepsilon}=0$ and
$\rho=1$, we conclude that, for $1<\rho<2$, the map ${\mathbb{C}}^{+}\ni
z\mapsto K_{0,\rho}(z)$ is Hölder continuous of order $(\rho-1)/3$, so is
${\mathbb{C}}^{-}\ni z\mapsto K_{0,\rho}(z)$ by taking the complex conjugate.
Finally, by the Hölder continuity, $\\{K_{0,\rho}(\lambda\pm
i{\varepsilon})\\}$ are Cauchy sequences on $B(L^{2})$ as
${\varepsilon}\searrow 0$, so the limits

$K_{0,\rho}(\lambda\pm i0)=\lim_{{\varepsilon}\searrow 0}K_{0,\rho}(\lambda\pm
i{\varepsilon})\in\mathbb{B}(L^{2})$

exist and Hölder continuous for all $\lambda\in{\mathbb{R}}$. ∎

##  4\. Proof of Theorem 1.4

Here we prove Theorem 1.4. Assume for contradiction that $Pu=\lambda u$ with
some $\lambda\in{\mathbb{R}}$ and non-trivial $u\in D(P)$. Since $P\in
C^{1}(A)$, the Virial theorem implies

$\displaystyle{\langle}[P,iA]u,u{\rangle}=0$ ( 4.1)

for all $r>0$. On the other hand, we find by 1.6, Lemmas 2.1 and 2.3 that

$\displaystyle{\langle}[P,iA]u,u{\rangle}$
$\displaystyle={\langle}G_{11}QD^{T}u,QD^{T}u{\rangle}+{\langle}G_{12}QD^{T}u,QD^{T}u{\rangle}$
$\displaystyle\geq
C_{0}{\|QD^{T}u\|}^{2}-C_{1}{\|Q{\langle}D{\rangle}QD^{T}u\|}^{2}$

with some positive constants $C_{0},C_{1}$ independent of $r$, where the right
hand side is equal to

$\displaystyle\int\left(C_{0}-C_{1}\frac{1+|\xi|^{2}}{r+|\xi|^{2}}\right)\frac{|\xi\widehat{u}(\xi)|^{2}}{r+|\xi|^{2}}d\xi=\int\left(C_{0}-C_{1}+C_{1}\frac{r-1}{r+|\xi|^{2}}\right)\frac{|\xi|^{2}}{r+|\xi|^{2}}|\widehat{u}(\xi)|^{2}d\xi.$

If $|\xi|\leq 1$, then

$-C_{1}+C_{1}\frac{r-1}{r+|\xi|^{2}}\geq-
C_{1}+C_{1}\frac{r-1}{r+1}=-\frac{2C_{1}}{r+1},$

so we find for sufficiently large $r$ that

$\displaystyle I_{1}:=\int_{|\xi|\leq
1}\left(C_{0}-C_{1}+C_{1}\frac{r-1}{r+|\xi|^{2}}\right)\frac{|\xi|^{2}}{r+|\xi|^{2}}|\widehat{u}(\xi)|^{2}d\xi\geq\frac{C_{0}}{2}\int_{|\xi|\leq
1}\frac{|\xi|^{2}|\widehat{u}(\xi)|^{2}}{r+|\xi|^{2}}d\xi\geq 0.$

If $|\xi|\geq 1$, then we have $1-r/(r+|\xi|^{2})\geq 1-r/(r+1)=1/(r+1)$ and
thus

$\displaystyle I_{2}:=\int_{|\xi|\geq
1}\left(C_{0}-C_{1}+C_{1}\frac{r-1}{r+|\xi|^{2}}\right)\frac{|\xi|^{2}}{r+|\xi|^{2}}|\widehat{u}(\xi)|^{2}d\xi\geq(r+1)^{-1}\\{C_{0}a-C_{1}(a-a_{r})\\},$

where $a={\|\mathds{1}_{\\{|\xi|\geq 1\\}}u\|}^{2}$ and
$a_{r}=(r-1){\|\mathds{1}_{\\{|\xi|\geq 1\\}}Qu\|}^{2}$. Since $a_{r}\to a$ as
$r\to\infty$ by the Lebesgue dominated convergence theorem, $I_{2}\geq
C_{0}a/(r+1)\geq 0$ for $r$ large enough. Hence $I_{1}=I_{2}=0$ for $r$ large
enough by 4.1. This shows $u\equiv 0$ almost everywhere, yielding a
contradiction.

## Acknowledgments

The authors would like to express his thank to Koichi Taira for helpful
discussions and remarks, especially regarding the proof of Lemma 2.5. The
author is partially supported by JSPS KAKENHI Grant Number JP21K03325.

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