# Semiclassical estimates for measure potentials on the real line

Andrés Larraín-Hubach  Department of Mathematics, University of Dayton,
Dayton, OH 45469-2316, USA<EMAIL_ADDRESS>and  Jacob Shapiro
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA
<EMAIL_ADDRESS>
###### Abstract.

We prove an explicit Carleman estimate for the semiclassical Schrödinger
operator $P=-h^{2}\partial^{2}_{x}+V(x,h)$ on $L^{2}(\mathbb{R})$, with
$V(x,h)$ a finite signed measure, and where $h>0$ is the semiclassical
parameter. The proof is a one dimensional instance of the spherical energy
method, which has been used to prove similar Carleman estimates in higher
dimensions and in more complicated geometries. The novelty of our result is
that the potential need not be absolutely continuous with respect to Lebesgue
measure. Two consequences of the Carleman estimate are the absence of positive
eigenvalues for $P$, and a limiting absorption resolvent estimate with sharp
$h$-dependence. The resolvent estimate implies exponential time-decay of the
local energy for solutions to the corresponding wave equation with a measure
potential, provided there are no negative eigenvalues and no zero resonance,
and provided the initial data have compact support.

##  1\. Introduction and statement of results

The goal of this note is to study the spectral and scattering theory for the
one dimensional semiclassical Schrödinger operator,

$P=P(h)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-h^{2}\partial_{x}^{2}+V(x;h):L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}),\qquad h>0,$ (1.1)

with potential $V=V(x;h)$ a real, finite signed Borel measure on $\mathbb{R}$,
which may depend on the semiclassical parameter $h$. Here and below,
$L^{2}(\mathbb{R})$ is the usual Hilbert space of equivalence classes of
functions $u:\mathbb{R}\to\mathbb{C}$ which are measurable with respect to the
Lebesgue sigma algebra on $\mathbb{R}$, and for which
$\int_{\mathbb{R}}|u|^{2}dx<\infty$, where $dx$ denotes Lebesgue measure.

We first address self-adjointness of (1.1) on $L^{2}(\mathbb{R})$. Some care
is needed, because while $u\in L^{2}(\mathbb{R})$ is specified only up to sets
of Lebesgue measure zero, the measure $V$ may have point masses or singular
continuous part. To handle this, we specify the domain of $\mathcal{D}$ of $P$
as a certain dense subspace of $L^{2}(\mathbb{R})$ which is contained in the
Sobolev space $H^{1}(\mathbb{R})$. Recall each $u\in H^{1}(\mathbb{R})$ has a
(unique) continuous, bounded representative, which we denote by $u_{c}$. Thus,
for $u\in H^{1}(\mathbb{R})$, we prescribe the product of $u$ and $V$ to be
the complex Borel measure $u_{c}V$, and define the expression

$Pu\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-h^{2}\partial^{2}_{x}u+u_{c}V$
(1.2)

in the sense of distributions on $\mathbb{R}$. In Section 3, we show that (for
all $h>0$) $P$ is self-adjoint with respect to

$\mathcal{D}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{u\in
H^{1}(\mathbb{R}):u^{\prime}\in L^{\infty}(\mathbb{R}),\text{ and }Pu\in
L^{2}(\mathbb{R})\\}.$ (1.3)

After establishing self-adjointness, in Section 4 we turn to proving our main
result, a weighted semiclassical Carleman estimate:

######  Theorem 1.1 (Carleman estimate).

Fix $\delta>0$. For all $E=E(h)>0$ (which may depend on $h$),
$\varepsilon\in[0,1]$, $h>0$, and $u\in\mathcal{D}$ with
$(|x|+1)^{(1+\delta)/2}(P-E\pm i\varepsilon)u\in L^{2}(\mathbb{R})$,

$\begin{split}\int_{\mathbb{R}}(|x|+1)^{-1-\delta}(E|u|^{2}&+|hu^{\prime}|^{2})dx\\\
&\leq C(V,E,h,\delta)\int_{\mathbb{R}}(|x|+1)^{1+\delta}|(P-E\pm
i\varepsilon)u|^{2}dx.\end{split}$ (1.4)

where

$\displaystyle
C(V,E,h,\delta)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{C_{1}(V,E,h,\delta)}\Big{(}\big{(}\frac{4}{h^{2}}+\frac{2}{h}+\frac{1}{Eh^{2}}\big{(}2+2E+\frac{\|V\|^{2}}{h^{2}}\big{)}^{2}e^{C_{1}(V,E,h,\delta)}\Big{)},$
(1.5) $\displaystyle
C_{1}(V,E,h,\delta)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2\delta^{-1}+E^{-1/2}h^{-1}\|V\|,$
(1.6)

where
$\|V\|\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|V|(\mathbb{R})$,
with $|V|$ the total variation of $V$.

######  Remark 1.2.

In the case that $E>0$ is fixed independent of $h$, and we restrict
$h\in(0,1]$, the constant (1.5) is bounded from above by the more succinct
expression

$\exp(\tilde{C}(E,\delta)(1+\|V\|)/h),\qquad\text{for some
$\tilde{C}(E,\delta)>0$ depending on $E$ and $\delta$.}$ (1.7)

Two consequences of Theorem 1.1 are the absence of positive eigenvalues of
$P$, and a weighted limiting absorption resolvent bound.

######  Corollary 1.3.

The operator $P$ on $L^{2}(\mathbb{R})$, given by (1.2) and equipped with
domain $\mathcal{D}$, has no positive eigenvalues.

######  Corollary 1.4.

Fix $\delta>0$. Then for all $E=E(h)>0$ (which may depend on $h$),
$\varepsilon\in(0,1]$, and $h>0$,

$\|(|x|+1)^{-\frac{1+\delta}{2}}(P(h)-E\pm
i\varepsilon)^{-1}(|x|+1)^{-\frac{1+\delta}{2}}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\leq\Big{(}\frac{C(V,E,h,\delta)}{E}\Big{)}^{1/2},$ (1.8)

where $C(V,E,h,\delta)$ is as given in (1.5).

Note that for $E>0$ fixed and the $h\in(0,1]$, the right side of (1.8) is
bounded from above by an expression of the form (1.7).

When $V$ is compactly supported, we prove a simpler Carleman estimate away
from the support of $V$, which yields an improvement to (1.8) for $h$ small.

######  Theorem 1.5 (exterior estimate).

Fix $\delta>0$ and $E>0$. Suppose $V$ is supported in $[-R_{0},R_{0}]$
(independent of $h$) for some $R_{0}>0$. Set
$h_{0}=2^{-1}\delta^{-1}(1+R_{0})^{-\delta}$. There exist $C$, depending only
on $R_{0},E$, and $\delta$ (see (5.5) below), so that for all
$\varepsilon\in(0,1]$ and $h\in(0,h_{0}]$,

$\|(|x|+1)^{-\frac{1+\delta}{2}}\mathbf{1}_{>R_{0}}(P-E\pm
i\varepsilon)^{-1}\mathbf{1}_{>R_{0}}(|x|+1)^{-\frac{1+\delta}{2}}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\leq\frac{C}{h}.$ (1.9)

Here, $\mathbf{1}_{>R_{0}}$ denotes the characteristic function of $\\{\pm
x>R_{0}\\}$.

Absence of positive eigenvalues is well known to hold for $V\in
L_{\operatorname{loc}}^{1}(\mathbb{R};\mathbb{R})$ with

$\lim_{|n|\to\infty}\int_{n}^{n+1}|V(x)|dx=0,$

see [Te14, Section 9.7]. Recall also the celebrated von Neumann-Wigner
potential $W$ [ReSi78, Section XIII.13], which obeys

$W(x)=\frac{-8\sin(2|x|)}{|x|}+O(|x|^{-2}),\qquad\text{as $|x|\to\infty$,}$

and has an eigenvalue at $E=1$. The $h$-dependencies in (1.8) and (1.9) are
sharp in general, and were proved previously for $V\in
L^{1}(\mathbb{R};\mathbb{R})$ [DaSh20]. The novelty of this work is that
Theorem 1.1 implies absence of positive eigenvalues, and optimal semiclassical
resolvent bounds, for potentials in one dimension which may not be absolutely
continuous with respect to Lebesgue measure.

When $V$ is smooth, the exponential bound (1.8) (with different constants) was
first proved by Burq [Bu98, Bu02], who also considered higher dimensions and
more general operators. Further proofs and generalizations can be found in
[CaVo02, Da14, RoTa15, DadH16, Sh19, Ga19, GaSh22b, Ob23]. In dimension $n>1$,
current results require at least $V\in
L_{\text{loc}}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$, with sufficient
decay toward infinity, to obtain a semiclassical resolvent estimate. And
often, only weaker versions of (1.8) are known [KlVo19, Vo19, Vo20a, Vo20b,
Vo20c, Vo20d, Sh20, Vo22, GaSh22a, DGS23], with $e^{C/h}$ replaced by
$e^{C/h^{\ell}}$ for some $\ell>1$.

When $V$ is smooth, the improvement (1.9) away from the support of $V$ was
first proved by Cardoso and Vodev [CaVo02], refining earlier work of Burq
[Bu02], and again analogous results hold in many settings [CaVo02, Da14,
RoTa15, DadH16, Sh19, GaSh22b, Ob23]. When the dimension $n>1$, Datchev and
Jin [DaJi20] showed the cutoff ${\bf 1}_{|x|>R_{0}}$ may need to be replaced
by ${\bf 1}_{|x|>R}$ with $R\gg R_{0}$, even when $V\in
C_{0}^{\infty}(\mathbb{R}^{n})$.

To prove Theorem 1.1, we employ a positive commutator-style argument in the
context of the so-called spherical energy method. This strategy has long been
used to prove semiclassical Carleman estimates [CaVo02, Da14, KlVo19, DaSh20,
GaSh22b, Ob23]. In fact, as we work in one dimension, it suffices to use the
pointwise energy

$\begin{gathered}F(x)=F_{\pm}[u](x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|hu^{\prime}(x)|^{2}+E|u(x)|^{2},\\\
u\in\mathcal{D}\text{ such that }(|x|+1)^{(1+\delta)/2}(P(h)-E\pm
i\varepsilon)u\in L^{2}(\mathbb{R}).\end{gathered}$ (1.10)

The goal is to construct a suitable weight $w(x)$ having locally bounded
variation, so that, roughly speaking, the distributional derivative of $wF$ is
bounded from above by a term involving
$2w\mathop{\rm Re}((P-E\pm i\varepsilon)u\overline{u}^{\prime})$, and bounded
from below by $w(E|u|^{2}+|hu^{\prime}|^{2})$ (see (4.6) for the precise
estimate). To attain the upper bound, $V$ needs to be reintroduced after
differentiation of $wF$, at the cost of a perturbation term (see (4.4), and
note that $V$ does appear in $F$ in the first place because its distributional
derivative may be irregular). This perturbation can be controlled, yielding
the desired the lower bound, by designing $w$ appropriately. In particular,
since $V$ may have discrete part $V_{d}$ (i.e., countably many point masses
which are absolutely summable), we use a family of weights $w_{\eta}(x)$
depending on a parameter $\eta>0$. Each $w_{\eta}$ controls a certain Gaussian
approximation of $|V_{d}|$ (see (4.2)). We then show that the needed estimates
hold uniformly as $\eta\to 0^{+}$.

When $V$ is compactly supported, it is well known that Corollary 1.4 is
related to the distribution of scattering resonances for the operator
$-\partial^{2}_{x}+V$. As in [SjZw91], we define the resonances of
$-\partial^{2}_{x}+V$ as the poles of the cutoff resolvent

$\chi(-\partial^{2}_{x}+V-\lambda^{2})^{-1}\chi:L^{2}(\mathbb{R})\to\mathcal{D},\qquad\chi\in
C^{\infty}_{0}(\mathbb{R};[0,1]),\chi\equiv 1\text{ on }\mathop{\rm supp}V,$
(1.11)

which continues meromorphically from $\mathop{\rm Im}\lambda\gg 1$ to the
complex plane. In Section 6, we combine (1.8) with a resolvent identity
argument of Vodev [Vo14, Theorem 1.5] to show

######  Theorem 1.6.

Suppose $V$ is a finite signed Borel measure on $\mathbb{R}$, which is
supported in $[-R_{0},R_{0}]$ for some $R_{0}>0$. Fix $\chi\in
C_{0}^{\infty}(\mathbb{R};[0,1])$ such that $\chi=1$ near $[-R_{0},R_{0}]$,
and fix $\lambda_{0}>0$. There exist $C,\varepsilon_{0}>0$ so that for all
$|\mathop{\rm Re}\lambda|\geq\lambda_{0}$, and $|\mathop{\rm
Im}\lambda|\leq\varepsilon_{0}$,

$\|\chi(-\partial^{2}_{x}+V-\lambda^{2})^{-1}\chi\|_{L^{2}\to H^{k}}\leq
C|\mathop{\rm Re}\lambda|^{k-1},\qquad
k=0,\,1\,(H^{0}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(\mathbb{R})),$
(1.12)

and

$\|\chi(-\partial^{2}_{x}+V-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\leq
C(|\mathop{\rm Re}\lambda|+1),$ (1.13)

where $\mathcal{D}$ is equipped with the graph norm
$\|u\|_{\mathcal{D}}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\|(-\partial^{2}_{x}+V)u\|^{2}_{L^{2}}+\|u\|^{2}_{L^{2}})^{1/2}$.

The existence of resonance free regions below the real axis is a long-studied
problem: [Ha82, Zw87, Hi99] treat the cases $V\in
L^{\infty}_{\text{comp}}(\mathbb{R})$, $V\in L_{\text{comp}}^{1}(\mathbb{R})$,
and $V$ exponentially decaying, respectively. More recent articles [Sa16,
DMW22, DaMa22] describe the distribution of resonances for thin barriers in
the semiclassical regime. To the authors’ knowledge, Theorem 1.6 is the first
demonstration of a resonance free strip for a class of potentials in one
dimension that can have singularly continuous part.

Estimates such as (1.12) and (1.13) yield regularity and decay results for
operators involving the measure $V$. As an illustration, in Section 7 we show
how (1.12) implies an exponential local energy decay rate, modulo negative
eigenvalues and a possible zero-resonance, for the associated wave equation,

$\begin{cases}(\partial_{t}^{2}-\partial_{x}^{2}+V(x))w(x,t)=0,\qquad(x,t)\in\mathbb{R}\times(0,\infty),\\\
w(x,0)=w_{0}(x),\\\ \partial_{t}w(x,0)=w_{1}(x),\\\ \mathop{\rm
supp}w_{0},\,\mathop{\rm supp}w_{1}\subseteq(-R,R),\qquad R>0.\end{cases}$
(1.14)

See Theorem 7.1 for a precise statement. Similar wave decay was previously
established for
$V\in L_{\text{comp}}^{\infty}(\mathbb{R};\mathbb{R})$ ([TaZw] and [DyZw19,
Theorem 2.9]). We also mention that exterior estimates like (1.9) have
application to integrated wave decay [DaJi20, Lemma 5].

There is a vast literature on Schrödinger operators whose coefficients are
measures. Thus, we will not attempt to give a comprehensive review here. For
the one dimensional case, we point the reader to [AmRe05, EcTe13, EKMT14],
which develop the Sturm-Liouville theory for such operators, and investigate
topics including boundary conditions, self-adjoint extensions, and inverse
spectral theory. The research monograph [AGHH05] gives a comprehensive
treatment of point interactions in three and fewer dimensions. Higher
dimensional studies include [BEKS92, Gal19].

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

Acknowledgements: It is a pleasure to thank Kiril Datchev for helpful
discussions. JS gratefully acknowledges support from ARC DP180100589, NSF DMS
2204322, and from a 2023 Fulbright Future Scholarship funded by the Kinghorn
Foundation and hosted by University of Melbourne.

##  2\. Review of BV

To keep the notation concise, for the rest of the article, we use “prime”
notation to denote differentiation with respect to $x$, e.g.,
$u^{\prime}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial_{x}u$.

In Section 2, we review the basics of functions of bounded variation (BV), and
collect four well-known Propositions concerning their calculus. This material
is relied upon frequently in later Sections. We give the proof of Proposition
2.2, while proofs of Propostions 2.1, 2.3, and 2.4 may be found in [DaSh22,
Appendix B].

Let $f:\mathbb{R}\to\mathbb{C}$ be a function of locally bounded variation.
For all $x\in\mathbb{R}$, put

$f^{L}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\delta\to
0^{+}}f(x-\delta),\qquad
f^{R}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\delta\to
0^{+}}f(x+\delta),\qquad
f^{A}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(f^{L}(x)+f^{R}(x))/2,$
(2.1)

where the limits exist because both the real and imaginary parts of $f$ are a
difference of two increasing functions. Recall that $f$ is differentiable
Lebesgue almost everywhere, so $f(x)=f^{L}(x)=f^{R}(x)=f^{A}(x)$ for almost
all $x\in\mathbb{R}$.

We may decompose $f$ as

$f=f_{r,+}-f_{r,-}+i(f_{i,+}-f_{i,-}),$ (2.2)

where the $f_{\sigma,\pm}$, $\sigma\in\\{r,i\\}$, are increasing functions on
$\mathbb{R}$. Each $f^{R}_{\sigma,\pm}$ uniquely determines a regular Borel
measure $\mu_{\sigma,\pm}$ on $\mathbb{R}$ satisfying
$\mu_{\sigma,\pm}(x_{1},x_{2}]=f^{R}_{\sigma,\pm}(x_{2})-f^{R}_{\sigma,\pm}(x_{1})$,
see [Fo07, Theorem 1.16]. We put

$df\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mu_{r,+}-\mu_{r,-}+i(\mu_{i,+}-\mu_{i,-}),$
(2.3)

which is a complex measure when restricted to any bounded Borel subset. For
any $a<b$,

$\begin{gathered}\int_{(a,b]}df=f^{R}(b)-f^{R}(a),\\\
\int_{(a,b)}df=f^{L}(b)-f^{R}(a).\end{gathered}$ (2.4)

######  Proposition 2.1 (integration by parts).

Let $f:\mathbb{R}\to\mathbb{C}$ have locally BV. For any $a<b$, and any
continuous $\varphi$ with $\varphi^{\prime}$ piecewise continuous,

$\int_{(a,b]}\varphi
df+\int_{(a,b]}\varphi^{\prime}fdx=f^{R}(b)\varphi(b)-f^{R}(a)\varphi(a).$
(2.5)

######  Proposition 2.2 (funadmental theorem of calculus).

Let $\mu_{\sigma,\pm}$, $\sigma\in\\{r,\,i\\}$ be positive Borel measures on
$\mathbb{R}$ which are finite on all bounded Borel subsets $\mathbb{R}$.
Suppose $u\in\mathcal{D}^{\prime}(\mathbb{R})$ has distributional derivative
equal to $\mu=\mu_{r,+}-\mu_{r,-}+i(\mu_{i,+}-\mu_{i,-})$. (For example, this
will hold for $u\in\mathcal{D}$, with $\mu=u_{c}V+gdx$ for some $g\in
L^{2}(\mathbb{R})$.) Then $u$ is a function of locally BV; For any
$a\in\mathbb{R}$, $u$ differs by a constant from the right continuous, locally
BV function

$f_{\mu}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{cases}\int_{[a,x]}d\mu&x\geq
a,\\\ -\int_{(x,a)}d\mu&x<a.\end{cases}$ (2.6)

###### Proof.

We need to show that the function (2.6) has distributional derivative $\mu$.
First, it is straightforward to check that
$f_{\mu}(x_{2})-f_{\mu}(x_{1})=\mu(x_{1},x_{2}]$ for all $x_{1}<x_{2}$. Hence
$df_{\mu}=\mu$. Then (2.5) implies

$-\int_{\mathbb{R}}\varphi^{\prime}f_{\mu}dx=\int_{\mathbb{R}}\varphi
d\mu,\qquad\varphi\in C^{\infty}_{0}(\mathbb{R}),$ (2.7)

where all boundary terms vanish, and the right side of (2.7) is finite, due to
the the compact support of $\varphi$.
∎

######  Proposition 2.3 (product rule).

Let $f,\,g:\mathbb{R}\to\mathbb{C}$ be functions of locally bounded variation.
Then

$d(fg)=f^{A}dg+g^{A}df$ (2.8)

as measures on a bounded Borel subset of $\mathbb{R}$.

######  Proposition 2.4 (chain rule).

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous and have locally bounded
variation. Then, as measures on a bounded Borel set of $\mathbb{R}$,

$d(e^{f})=e^{f}df.$ (2.9)

##  3\. Self-adjointness for $V$ a measure

The goal of Section 3 is to show $(P,\mathcal{D})$ is self-adjoint on
$L^{2}(\mathbb{R})$. By the end, we will demonstrate that $(P,\mathcal{D})$ is
merely the self-adjoint operator naturally associated to the quadratic form

$q(u,v)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int
h^{2}\overline{u}^{\prime}v^{\prime}dx+\int\overline{u_{c}}v_{c}V,\qquad
u,\,v\in H^{1}(\mathbb{R}).$ (3.1)

######  Lemma 3.1.

Let $V=V(x;h)$ be a real, finite signed Borel measure on $\mathbb{R}$. Then
$\mathcal{D}$ specified by (1.3) is dense in $L^{2}(\mathbb{R})$. The operator
$P:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ given by (1.2) with domain
$\mathcal{D}$ is self-adjoint.

###### Proof.

Throughout the proof, we work with $u\in L^{2}(\mathbb{R})$ that have locally
integrable distributional derivative $u^{\prime}$. Such $u$ have a (unique,
locally absolutely) continuous representative $u_{c}$, hence we can define the
product $u_{c}V$ as a distribution on $\mathbb{R}$, since it is a complex
measure when restricted to bounded Borel subsets of $\mathbb{R}$.

Let $\mathcal{D}_{\textrm{max}}\supseteq\mathcal{D}$ be the set of all $u\in
L^{2}(\mathbb{R})$ such that $u^{\prime}\in L^{1}_{\textrm{loc}}(\mathbb{R})$
and
$Pu\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-h^{2}u^{\prime\prime}+u_{c}V\in
L^{2}(\mathbb{R})$. By Proposition 2.2, for any
$u\in\mathcal{D}_{\textrm{max}}$, we may fix a representative
$u_{\text{bv}}^{\prime}$ for $u^{\prime}$ that has locally bounded variation.
If necessary, we redefine $u_{\text{bv}}^{\prime}$ on a set of Lebesgue
measure zero so that
$u_{\text{bv}}^{\prime}(x)=(u_{\text{bv}}^{\prime})^{A}(x)$ for all
$x\in\mathbb{R}$ (this updated $u^{\prime}_{\text{bv}}$ still has locally BV).

In the computations to follow, we always work with the representatives $u_{c}$
and $u_{\text{bv}}^{\prime}$, but we drop subscripts to keep notation concise.
This convention ensures that expressions like $u^{\prime}V$ are well defined
as complex Borel measures (on bounded Borel subsets), since locally BV
functions are Borel measurable. It also simplifies some calculations that
involve (2.4) or (2.8).

Our first step is prove $\mathcal{D}_{\textrm{max}}\subseteq\mathcal{D}$.
Since the reverse containment is trivial, we will conclude
$\mathcal{D}_{\textrm{max}}=\mathcal{D}$. Indeed, for
$u\in\mathcal{D}_{\textrm{max}}$ and any $a>0$,

$\begin{split}\int_{(-a,a)}|u^{\prime}|^{2}dx&=\int_{(-a,a)}u^{\prime}d(\overline{u})\\\
&=\int_{(-a,a)}d(u^{\prime}\overline{u})-\overline{u}d(u^{\prime})\\\
&=(u^{\prime}\bar{u})(a)-(u^{\prime}\bar{u})(-a)+h^{-2}\int_{(-a,a)}\overline{u}Pudx-h^{-2}\int_{(-a,a)}\overline{u}uV\\\
&\leq
2\sup_{(-a,a)}|u^{\prime}|\sup_{(-a,a)}|u|+h^{-2}\|V\|\sup_{(-a,a)}|u|^{2}+h^{-2}\|Pu\|_{L^{2}}\|u\|_{L^{2}},\end{split}$
(3.2)

where
$\|V\|\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|V|(\mathbb{R})$,
with $|V|$ the total variation of $V$. The second line of (3.2) follows from
(2.8) and $u=u^{A}$, $u^{\prime}=(u^{\prime})^{A}$; the third line follows
from the fact that $-h^{2}d(u^{\prime})=Pu-uV$ as Borel measures, which is a
consequence of (2.7).

Since $u$ is locally absolutely continuous,

$\sup_{(-a,a)}|u|^{2}=\sup_{x\in(-a,a)}\Big{(}|u(0)|^{2}+2\mathop{\rm
Re}\int_{0}^{x}u^{\prime}\overline{u}dx\Big{)}\leq|u(0)|^{2}+2\Big{(}\int_{(-a,a)}|u^{\prime}|^{2}dx\Big{)}^{1/2}\|u\|_{L^{2}}.$
(3.3)

Furthermore, if $x\in(0,a)$, then by (2.4), (2.8) and
$u^{\prime}=(u^{\prime})^{A}$,

$\begin{split}(u^{\prime}&\overline{u}^{\prime})(x)\\\
&=(u^{\prime}\overline{u}^{\prime})(0)+\int_{(0,x]}d(u^{\prime}\overline{u}^{\prime})\\\
&=|u^{\prime}|^{2}(0)-2h^{-2}\mathop{\rm
Re}\Big{(}\int_{(0,x]}\overline{u}^{\prime}Pudx-\int_{(0,x]}\overline{u}^{\prime}uV\Big{)}\\\
&\leq|u^{\prime}|^{2}(0)+2h^{-2}\|V\|\sup_{(-a,a)}|u|\sup_{(-a,a)}|u^{\prime}|+2h^{-2}\|Pu\|_{L^{2}}\Big{(}\int_{(-a,a)}|u^{\prime}|^{2}dx\Big{)}^{1/2},\end{split}$

while if $x\in(-a,0)$, we similarly find

$\begin{split}(u^{\prime}&\overline{u}^{\prime})(x)\\\
&=(u^{\prime}\overline{u}^{\prime})(0)-\int_{(x,0]}d(u^{\prime}\overline{u}^{\prime})\\\
&\leq|u^{\prime}|^{2}(0)+2h^{-2}\|V\|\sup_{(-a,a)}|u|\sup_{(-a,a)}|u^{\prime}|+2h^{-2}\|Pu\|_{L^{2}}\Big{(}\int_{(-a,a)}|u^{\prime}|^{2}dx\Big{)}^{1/2}.\end{split}$

We thus arrive at a system of inequalities of the form $x^{2}\leq
2yz+Ay^{2}+B$, $y^{2}\leq C+Dx$, $z^{2}\leq E+Fyz+Gx$, where
$x\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{(-a,a)}|u^{\prime}|^{2}$,
$y\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{(-a,a)}|u|^{2}$,
and
$z\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{(-a,a)}|u^{\prime}|^{2}$.
After using the second inequality to eliminate $y$, we obtain a system in $x$
and $z$ with quadratic left hand sides and subquadratic right hand sides.
Hence $x$, $y$, and $z$ are each bounded in terms of $A,B,\dots,G$. Letting
$a\to\infty$, we conclude that $u^{\prime}\in L^{2}(\mathbb{R})$ and
$u,\,u^{\prime}\in L^{\infty}(\mathbb{R})$. Hence
$\mathcal{D}_{\textrm{max}}\subseteq\mathcal{D}$ is desired.

Next, we equip $P$ with the domain $\mathcal{D}_{\textrm{max}}=\mathcal{D}$,
and show that $P$ is symmetric. Let $u,v\in\mathcal{D}$, and take
$\\{\varphi_{k}\\}_{k=1}^{\infty}\subseteq C^{\infty}_{0}(\mathbb{R})$
converging to $v$ in $H^{1}(\mathbb{R})$. Using the distributional definition
of $Pu$,

$\begin{split}\langle Pu,v\rangle_{L^{2}}&=\lim_{k\to\infty}\langle
Pu,\varphi_{k}\rangle_{L^{2}}\\\
&=\lim_{k\to\infty}\int\overline{u}(-h^{2}\varphi_{k}^{\prime\prime})dx+\int\overline{u}\varphi_{k}V\\\
&=\lim_{k\to\infty}\int
h^{2}\overline{u}^{\prime}\varphi_{k}^{\prime}dx+\int\overline{u}\varphi_{k}V\\\
&=\int h^{2}\overline{u}^{\prime}v^{\prime}dx+\int\overline{u}vV,\end{split}$
(3.4)

where the last equal sign follows since $|V|$ is finite and
$\|w\|^{2}_{L^{\infty}}\leq\|w\|_{L^{2}}\|w^{\prime}\|_{L^{2}}$ for any $w\in
H^{1}(\mathbb{R})$. Approximating $u\in H^{1}(\mathbb{R})$ by
$C^{\infty}_{0}(\mathbb{R})$-functions, we similarly have $\langle
u,Pv\rangle_{L^{2}}=\int
h^{2}\overline{u}^{\prime}v^{\prime}dx+\int\overline{u}vV$. Thus $P$ is
symmetric.

The last step is to establish that $(P,\mathcal{D})$ is densely defined and
$P^{*}\subseteq P$. For this, define on $H^{1}(\mathbb{R})$ the quadratic form
(3.1). Since, for any $\gamma>0$,

$\begin{split}\left|\int V|u|^{2}\right|&\leq\|V\|\|u\|_{L^{\infty}}^{2}\\\
&\leq\|V\|\|u\|_{L^{2}}\|u^{\prime}\|_{L^{2}}\\\
&\leq\|V\|\left(\tfrac{1}{2\gamma}\|u\|^{2}_{L^{2}}+\tfrac{\gamma}{2}\|u^{\prime}\|^{2}_{L^{2}}\right),\end{split}$
(3.5)

setting $\gamma=h^{2}/\|V\|$ yields

$-\tfrac{\|V\|^{2}}{2h^{2}}\|u\|^{2}_{L^{2}}+\tfrac{h^{2}}{2}\|u^{\prime}\|^{2}_{L^{2}}\leq
q(u,u)\leq\tfrac{\|V\|^{2}}{2h^{2}}\|u\|^{2}_{L^{2}}+\tfrac{3h^{2}}{2}\|u^{\prime}\|^{2}_{L^{2}}.$
(3.6)

We thus conclude $q$ is semibounded and closed.

By Friedrichs’ result [Te14, Theorem 2.14], there is a unique (densely
defined) self-adjoint operator $(A,\mathcal{D}_{1})$ with

$\begin{gathered}\mathcal{D}_{1}=\\{u\in H^{1}(\mathbb{R}):\text{there exists
$\tilde{u}\in L^{2}$ with $q(u,v)=\langle\tilde{u},v\rangle_{L^{2}}$ for all
$v\in H^{1}(\mathbb{R})$}\\},\\\ Au=\tilde{u}.\end{gathered}$

Revisiting the calculation (3.4), we see that for any $u\in\mathcal{D}_{1}$,
$\tilde{u}=-h^{2}u^{\prime\prime}+uV$ in the distributional sense. Thus
$(A,\mathcal{D}_{1})\subseteq(P,\mathcal{D}_{\max})$, so $P^{*}\subseteq
A^{*}=A\subseteq P$. Since we already showed $P\subseteq P^{*}$
(symmetricity), we conclude $P^{*}=P$ as desired.
∎

##  4\. Carleman estimate

The purpose of this Section is to prove Theorem 1.1. As discussed in Section
1, we do so by means of a positive commutator argument that leverages the
energy method.

###### Proof of Theorem 1.1.

Our starting point is the pointwise energy $F$ given by (1.10). As in the
proof of Lemma 3.1, we fix with a continuous representative of
$u\in\mathcal{D}$, and fix a representative of $u^{\prime}\in
L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ that has locally bounded
variation and $(u^{\prime})^{A}=u^{\prime}$ (thus $F^{A}=F$).

Since the measure $V=V(x;h)$ is finite, it can have only countably many point
masses $\\{x_{j}\\}_{j}$, and moreover $\sum_{j}|V_{j}|<\infty$, where
$V_{j}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=V(\\{x_{j}\\})$.
Let us decompose

$V=V_{c}+V_{d},\qquad V_{d}=\sum_{j}V_{j}\delta_{x_{j}},$

into its discrete and continuous parts. Here, $\delta_{x_{j}}$ denotes the
dirac measure concentrated at $x_{j}$. A key technical feature of the ensuing
calculations is our use of the weight function

$w=w_{\eta}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\exp\Big{(}\int_{-\infty}^{x}\big{[}\frac{1}{E^{1/2}h}|V_{c}|+(\frac{1}{E^{1/2}h}V_{d,\eta}(x^{\prime})+(|x^{\prime}|+1)^{-1-\delta})dx^{\prime}\big{]}\Big{)},\qquad\eta>0,$
(4.1)

where $|V_{c}|$ denotes the total variation of $V_{c}$, and

$V_{d,\eta}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi^{-1/2}\eta^{-1}\sum_{j}|V_{j}|e^{-((x-x_{j})/\eta)^{2}}.$
(4.2)

The exponent of $w$ is a continuous function of, thus we may compute $dw$
using (2.9). Nearing the end of the argument, we manage to control a term
involving $|V_{d}|$ by sending $\eta\to 0^{+}$, essentially using that
$\pi^{-1/2}\eta^{-1}\sum_{j}|V_{j}|e^{-((x-x_{j})/\eta)^{2}}\to\sum_{j}|V_{j}|\delta_{x_{j}}$
in the distribution sense. We note also that

$\sup_{\mathbb{R}}|w_{\eta}(x)|\leq e^{C_{1}(V,E,h,\delta)},$ (4.3)

with $C_{1}(V,E,h,\delta)$ given by (1.6).

From (2.8) and $u^{A}=u$, $(u^{\prime})^{A}=u^{\prime}$, we find, in the sense
of measures on $\mathbb{R}$,

$\begin{split}dF&=2h^{2}\mathop{\rm
Re}(\overline{u}^{\prime}d(u^{\prime}))+2E\mathop{\rm
Re}\left(u\overline{u}^{\prime}\right)\\\ &=-2\mathop{\rm Re}(((P-E\pm
i\varepsilon)u)\overline{u}^{\prime})\mp 2\varepsilon\mathop{\rm
Im}\left(u\overline{u}^{\prime}\right)+2\mathop{\rm
Re}(u\overline{u}^{\prime}V),\end{split}$

where, to get the second line, we used that $-h^{2}d(u^{\prime})=Pu-uV$ as
Borel measures. Using (2.8) again, this time to expand $d(wF)$,

$\begin{split}d(wF)&=Fdw+wdF\\\ &=|hu^{\prime}|^{2}dw+E|u|^{2}dw\\\
&-2w\mathop{\rm Re}(((P-E\pm i\varepsilon)u)\overline{u}^{\prime})\mp
2\varepsilon w\mathop{\rm Im}\left(u\overline{u}^{\prime}\right)+2\mathop{\rm
Re}w(u\overline{u}^{\prime}V)\\\ &\geq-2w\mathop{\rm Re}(((P-E\pm
i\varepsilon)u)\overline{u}^{\prime})\mp 2\varepsilon w\mathop{\rm
Im}\left(u\overline{u}^{\prime}\right)\\\
&+(|hu^{\prime}|^{2}+E|u|^{2})dw-h^{-1}w(E^{1/2}|u|^{2}+E^{-1/2}|hu^{\prime}|^{2})(|V_{c}|+\sum_{j}|V_{j}|\delta_{x_{j}}).\end{split}$
(4.4)

By (2.9) and (4.1),

$dw_{\eta}=h^{-1}w_{\eta}(E^{-1/2}|V_{c}|+E^{-1/2}\eta^{-1}\pi^{-1/2}\sum_{j}|V_{j}|e^{-((x-x_{j})/\eta)^{2}}dx+h(|x|+1)^{-1-\delta}dx).$
(4.5)

Plugging (4.5) into the fifth line of (4.4) implies,

$\begin{split}d(wF)&\geq-2w\mathop{\rm Re}(((P-E\pm
i\varepsilon)u)\overline{u}^{\prime})\mp 2\varepsilon w\mathop{\rm
Im}\left(u\overline{u}^{\prime}\right)\\\
&+w(|x|+1)^{-1-\delta}(E|u|^{2}+|hu^{\prime}|^{2})dx\\\
&+\sum_{j}h^{-1}w(E^{1/2}|u|^{2}+E^{-1/2}|hu^{\prime}|^{2})|V_{j}|(\eta^{-1}\pi^{-1/2}e^{-((x-x_{j})/\eta)^{2}}dx-\delta_{x_{j}})\end{split}$
(4.6)

Next, we note there exist sequences $\\{a^{\pm}_{n}\\}_{n=1}^{\infty}$ tending
to $\pm\infty$, along which
$F(a^{\pm}_{n})=F^{R}(a^{\pm}_{n})=F^{L}(a^{\pm}_{n})\to 0$. This is because
$F(x)\in L^{1}(\mathbb{R})$ and is continuous off of a countable set. So, we
integrate both sides of (4.6) over $(a^{-}_{n},a^{+}_{n}]$ and send
$n\to\infty$. By (2.4), the left side of (4.6) becomes zero. Hence, from (4.3)
and $w\geq 1$,

$\begin{split}\int&(|x|+1)^{-1-\delta}(E|u|^{2}+|hu^{\prime}|^{2})dx\\\
&+\sum_{j}h^{-1}|V_{j}|\int
w(E^{1/2}|u|^{2}+E^{-1/2}|hu^{\prime}|^{2})(\eta^{-1}\pi^{-1/2}e^{-((x-x_{j})/\eta)^{2}}dx-\delta_{x_{j}})\\\
&\leq e^{C_{1}(V,E,h,\delta)}\big{(}\int\frac{1}{\gamma
h^{2}}(|x|+1)^{1+\delta}|(P-E\pm
i\varepsilon)u|^{2}+\gamma(|x|+1)^{-1-\delta}|hu^{\prime}|^{2}\\\
&+2\varepsilon\int|uu^{\prime}|dx\big{)},\qquad\gamma,\,h>0.\end{split}$ (4.7)

The goal of the following calculations is to show that second line of (4.7) is
nonnegative the limit that $\eta\to 0^{+}$. First notice that as $\eta\to
0^{+}$,

$\begin{gathered}\int_{-\infty}^{x_{j}}V_{d,\eta}(x^{\prime})dx^{\prime}=\pi^{-1/2}\sum_{\ell}|V_{\ell}|\int_{-\infty}^{(x_{j}-x_{\ell})/\eta}e^{(-x^{\prime})^{2}}dx^{\prime}\to\frac{1}{2}|V_{j}|+\sum_{x_{\ell}<x_{j}}|V_{\ell}|,\\\
\int_{-\infty}^{x_{j}+\eta
x}V_{d,\eta}(x^{\prime})dx^{\prime}=\pi^{-1/2}|V_{\ell}|\sum_{\ell}\int_{-\infty}^{\frac{x_{j}-x_{\ell}}{\eta}+x}e^{(-x^{\prime})^{2}}dx^{\prime}\to\pi^{-1/2}|V_{j}|\int_{-\infty}^{x}e^{(-x^{\prime})^{2}}dx^{\prime}+\sum_{x_{\ell}<x_{j}}|V_{\ell}|.\end{gathered}$

This implies

$\begin{gathered}w_{\eta}(x_{j})\to
e^{\Gamma(E,h,j)}\exp\Big{(}\frac{|V_{j}|}{2E^{1/2}h}\Big{)}\\\
w_{\eta}(x_{j}+\eta x)\to e^{\Gamma(E,h,j)}\exp\Big{(}\frac{|V_{j}|}{(\pi
E)^{1/2}h}\int_{-\infty}^{x}e^{(-x^{\prime})^{2}}dx^{\prime}\Big{)},\end{gathered}$

where

$e^{\Gamma(E,h,j)}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\exp\Big{(}\frac{1}{E^{1/2}h}\sum_{x_{\ell}<x_{j}}|V_{\ell}|+\int_{-\infty}^{x_{j}}\big{[}\frac{1}{E^{1/2}h}|V_{c}|+(|x^{\prime}|+1)^{-1-\delta})dx^{\prime}\big{]}\Big{)}.$

Therefore,

$\begin{split}h^{-1}|V_{j}|\int&w_{\eta}(E^{1/2}|u|^{2}+E^{-1/2}|hu^{\prime}|^{2})\delta_{x_{j}}\\\
&=h^{-1}|V_{j}|w_{\eta}(x_{j})(E^{1/2}|u(x_{j})|^{2}+E^{-1/2}|hu^{\prime}(x_{j})|^{2}),\\\
&\to
h^{-1}|V_{j}|(E^{1/2}|u(x_{j})|^{2}+E^{-1/2}|hu^{\prime}(x_{j})|^{2})e^{\Gamma(E,h,j)}\exp\Big{(}\frac{|V_{j}|}{2E^{1/2}h}\Big{)},\end{split}$

while

$\begin{split}h^{-1}&\eta^{-1}\pi^{-1/2}|V_{j}|\int
w_{\eta}(E^{1/2}|u|^{2}+E^{-1/2}|hu^{\prime}|^{2})e^{-((x-x_{j})/\eta)^{2}}dx\\\
&=h^{-1}\pi^{-1/2}|V_{j}|\int w_{\eta}(x_{j}+\eta x)(E^{1/2}|u(x_{j}+\eta
x)|^{2}+E^{-1/2}|hu^{\prime}(x_{j}+\eta x)|^{2})e^{-x^{2}}dx,\\\
&\to(E^{1/2}|u(x_{j})|^{2}+E^{-1/2}|hu^{\prime}(x_{j})|^{2})e^{\Gamma(E,h,j)}\\\
&\cdot h^{-1}\pi^{-1/2}|V_{j}|\int\exp\Big{(}\frac{|V_{j}|}{(\pi
E)^{1/2}h}\int_{-\infty}^{x}e^{(-x^{\prime})^{2}}dx^{\prime}\Big{)}e^{-x^{2}}dx\\\
&=E^{1/2}(E^{1/2}|u(x_{j})|^{2}+E^{-1/2}|hu^{\prime}(x_{j})|^{2})e^{\Gamma(E,h,j)}\big{(}\exp\Big{(}\frac{|V_{j}|}{E^{1/2}h}\Big{)}-1\big{)}.\end{split}$

In summary, we have shown that the second line of (4.7), upon sending $\eta\to
0^{+}$, converges to

$\sum_{j}(E|u(x_{j})|^{2}+|hu^{\prime}(x_{j})|^{2})e^{\Gamma(E,h,j)}\big{(}\exp\Big{(}\frac{|V_{j}|}{E^{1/2}h}\Big{)}-1-\frac{|V_{j}|}{E^{1/2}h}\exp\Big{(}\frac{|V_{j}|}{2E^{1/2}h}\Big{)}\big{)}\geq
0.$ (4.8)

The nonnegativity follows from the fact that $e^{x}-1-xe^{x/2}\geq 0$ for all
$x\geq 0$.

Returning to (4.7), we fix $\gamma=2^{-1}e^{-C_{1}(V,E,h,\delta)}$, so that we
may absorb the first term in line three into the left side, and invoke (4.8),
implying

$\begin{split}\int(|x|&+1)^{-1-\delta}(E|u|^{2}+\tfrac{1}{2}|hu^{\prime}|^{2})dx\\\
&\leq
e^{C_{1}(V,E,h,\delta)}\Big{(}\frac{2e^{C_{1}(V,E,h,\delta)}}{h^{2}}\int(|x|+1)^{1+\delta}|(P-E\pm
i\varepsilon)u|^{2}dx+2\varepsilon\int|uu^{\prime}|dx\Big{)},\qquad
h>0.\end{split}$ (4.9)

For the term in (4.9) having the factor of $\varepsilon$:

$2\int|uu^{\prime}|dx\leq\frac{1}{h}\int|u|^{2}dx+\frac{1}{h}\int|hu^{\prime}|^{2}dx,\qquad
h>0,$ (4.10)

and

$\begin{split}\int|hu^{\prime}|^{2}dx&=\mathop{\rm Re}\int((P-E\pm
i\varepsilon)u)\overline{u}dx+E\int|u|^{2}dx+\int|u|^{2}V\\\
&\leq\frac{1}{2}\int|(P-E\pm
i\varepsilon)u|^{2}dx+\big{(}\frac{1}{2}+E\big{)}\int|u|^{2}dx+\|V\|\|u\|_{L^{2}}\|u^{\prime}\|_{L^{2}}\\\
&\leq\frac{1}{2}\int|(P-E\pm
i\varepsilon)u|^{2}dx+\big{(}\frac{1}{2}+E+\frac{\|V\|}{2\gamma
h^{2}}\big{)}\int|u|^{2}dx\\\
&+\frac{\gamma}{2}\|V\|\int|hu^{\prime}|^{2}dx,\qquad\gamma,\,h>0.\end{split}$
(4.11)

Fixing $\gamma=\|V\|^{-1}$ in (4.11), implies

$\int|hu^{\prime}|^{2}dx\leq\int|(P-E\pm
i\varepsilon)u|^{2}dx+(1+2E+\frac{\|V\|^{2}}{h^{2}})\int|u|^{2}dx,\qquad h>0.$
(4.12)

Now, replace $\int|hu^{\prime}|^{2}dx$ in (4.10) by the right side of (4.12).
From this, we get a bound for $2\int|uu^{\prime}|dx$, which we insert into the
in the last line of (4.9). We conclude

$\begin{split}\int(|x|&+1)^{-1-\delta}(E|u|^{2}+\tfrac{1}{2}|hu^{\prime}|^{2})dx\\\
&\leq
e^{C_{1}(V,E,h,\delta)}\Big{(}\big{(}\frac{2e^{C_{1}(V,E,h,\delta)}}{h^{2}}+\frac{1}{h}\big{)}\int(|x|+1)^{1+\delta}|(P-E\pm
i\varepsilon)u|^{2}dx\\\
&+\frac{\varepsilon}{h}\big{(}2+2E+\frac{\|V\|^{2}}{h^{2}}\big{)}\int|u|^{2}dx\Big{)},\qquad\varepsilon\in[0,1],\,h>0.\end{split}$
(4.13)

We absorb the last term of (4.13) into the left side by estimating

$\begin{split}\varepsilon\|u\|^{2}_{L^{2}}&=\mp\mathop{\rm Im}\langle(P-E\pm
i\varepsilon)u,u\rangle_{L^{2}}\\\
&\leq\tfrac{\gamma^{-1}}{2}\|(|x|+1)^{\frac{1+\delta}{2}}(P-E\pm
i\varepsilon)u\|^{2}_{L^{2}}+\tfrac{\gamma}{2}\|(|x|+1)^{-\frac{1+\delta}{2}}u\|^{2}_{L^{2}},\end{split}$
(4.14)

and then choosing
$\gamma=Eh^{-1}(2+2E+h^{-2}\|V\|^{2})^{-1}e^{-C_{1}(V,E,h,\delta)}$. We then
have

$\begin{split}\int(|x|&+1)^{-1-\delta}(\tfrac{E}{2}|u|^{2}+\tfrac{1}{2}|hu^{\prime}|^{2})dx\\\
&\leq
e^{C_{1}(V,E,h,\delta)}\Big{(}\frac{2e^{C_{1}(V,E,h,\delta)}}{h^{2}}+\frac{1}{h}+\frac{1}{2Eh^{2}}\big{(}2+2E+\frac{\|V\|^{2}}{h^{2}}\big{)}^{2}e^{C_{1}(V,E,h,\delta)}\Big{)}\\\
&\cdot\int(|x|+1)^{1+\delta}|(P-E\pm
i\varepsilon)u|^{2}dx,\qquad\varepsilon\in[0,1],\,h>0.\end{split}$ (4.15)

which implies (1.4).
∎

##  5\. Exterior estimate

###### Proof of Theorem 1.5.

We again start from (1.10), considering the case $\varepsilon\in(0,1]$ and
putting

$f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(P(h)-E\pm
i\varepsilon)^{-1}(|x|+1)^{-(1+\delta)/2}u.$

This time, we pair $F$ with a much simpler weight $w$. In particular we take
$w$ to be the continuous, odd function vanishing on $[-R_{0},R_{0}]$, and
obeying

$w(x)=1-\frac{(1+R_{0})^{\delta}}{(1+x)^{\delta}},\quad x>R_{0}\implies
dw=w^{\prime}(x)=\delta(1+R_{0})^{\delta}(1+|x|)^{-1-\delta}\textbf{1}_{\geq
R_{0}}.$

Note that the same $w$ was used in the proof of [DaSh20, Theorem 2]. Since
$wV=0$, we find, proceeding as in (4.4),

$\begin{split}d(wF)=-2w^{A}\mathop{\rm Re}(((P-E\pm
i\varepsilon)u)\overline{u}^{\prime})\mp 2\varepsilon w^{A}\mathop{\rm
Im}\left(u\overline{u}^{\prime}\right)+|hu^{\prime}|^{2}dw+E|u|^{2}dw,\end{split}$
(5.1)

and thus

$\begin{split}\delta(1+R_{0})^{\delta}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}(|x|&+1)^{-1-\delta}\big{(}E|u|^{2}+|hu^{\prime}|^{2}\big{)}\\\
&\leq\frac{1}{\gamma
h^{2}}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|f|^{2}+\gamma\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}(|x|+1)^{-1-\delta}|hu^{\prime}|^{2}\\\
&+2\varepsilon\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w|uu^{\prime}|\qquad\gamma,\,h>0.\end{split}$
(5.2)

Taking $\gamma=2^{-1}\delta(1+R_{0})^{\delta}$, we absorb the second term on
the right side of (5.2) into the left side. To handle the term involving
$\varepsilon$, we proceed to find

$2\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w|uu^{\prime}|\leq\frac{1}{h}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|u|^{2}+\frac{1}{h}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w^{2}|hu^{\prime}|^{2},\qquad
h>0,$ (5.3)

and

$\begin{split}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}&w^{2}|hu^{\prime}|^{2}=2h^{2}\mathop{\rm
Re}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}ww^{\prime}u^{\prime}\overline{u}+\mathop{\rm
Re}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w^{2}(-h^{2}u^{\prime\prime})\overline{u}\\\
&\leq\delta(1+R_{0})^{\delta}h\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}\big{(}|u|^{2}+w^{2}|hu^{\prime}|^{2}\big{)}\\\
&+\mathop{\rm Re}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w^{2}((P-E\pm
i\varepsilon)u)\overline{u}+E\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|u|^{2}\\\
&\leq\frac{1}{2}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|f|^{2}+\big{(}\frac{1}{2}+E+\delta(1+R_{0})^{\delta}h)\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|u|^{2}\\\
&+\delta(1+R_{0})^{\delta}h\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w^{2}|hu^{\prime}|^{2},\qquad
h>0.\end{split}$ (5.4)

Putting
$h_{0}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=2^{-1}\delta^{-1}(1+R_{0})^{-\delta}$
and restricting $h\in(0,h_{0}]$ in (5.4) allows us to bound
$\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w^{2}|hu^{\prime}|^{2}$ in (5.3) by
twice the fourth line of (5.4). Inserting the resulting estimate for
$2\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}w|u\overline{u}^{\prime}|$ into the
right side of (5.2) yields, for $\varepsilon\in(0,1]$ and $h\in(0,h_{0}]$,

$\begin{split}&\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}(|x|+1)^{-1-\delta}|u|^{2}\\\
&\leq\big{(}\frac{2}{\delta^{2}E(1+R_{0})^{2\delta}h^{2}}+\frac{1}{E\delta(1+R_{0})^{\delta}h}\big{)}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|f|^{2}+\varepsilon\frac{3+2E}{E\delta(1+R_{0})^{\delta}h}\int_{\mathbb{R}\setminus[-R_{0},R_{0}]}|u|^{2}.\\\
\end{split}$ (5.5)

The last term in line two of (5.5) may be estimated in manner similar (4.14),
leading to (1.9).
∎

##  6\. Uniform resolvent estimate and resonance free strips

In this Section, we prove Theorem 1.6 as an application of Corollary 1.4. We
are concerned with the self-adjoint operator

$H\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\partial^{2}_{x}+V:\mathcal{D}\to
L^{2}(\mathbb{R}),$ (6.1)

where $V$ remains a finite signed Borel measure, and has support in
$[-R_{0},R_{0}]$ for some $R_{0}>0$.

In this situation, $H$ is a black box Hamiltonian in the sense of Sjöstrand
and Zworski [SjZw91], as defined in [DyZw19, Definition 4.1]. More precisely,
in our setting this means the following. First, if $u\in\mathcal{D}$, then
$u|_{\mathbb{R}\setminus[-R_{0},R_{0}]}\in
H^{2}(\mathbb{R}\setminus[-R_{0},R_{0}])$. Second, for any $u\in\mathcal{D}$,
we have
$(Hu)|_{\mathbb{R}\setminus[-R_{0},R_{0}]}=-(u|_{\mathbb{R}\setminus[-R_{0},R_{0}]})^{\prime\prime}$.
Third, any $u\in H^{2}(\mathbb{R})$ which vanishes on a neighborhood of
$[-R_{0},R_{0}]$ is also in $\mathcal{D}$. Fourth,
$\textbf{1}_{[-R_{0},R_{0}]}(H+i)^{-1}$ is compact on $\mathcal{H}$; this last
condition follows from the fact that $\mathcal{D}\subseteq H^{1}(\mathbb{R})$.

Then, by the analytic Fredholm theorem (see [DyZw19, Theorem 4.4]), we have
the following. In $\mathop{\rm Im}\lambda>0$, the resolvent
$(H-\lambda^{2})^{-1}$ is meromorphic $L^{2}(\mathbb{R})\to\mathcal{D}$;
$\lambda$ is a pole of $(H-\lambda^{2})^{-1}$, if and only if $\lambda^{2}<0$
is an eigenvalue of $H$. Furthermore, for $\chi\in
C_{0}^{\infty}(\mathbb{R};[0,1])$ with $\chi=1$ near $[-R_{0},R_{0}]$, the
cutoff resolvent $\chi(H-\lambda^{2})^{-1}\chi$ continues meromorphically
$L^{2}(\mathbb{R})\to\mathcal{D}$ from $\mathop{\rm Im}\lambda>0$ to
$\mathbb{C}$. The poles of the continuation are known as its resonances.

###### Proof of Theorem 1.6.

Throughout the proof, we use $C(\|V\|,\lambda_{0})$ to denote a positive
constant which may depend on $\|V\|$ and $\lambda_{0}$, and whose value may
change from line to line, but is always independent of $\lambda$.

We first show (1.12) for $k=0$, $\mathop{\rm Im}\lambda>0$, and $|\mathop{\rm
Re}\lambda|\geq\lambda_{0}$. In this case, let us expand

$\begin{split}\chi(H&-\lambda^{2})^{-1}\chi\\\
&=\chi(\partial^{2}_{x}+V-(\mathop{\rm Re}\lambda)^{2}+(\mathop{\rm
Im}\lambda)^{2}-i2\mathop{\rm Re}\lambda\cdot\mathop{\rm
Im}\lambda)^{-1}\chi\\\ &=(\mathop{\rm Re}\lambda)^{-2}\chi((\mathop{\rm
Re}\lambda)^{-2}\partial^{2}_{x}+(\mathop{\rm Re}\lambda)^{-2}V-1+(\mathop{\rm
Im}\lambda)^{2}(\mathop{\rm Re}\lambda)^{-2}-i2(\mathop{\rm
Re}\lambda)^{-1}\mathop{\rm Im}\lambda)^{-1}\chi.\end{split}$ (6.2)

If $\mathop{\rm Im}\lambda\geq|\mathop{\rm Re}\lambda|/2$, then by the
spectral theorem for self-adjoint operators,

$\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to L^{2}}\leq|\mathop{\rm
Re}\lambda|^{-2}\leq\lambda_{0}^{-1}|\mathop{\rm
Re}\lambda|^{-1},\qquad\mathop{\rm Im}\lambda\geq|\mathop{\rm
Re}\lambda|/2,\,|\mathop{\rm Re}\lambda|\geq\lambda_{0}.$ (6.3)

If $\mathop{\rm Im}\lambda<|\mathop{\rm Re}\lambda|/2$, we apply (1.8) to
(6.2) (the notational correspondence is $\delta=1$, $E=1-(\mathop{\rm
Im}\lambda)^{2}(\mathop{\rm Re}\lambda)^{-2}\geq 3/4$,
$\varepsilon=2|\mathop{\rm Re}\lambda|^{-1}\mathop{\rm Im}\lambda\in(0,1]$,
$h=|\mathop{\rm Re}\lambda|^{-1}$, and $V(x,h)=h^{2}V$). Therefore,

$\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to L^{2}}\leq
C(\|V\|,\lambda_{0})|\mathop{\rm Re}\lambda|^{-1},\qquad\mathop{\rm
Im}\lambda>0,\,|\mathop{\rm Re}\lambda|\geq\lambda_{0}.$ (6.4)

Next, we adapt the proof of [BGT04, Proposition 2.5] to show

$\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to H^{1}}\leq
C(\|V\|,\lambda_{0}),\qquad 0<\mathop{\rm Im}\lambda\leq 1,\,|\mathop{\rm
Re}\lambda|\geq\lambda_{0}.$ (6.5)

We employ the notation,

$(H-\lambda^{2})u=\chi f,\qquad 0<\mathop{\rm Im}\lambda\leq 1,\,|\mathop{\rm
Re}\lambda|\geq\lambda_{0},\,f\in L^{2}(\mathbb{R}),\,u\in\mathcal{D},$ (6.6)

and make use of additional cutoffs

$\chi_{1},\,\chi_{2}\in C^{\infty}_{0}(\mathbb{R};[0,1]),\quad\chi_{1}=1\text{
on }\mathop{\rm supp}\chi,\quad\chi_{2}=1\text{ on }\mathop{\rm
supp}\chi_{1}.$ (6.7)

Observe

$\|\chi(H-\lambda^{2})^{-1}\chi f\|_{H^{1}}\leq\|\chi u\|_{L^{2}}+\|(\chi
u)^{\prime}\|_{L^{2}}\leq
C(\|\chi_{2}u\|_{L^{2}}+\|\chi_{1}u^{\prime}\|_{L^{2}}),$

where here and below, $C$ is a positive constant, which may depend on $\|V\|$
and the derivatives of the cutoffs, and which may change between lines, but
stays independent of $\lambda$. So, by (6.4) it suffices to show

$\|\chi_{1}u^{\prime}\|^{2}_{L^{2}}\leq C\big{(}(|\mathop{\rm
Re}\lambda|+1)^{2}\|\chi_{2}u\|_{L^{2}}^{2}+\|\chi_{2}f\|_{L^{2}}^{2}\big{)}.$
(6.8)

Multiplying (6.6) by $\chi^{2}_{1}\overline{u}$ and integrating gives

$\int\chi_{1}^{2}\chi
f\overline{u}dx=\int\chi_{1}^{2}|u^{\prime}|^{2}dx+2\int\chi_{1}^{\prime}\overline{u}\chi_{1}u^{\prime}dx+\int\chi^{2}_{1}|u|^{2}V-\lambda^{2}\int\chi^{2}_{1}|u|^{2}dx,$

where (3.4) was used, and consequently

$\begin{split}\int\chi_{1}^{2}|u^{\prime}|^{2}dx&\leq\int|\chi_{1}^{2}\chi
f\overline{u}|dx+2\int|\chi_{1}^{\prime}u^{\prime}\chi_{1}\overline{u}|dx+\|V\|\|\chi_{1}u\|_{L^{2}}\|(\chi_{1}u)^{\prime}\|_{L^{2}}\\\
&+(|\mathop{\rm Re}\lambda|+1)^{2}\int\chi^{2}_{1}|u|^{2}dx\\\ &\leq
C\big{(}\|\chi_{2}f\|^{2}_{L^{2}}+(|\mathop{\rm
Re}\lambda|+1)^{2}\|\chi_{2}u\|^{2}_{L^{2}}\big{)}+\frac{1}{2}\int\chi_{1}^{2}|u^{\prime}|^{2}dx.\end{split}$
(6.9)

Absorbing the last term on the right side into the left side confirms (6.8).

With (6.4) and (6.5), for $0<\mathop{\rm Im}\lambda\leq 1,\,|\mathop{\rm
Re}\lambda|\geq\lambda_{0}$, and $f\in L^{2}(\mathbb{R})$,

$\begin{split}\|\chi(H-\lambda^{2})^{-1}\chi
f\|_{\mathcal{D}}&\leq\|\chi(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}+\|H\chi(H-\lambda^{2})^{-1}\chi f\|_{L^{2}}\\\
&\leq\|\chi(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}+\|[-\partial^{2}_{x},\chi]\chi_{1}(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}\\\ &+\|\chi((H-\lambda^{2})+\lambda^{2})(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}\\\ &\leq\|\chi(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}+\|f\|_{L^{2}}\\\
&+\|[-\partial^{2}_{x},\chi]\chi_{1}(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}+(|\mathop{\rm Re}\lambda|+1)^{2}\|\chi(H-\lambda^{2})^{-1}\chi
f\|_{L^{2}}\\\ &\leq C(\|V\|,\lambda_{0})(|\mathop{\rm
Re}\lambda|+1)\|f\|_{L^{2}}.\end{split}$

This implies (1.13) for $0<\mathop{\rm Im}\lambda\leq 1,\,|\mathop{\rm
Re}\lambda|\geq\lambda_{0}$, and that continued resolvent
$L^{2}(\mathbb{R})\to\mathcal{D}$ has no poles in $\mathbb{R}\setminus\\{0\\}$
(since $\lambda_{0}>0$ is arbitrary).

Now, we turn to showing (1.13) in strips in the lower half plane. For this, we
use a resolvent identity argument due to Vodev [Vo14, Theorem 1.5], adapted to
the non-semiclassical case. It yields holomorphicity of
$\chi(H-\lambda^{2})^{-1}\chi:L^{2}(\mathbb{R})\to\mathcal{D}$ in
$|\mathop{\rm Re}\lambda|\geq\lambda_{0},\,-\varepsilon_{0}<\mathop{\rm
Im}\lambda<0$ ($\varepsilon_{0}>0$ sufficiently small), with bounds in these
strips of the form (1.12) and (1.13).

Fix $\tilde{\chi}\in C_{0}^{\infty}(\mathbb{R};[0,1])$ such that
$\tilde{\chi}=1$ near $[-R_{0},R_{0}]$ and $\chi=1$ near $\mathop{\rm
supp}\tilde{\chi}$. We are going to develop several resolvent identities, and
let us work initially with $\lambda,\mu$ such that $\mathop{\rm Im}\lambda,\
\mathop{\rm Im}\mu>0$, $|\mathop{\rm Re}\lambda|,\,|\mathop{\rm
Re}\mu|\geq\lambda_{0}$ (before sending these parameters into the the lower
half plane). By the first resolvent identity,

$\begin{split}(H-\lambda^{2})^{-1}-(H-\mu^{2})^{-1}&=(\lambda^{2}-\mu^{2})(H-\lambda^{2})^{-1}(H-\mu^{2})^{-1}\\\
&=(\lambda^{2}-\mu^{2})(H-\lambda^{2})^{-1}\tilde{\chi}(2-\tilde{\chi})(H-\mu^{2})^{-1}\\\
&+(\lambda^{2}-\mu^{2})(H-\lambda^{2})^{-1}(1-\tilde{\chi})^{2}(H-\mu^{2})^{-1}.\end{split}$

As operators on $H^{2}(\mathbb{R})$,

$\begin{split}(1-\tilde{\chi})&(-\partial^{2}_{x}-\lambda^{2})-(H-\lambda^{2})(1-\tilde{\chi})=[-\partial^{2}_{x},\tilde{\chi}]\implies\\\
&(H-\lambda^{2})^{-1}(1-\tilde{\chi})-(1-\tilde{\chi})(-\partial^{2}_{x}-\lambda^{2})^{-1}=(H-\lambda^{2})^{-1}[-\partial^{2}_{x},\tilde{\chi}](-\partial^{2}_{x}-\lambda^{2})^{-1},\end{split}$

while as operators on on $\mathcal{D}$,

$\begin{split}(-\partial^{2}_{x}-\mu^{2})&(1-\tilde{\chi})-(1-\tilde{\chi})(H-\mu^{2})=[\partial^{2}_{x},\tilde{\chi}]\implies\\\
&(1-\tilde{\chi})(H-\mu^{2})^{-1}-(-\partial^{2}_{x}-\mu^{2})^{-1}(1-\tilde{\chi})=(-\partial^{2}_{x}-\mu^{2})^{-1}[\partial^{2}_{x},\tilde{\chi}](H-\mu^{2})^{-1}.\end{split}$

Using $\chi=1$ on $\mathop{\rm supp}\tilde{\chi}$ and the three previous
calculations,

$\begin{split}\chi&(H-\lambda^{2})^{-1}\chi-\chi(H-\mu^{2})^{-1}\chi=(\lambda^{2}-\mu^{2})\chi(H-\lambda^{2})^{-1}\chi\tilde{\chi}(2-\tilde{\chi})(H-\mu^{2})^{-1}\chi\\\
&+(\lambda^{2}-\mu^{2})\chi((1-\tilde{\chi})(-\partial^{2}_{x}-\lambda^{2})^{-1}+(H-\lambda^{2})^{-1}[-\partial^{2}_{x},\tilde{\chi}](-\partial^{2}_{x}-\lambda^{2})^{-1})\\\
&\cdot((-\partial^{2}_{x}-\mu^{2})^{-1}(1-\tilde{\chi})+(-\partial^{2}_{x}-\mu^{2})^{-1}[\partial^{2}_{x},\tilde{\chi}](H-\mu^{2})^{-1})\chi\\\
&=(\lambda^{2}-\mu^{2})\chi(H-\lambda^{2})^{-1}\chi\tilde{\chi}(2-\tilde{\chi})\chi(H-\mu^{2})^{-1}\chi\\\
&+(1-\tilde{\chi}-\chi(H-\lambda^{2})^{-1}\chi[\partial^{2}_{x},\tilde{\chi}])\\\
&\cdot(\chi(-\partial^{2}_{x}-\lambda^{2})^{-1}\chi-\chi(-\partial^{2}_{x}-\mu^{2})^{-1}\chi)\\\
&\cdot(1-\tilde{\chi}+[\partial^{2}_{x},\tilde{\chi}]\chi(H-\mu^{2})^{-1}\chi).\end{split}$
(6.10)

To get the equality sign in line four of (6.10), we expanded the terms
appearing in lines two and three, and repeatedly applied the first resolvent
identity to
$(\lambda^{2}-\mu^{2})(-\partial^{2}_{x}-\lambda^{2})^{-1}(-\partial^{2}_{x}-\mu^{2})^{-1}$.

Before proceeding to use (6.10) to estimate
$\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}$ in the lower half
plane, we quote a well known estimate for the difference of continued free
resolvents (see [Vo14, Section 5] or [Sh18, Section 3.2]):

$\begin{gathered}\|\chi(-\partial^{2}_{x}-\lambda^{2})^{-1}\chi-\chi(-\partial^{2}_{x}-\mu^{2})^{-1}\chi\|_{H^{k_{1}}\to
H^{k_{2}}}\leq
C(\lambda_{0})|\lambda-\mu|\sup_{\lambda^{\prime}\in\Gamma_{\lambda,\mu}}|\lambda^{\prime}|^{k_{2}-k_{1}-1},\\\
k_{1}\in\\{0,1\\},\quad k_{2}\in\\{0,1,2\\},\quad|\mathop{\rm
Re}\lambda|,\,|\mathop{\rm Re}\mu|\geq\lambda_{0},\quad\mathop{\rm
Im}\lambda,\mathop{\rm Im}\mu\geq-1,\end{gathered}$ (6.11)

where $\Gamma_{\lambda,\mu}$ denotes the line segment connecting $\lambda$ and
$\mu$.

The identity (6.10) continues to hold after meromorphically continuing both
sides ($L^{2}\to\mathcal{D}$) to $\lambda,\,\mu\in\mathbb{C}$. Now, fix
$\mu\in\mathbb{R}$ with $|\mu|\geq\lambda_{0}$; Assume $\lambda$ in the lower
half plane is not a pole of the continued cutoff resolvent, and obeys
$|\lambda-\mu|\leq\min(1,\lambda_{0}/2,\gamma)$, for suitable $0<\gamma\ll 1$
to be chosen. Then $\|\chi(H-\mu)^{-1}\chi\|_{L^{2}\to H^{k}}\leq
C(\|V\|,\lambda_{0})|\mu|^{k-1}$, $k=0,\,1$,
$\|\chi(H-\mu)^{-1}\chi\|_{L^{2}\to\mathcal{D}}\leq
C(\|V\|,\lambda_{0})|\mu|$, (6.10), and (6.11) imply

$\begin{split}\|\chi&(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\leq
C(\|V\|,\lambda_{0})\Big{(}|\mu|+|\lambda^{2}-\mu^{2}||\mu|^{-1}\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\\\
&+\|(1-\tilde{\chi})\chi((-h^{2}\partial^{2}_{x}-\lambda^{2})^{-1}-(-\partial^{2}_{x}-\mu^{2})^{-1})\chi\|_{L^{2}\to\mathcal{D}}\\\
&+\|\chi((-h^{2}\partial^{2}_{x}-\lambda^{2})^{-1}-(-\partial^{2}_{x}-\mu^{2})^{-1})\chi\|_{L^{2}\to
H^{1}}\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\\\
&+|\mu|\|\chi((-h^{2}\partial^{2}_{x}-\lambda^{2})^{-1}-(-\partial^{2}_{x}-\mu^{2})^{-1})\chi\|_{L^{2}\to
L^{2}}\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\Big{)}\\\ &\leq
C(\|V\|,\lambda_{0})\Big{(}|\mu|+\gamma\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\Big{)}.\\\
\end{split}$ (6.12)

Fixing $\gamma$ small enough (depending on $\lambda_{0}$) allows us to absorb
the term involving $\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}$ on
the right side of (6.12) into the left side. This precludes resonances in the
region $\mathop{\rm Im}\lambda\leq 0$,
$|\lambda-\mu|\leq\min(1,\lambda_{0}/2,\gamma)$, and in this region we have
$\|\chi(H-\lambda^{2})^{-1}\chi\|_{L^{2}\to\mathcal{D}}\leq
C(\|V\|,\lambda_{0})|\mu|$.

Starting from (6.10), we use the same strategy to show (1.12) in strips in the
lower half plane. Thanks to (6.4), (6.5), and (6.11), more negative powers of
$|\mu|$ appear while making an estimate similar to (6.12), since now we need
only use operator norms $L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ or
$L^{2}(\mathbb{R})\to H^{1}(\mathbb{R})$.
∎

To conclude this section, we consider the two by two matrix operator

$G\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-i\begin{pmatrix}0&1\\\
-H&0\end{pmatrix}:\mathcal{D}\oplus L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})\oplus L^{2}(\mathbb{R}),$

which arises naturally from rewriting (1.14) as a first order system. A short
computation yields

$(G+\lambda)^{-1}=\begin{pmatrix}-\lambda(H-\lambda^{2})^{-1}&-i(H-\lambda^{2})^{-1}\\\
i\lambda^{2}(H-\lambda^{2})^{-1}+i&-\lambda(H-\lambda^{2})^{-1}\end{pmatrix}\quad\text{when
}\mathop{\rm Im}\lambda>0\text{ and }(H-\lambda^{2})^{-1}\text{exists}.$
(6.13)

The following Corollary of Theorem 1.6 is essentially well-known, and is an
important input to the proof of Theorem 7.1 in Section 7.

######  Corollary 6.1.

Let $\chi\in C^{\infty}_{0}(\mathbb{R};[0,1])$ be identically one near
$[-R_{0},R_{0}]$. The operator

$\begin{split}\chi(G+&\lambda)^{-1}\chi\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\\
&\begin{pmatrix}-\lambda\chi(H-\lambda^{2})^{-1}\chi&-i\chi(H-\lambda^{2})^{-1}\chi\\\
i\lambda^{2}\chi(H-\lambda^{2})^{-1}\chi+i\chi^{2}&-\lambda\chi(H-\lambda^{2})^{-1}\chi\end{pmatrix}:L^{2}(\mathbb{R})\oplus
L^{2}(\mathbb{R}),\to\mathcal{D}\oplus L^{2}(\mathbb{R})\end{split}$ (6.14)

continues meromorphically from $\mathop{\rm Im}\lambda>0$ to $\mathbb{C}$,
without poles on $\mathbb{R}\setminus\\{0\\}$. For any $\lambda_{0}>0$, there
exist $C,\,\lambda_{0},\,\varepsilon_{0}>0$ so that

$\|\chi(G+\lambda)^{-1}\chi\|_{H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})\to
H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})}\leq C,\qquad|\mathop{\rm
Re}\lambda|\geq\lambda_{0},\,|\mathop{\rm Im}\lambda|\leq\varepsilon_{0}.$
(6.15)

If $\chi(G+\lambda)^{-1}\chi$ has a pole at $\lambda=0$, it is a simple pole.
More precisely, if $w_{0}\in H^{1}(\mathbb{R})$ and $w_{1}\in
L^{2}(\mathbb{R})$, then

$\lim_{\lambda\to 0}\lambda\chi(G+\lambda)^{-1}\chi\begin{pmatrix}w_{0}\\\
w_{1}\end{pmatrix}=\begin{pmatrix}-i\lim_{\lambda\to
0}\lambda\chi(H-\lambda^{2})^{-1}\chi w_{1}\\\
0\end{pmatrix}=\begin{pmatrix}\langle\chi u_{0},w_{1}\rangle_{L^{2}}\chi
u_{0}\\\ 0\end{pmatrix}.$ (6.16)

for some real valued $u_{0}\in H^{1}_{\operatorname{loc}}(\mathbb{R})\cap
H^{2}_{\operatorname{loc}}(\mathbb{R}\setminus[-R_{0},R_{0}])$ with $Hu_{0}=0$
in the sense of distributions.

###### Proof.

By the blackbox formalism (see [DyZw19, Definition 4.1 and Theorem 4.4]) and
Theorem 1.6, $\chi(H-\lambda^{2})^{-1}\chi:L^{2}(\mathbb{R})\to\mathcal{D}$
continues meromorphically from $\mathop{\rm Im}\lambda>0$ to $\mathbb{C}$, and
has no poles in $\mathbb{R}\setminus\\{0\\}$. This implies that each entry of
(6.14) continues meromorphically as an operator between the appropriate
spaces, again without poles in $\mathbb{R}\setminus\\{0\\}$.

With (1.12) already in hand, to establish (6.15), we need to show for any
$\lambda_{0}>0$, there exist $C,\,\varepsilon_{0}>0$ so that

$\displaystyle\|\lambda^{2}\chi(H-\lambda^{2})^{-1}\chi+\chi^{2}\|_{H^{1}\to
L^{2}}=\|\chi H(H-\lambda^{2})^{-1}\chi\|_{H^{1}\to L^{2}}\leq C,$ (6.17)
$\displaystyle\|\lambda\chi(H-\lambda^{2})^{-1}\chi\|_{H^{1}\to H^{1}}\leq C,$
(6.18)

for $|\mathop{\rm Re}\lambda|\geq\lambda_{0}$ and $|\mathop{\rm
Im}\lambda|\leq\varepsilon_{0}$. First, we first prove (6.17) for
$|\mathop{\rm Re}\lambda|\geq\lambda_{0}$ and $0<\mathop{\rm
Im}\lambda\leq\varepsilon_{0}$, and then handle the remaining cases.

Let us use the notation

$u=(H-\lambda^{2})^{-1}\chi f\in\mathcal{D},\quad f\in
H^{1}(\mathbb{R}),\quad|\mathop{\rm Re}\lambda|\geq\lambda_{0}\text{ and
}\mathop{\rm Im}\lambda>0.$ (6.19)

Let $\chi_{1}\in C_{0}^{\infty}(\mathbb{R})$ with $\chi_{1}=1$ near
$\mathop{\rm supp}\chi$. As we showed in the proof of Lemma 3.1, the form
domain of $(H,\mathcal{D})$ is $H^{1}(\mathbb{R})$, so there exists a sequence
$f_{k}\in\mathcal{D}$ converging to $f$ in $H^{1}(\mathbb{R})$, and
corresponding functions
$u_{k}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(H-\lambda^{2})^{-1}\chi
f_{k}$ converging to $u$ in $(\mathcal{D},\|\cdot\|_{\mathcal{D}})$. Since
$Hu_{k}=(H-\lambda^{2})^{-1}\chi_{1}H\chi f_{k}$,

$\begin{split}\|\chi Hu\|_{L^{2}}=\lim_{k\to\infty}\|\chi
Hu_{k}\|_{L^{2}}\leq\lim_{k\to\infty}\|\chi_{1}(H-\lambda^{2})^{-1}\chi_{1}H\chi
f_{k}\|_{L^{2}}.\end{split}$ (6.20)

Furthermore, by (3.4), there exists $C(\|V\|)>0$ depending on $\|V\|$ so that
for any $v\in L^{2}(\mathbb{R})$,

$\begin{split}|\langle\chi_{1}(H-\lambda^{2})^{-1}\chi_{1}H\chi
f_{k},v\rangle_{L^{2}}|&=|\langle H\chi
f_{k},\chi_{1}(H-(-\overline{\lambda})^{2})^{-1}\chi_{1}v\rangle_{L^{2}}\\\
&\leq C(\|V\|)\|\chi
f_{k}\|_{H^{1}}\|\chi_{1}(H-(-\overline{\lambda})^{2})^{-1}\chi_{1}v\|_{H^{1}}.\end{split}$

Since (1.12) gives
$\|\chi_{1}(H-(-\overline{\lambda})^{2})^{-1}\chi_{1}v\|_{H^{1}}\leq
C(\|V\|,\lambda_{0})\|v\|_{L^{2}}$ (provided $\mathop{\rm Im}\lambda$ is
small), we conclude $\|\chi_{1}(H-\lambda^{2})^{-1}\chi_{1}H\chi
f_{k}\|_{L^{2}}\leq C(\|V\|,\lambda_{0})\|\chi f_{k}\|_{H^{1}}$. Returning to
(6.20), we now find

$\|\chi Hu\|_{L^{2}}\leq C(\|V\|,\lambda_{0})\lim_{k\to\infty}\|\chi
f_{k}\|_{H^{1}}\leq C(\|V\|,\lambda_{0})\|f\|_{H^{1}},\qquad|\mathop{\rm
Re}\lambda|\geq\lambda_{0},\,0<\mathop{\rm Im}\lambda\leq\varepsilon_{0}.$

In turn,

$\begin{gathered}\|\lambda^{2}\chi(H-\lambda^{2})^{-1}\chi\|_{H^{1}\to
L^{2}}=\|\chi H(H-\lambda^{2})^{-1}\chi-\chi^{2}\|_{H^{1}\to L^{2}}\leq
C(\|V\|,\lambda_{0}),\\\ |\mathop{\rm Re}\lambda|\geq\lambda_{0}\text{ and
}0<\mathop{\rm Im}\lambda\leq\varepsilon_{0}.\end{gathered}$ (6.21)

which is (6.17) for $|\mathop{\rm Re}\lambda|\geq\lambda_{0}$ and
$0<\mathop{\rm Im}\lambda\leq\varepsilon_{0}$.

Next, with $u$ as in (6.19), we slightly modify the method of estimation in
(6.9), this time finding

$\int\chi_{1}^{2}|u^{\prime}|^{2}dx\leq
C(\|V\|,\lambda_{0})\big{(}|\mathop{\rm
Re}\lambda|+1)^{-2}\|\chi_{2}f\|^{2}_{L^{2}}+(|\mathop{\rm
Re}\lambda|+1)^{2}\|\chi_{2}u\|^{2}_{L^{2}}\big{)},$

and where we recall from (6.7) that $\chi_{2}=1$ on $\mathop{\rm
supp}\chi_{1}$. Combining this with (6.21) establishes (6.18) when
$|\mathop{\rm Re}\lambda|\geq\lambda_{0}$ and $0<\mathop{\rm
Im}\lambda\leq\varepsilon_{0}$

To show (6.18) and (6.17) hold for $|\mathop{\rm Re}\lambda|\geq\lambda_{0}$
and $-\varepsilon_{0}<\mathop{\rm Im}\lambda<0$, we revisit (6.10) and
multiply by the appropriate power of $\lambda$. We then perform an estimate
similar to (6.12). As needed, we invoke (6.11) and
$\|\mu^{2}\chi(H-\mu^{2})^{-1}\chi\|_{H^{1}\to
L^{2}},\,\|\mu\chi(H-\mu^{2})^{-1}\chi\|_{H^{1}\to H^{1}}\leq C$
($\mu\in\mathbb{R},|\mu|\geq\lambda_{0}$).

Finally, to show (6.16), we proceed as in the proof of [DyZw19, Theorem 2.7].
We omit the details, but take care to note that this argument does require
that for each $x_{0}\in\mathbb{R}\setminus[-R_{0},R_{0}]$ and
$a,b\in\mathbb{C}$, there is a unique solution $f$ to $Hf=0$ satisfying
$f(x_{0})=a$ and $f^{\prime}(x_{0})=b$; Even though $V$ is only a measure in
our setting, such well-posedness still holds for the initial value problem,
see [EcTe13, Theorem 3.1]. (In general it is not necessary to prescribe the
initial conditions outside the support of the measure, but this is sufficient
for our purpose). The result is that near $\lambda=0$,

$\chi(H-\lambda^{2})^{-1}\chi w_{1}=\frac{i}{\lambda}\langle\chi
u_{0},w_{1}\rangle_{L^{2}}\chi u_{0}+A(\lambda)w_{1},\qquad w_{1}\in
L^{2}(\mathbb{R}),$

where $A(\lambda):L^{2}(\mathbb{R})\to\mathcal{D}$ is holomorphic near zero,
and for some $u_{0}\in H^{1}_{\operatorname{loc}}(\mathbb{R})\cap
H^{2}_{\operatorname{loc}}(\mathbb{R}\setminus[-R_{0},R_{0}])$ with $Hu_{0}=0$
in the sense of distributions. Hence we have (6.16).
∎

##  7\. wave decay

In this Section, we combine Corollary 6.1 with an argument similar to those
appearing in [Vo99, Section 3] and [DaSh22, Section 4]. We establish
exponential local energy decay, modulo negative eigenvalues and a possible
zero resonance, for solutions of the wave equation (1.14).

First, we represent the solution to (1.14) via the spectral theorem of for
self-adjoint operators. Additionally, recall that the proof of Lemma 3.1 shows
the form domain of $(H,\mathcal{D})$ (i.e., the domain of $|H|^{1/2}$) is
$H^{1}(\mathbb{R})$. Thus, given initial conditions $w_{0}\in\mathcal{D}$,
$w_{1}\in H^{1}(\mathbb{R})$, the unique function $w\in
C^{2}((0,\infty),\mathcal{H})$ with $w(0)=w_{0}$, $\partial_{t}w(0)=w_{1}$,
$w(t)\in\mathcal{D}(H)$ for all $t>0$, and $\partial^{2}_{t}w(t)+Hw(t)=0$, is

$\begin{gathered}w(t)=w(\cdot,t)=\mathbf{1}_{\geq
0}(H)w(\cdot,t)+\mathbf{1}_{<0}(H)w(\cdot,t),\\\ \mathbf{1}_{\geq
0}(H)w(\cdot,t)=\mathbf{1}_{\geq
0}(H)\big{(}\cos(t|H|^{1/2})w_{0}+\frac{\sin(t|H|^{1/2})}{|H|^{1/2}}w_{1}\big{)},\\\
\mathbf{1}_{<0}(H)w(\cdot,t)=\mathbf{1}_{<0}(H)\big{(}\cos(it|H|^{1/2})w_{0}+\frac{\sin(it|H|^{1/2})}{i|H|^{1/2}}w_{1}\big{)}.\\\
\end{gathered}$ (7.1)

######  Theorem 7.1.

Suppose $w_{0}\in\mathcal{D},\,w_{1}\in H^{1}(\mathbb{R})$, and $\mathop{\rm
supp}w_{0},\,\mathop{\rm supp}w_{1}\subseteq(-R,R)$ for some $R>0$. Let $w(t)$
be given by (7.1). For any $R_{1}>0$, there exist $C,c>0$ so that

$\begin{gathered}\|\mathbf{1}_{\geq
0}(H)w(\cdot,t)-w_{\infty}(x)\|_{H^{1}(-R_{1},R_{1})}+\|\partial_{t}\mathbf{1}_{\geq
0}(H)w(\cdot,t)\|_{L^{2}(-R_{1},R_{1})}\\\ \leq
Ce^{-ct}(\|w_{0}\|_{H^{1}(\mathbb{R})}+\|w_{1}\|_{L^{2}(\mathbb{R})}),\qquad
t>0.\end{gathered}$ (7.2)

Furthermore, if $\chi\in C^{\infty}_{0}(\mathbb{R};[0,1])$ is identically one
near $[-R_{1},R_{1}]\cup[-R_{0},R_{0}]\cup[-R,R]$, then the function
$w_{\infty}(x)$ may be written as

$w_{\infty}(x)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\chi(x)u_{0}(x)\int_{\mathbb{R}}\chi
u_{0}w_{1},$ (7.3)

for some real valued $u_{0}\in H^{1}_{\operatorname{loc}}(\mathbb{R})\cap
H^{2}_{\operatorname{loc}}(\mathbb{R}\setminus[-R_{0},R_{0}])$ with $Hu_{0}=0$
in the sense of distributions (and if the continued operator (6.14) does not
have a pole at $\lambda=0$, we may take $u_{0}\equiv 0$.)

###### Proof.

Choose $\chi\in C_{0}^{\infty}(\mathbb{R};[0,1])$ such that $\chi=1$ near
$[-R_{1},R_{1}]\cup[-R,R]\cup[R_{0},R_{0}]$ (where as before $\mathop{\rm
supp}V\subseteq[-R_{0},R_{0}]$). From Corollary 6.1, for any $\lambda_{0}>0$,
there exist $C$, $\varepsilon_{0}>0$ such that

$\|\chi(G+\lambda)^{-1}\chi f\|\leq C\|f\|,$ (7.4)

whenever $|\mathop{\rm Re}\lambda|\geq\lambda_{0}$ and $|\mathop{\rm
Im}\lambda|\leq\varepsilon_{0}$, where here and for the rest of this Section,
all norms are $H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})$ unless otherwise
specified.

We have

$\begin{gathered}\mathbf{1}_{\geq 0}(H)w(t)=\mathbf{1}_{\geq
0}(H)\big{(}\cos(t|H|^{1/2})w_{0}+\sin(t|H|^{1/2})|H|^{-1/2}w_{1}\big{)},\\\
\partial_{t}\mathbf{1}_{\geq 0}(H)w(t)=\mathbf{1}_{\geq
0}(H)\big{(}-\sin(t|H|^{1/2})|H|^{1/2}w_{0}+\cos(t|H|^{1/2})w_{1}\big{)},\\\
\partial_{t}^{2}\mathbf{1}_{\geq 0}(H)w(t)=-H\mathbf{1}_{\geq
0}(H)w(t).\end{gathered}$

Consequently, after defining

$f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\begin{array}[]{c}w_{0}\\\
w_{1}\end{array}\right),\qquad
U(t)f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\begin{array}[]{c}\mathbf{1}_{\geq
0}(H)w(t)\\\ \partial_{t}\mathbf{1}_{\geq 0}(H)w(t)\end{array}\right),$

we have

$\|U(t)f\|\leq C\|f\|,\qquad\partial_{t}U(t)f=iGU(t)f,\qquad
U(t)U(s)f=U(t+s)f,$ (7.5)

for all real $t$ and $s$, and for some $C>0$ independent of $t$ and $f$.

Take $\varphi\in C^{\infty}(\mathbb{R};[0,1])$ which is $0$ near $(-\infty,1]$
and $1$ near $[2,\infty)$ and put

$W(t)f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\varphi(t)U(t)f=\int_{\mathop{\rm
Im}\lambda=\varepsilon}e^{-it\lambda}\check{W}(\lambda)\,d\lambda,\qquad\check{W}(\lambda)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{2\pi}\int_{\mathbb{R}}e^{is\lambda}W(s)fds,\qquad\varepsilon>0.$

We compute $\partial_{t}W(t)f=\varphi^{\prime}(t)U(t)f+iGW(t)f$, and therefore
find

$W(t)f=\int_{\mathop{\rm
Im}\lambda=\varepsilon}e^{-it\lambda}(G+\lambda)^{-1}(i\varphi^{\prime}Uf)\check{~{}}(\lambda)\,d\lambda,\qquad\varepsilon>0.$
(7.6)

Since $w_{0}$, $w_{1}$ and $V$ have compact support, finite propagation speeds
holds for the solution (7.1). Therefore, increasing $R>0$ if necessary, we
have that, $x\mapsto U(t)f$ is supported in $(-R,R)$ for all $t\in[0,2]$. By
continuity of integration, the same is true of
$x\mapsto(i\varphi^{\prime}Uf)\check{~{}}(\lambda)$ for every $\lambda$. Hence
$\lambda\mapsto(i\varphi^{\prime}Uf)\check{~{}}(\lambda)$ is entire and
rapidly decaying as $|\mathop{\rm Re}\lambda|\to\infty$ with $|\mathop{\rm
Im}\lambda|$ remaining bounded and further
$(i\varphi^{\prime}Uf)\check{~{}}(\lambda)=\chi(i\varphi^{\prime}Uf)\check{~{}}(\lambda)$.

By (7.4), there exists $\varepsilon>0$ small enough so that, within the strip
$|\mathop{\rm Im}\lambda|<2\varepsilon$, either $\chi(G+\lambda)^{-1}\chi$ has
no poles, or just a pole at $\lambda=0$. Deforming the contour in (7.6), by
the residue theorem, we find

$\begin{split}\chi W&(t)f\\\ &=-2\pi i\mathop{\rm
Res}{}_{\lambda=0}(e^{-it\lambda}\chi(G+\lambda)^{-1}\chi(i\varphi^{\prime}Uf)\check{~{}}(\lambda))+\int_{\mathop{\rm
Im}\lambda=-\varepsilon}e^{-it\lambda}\chi(G+\lambda)^{-1}\chi(i\varphi^{\prime}Uf)\check{~{}}(\lambda)d\lambda.\\\
&=\lim_{\lambda\to
0}\lambda\chi(G+\lambda)^{-1}\chi\int_{\mathbb{R}}\varphi^{\prime}(s)U(s)f\,ds+\int_{\mathop{\rm
Im}\lambda=-\varepsilon}e^{-it\lambda}\chi(G+\lambda)^{-1}\chi(i\varphi^{\prime}Uf)\check{~{}}(\lambda)d\lambda.\end{split}$

To simplify this, use (6.16) and put

$W_{1}(t)f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{-\infty}^{\infty}e^{-it\lambda}\chi(G+\lambda-i\varepsilon)^{-1}\chi(i\varphi^{\prime}Uf)\check{~{}}(\lambda-i\varepsilon)\,d\lambda,$

to obtain

$\chi W(t)f=\begin{pmatrix}\chi
u_{0}\int_{\mathbb{R}}\int_{0}^{2}\chi(x)u_{0}(x)\varphi^{\prime}(s)\partial_{s}w(s,x)dsdx\\\
0\end{pmatrix}+e^{-\varepsilon t}W_{1}(t)f.$

To simplify the first term, we integrate by parts in $s$, using
$\varphi^{\prime}=-(1-\varphi)^{\prime}$, to obtain

$\int_{\mathbb{R}}\int_{0}^{2}\chi(x)u_{0}(x)\varphi^{\prime}(s)\partial_{s}w(s,x)\,ds\,dx=\int_{\mathbb{R}}\chi
u_{0}w_{1}+\int_{\mathbb{R}}\int_{0}^{2}\chi(x)(1-\varphi(s))\partial_{s}^{2}w(s,x)\,ds\,dx.$

Now observe that $\partial_{s}^{2}w=-Hw$, $\chi u_{0}\in\mathcal{D}$, so

$\langle\chi u_{0},Hw(s)\rangle_{L^{2}}=\langle H\chi
u_{0},w(s)\rangle_{L^{2}}=\langle([H,\chi]+H)u_{0},w(s)\rangle_{L^{2}}=0,\qquad\text{
for }s\in[0,2],$

the last equality following from $\chi=1$ near $[-R,R]$ and $\mathop{\rm
supp}w(s)\subseteq(-R,R)$ for $s\in[0,2]$). Thus

$\chi W(t)f=\begin{pmatrix}\langle\chi u_{0},w_{1}\rangle_{L^{2}}\chi u_{0}\\\
0\end{pmatrix}+e^{-\varepsilon t}W_{1}(t)f.$

It now suffices to show that

$\|W_{1}(t)f\|\leq Ce^{\varepsilon t/2}\|f\|.$

To prove this, we first use Plancherel’s theorem, along with the fact that by
(7.5), the operator norm $\|U(t)\|_{H^{1}(\mathbb{R})\oplus
L^{2}(\mathbb{R})\to H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})}$ is uniformly
bounded for all $t\in\mathbb{R}$, as well as the fact that by Corollary 6.1,
the operator norm
$\|\chi(G+\lambda-i\varepsilon)^{-1}\chi\|_{H^{1}(\mathbb{R})\oplus
L^{2}(\mathbb{R})\to H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})}$ is uniformly
bounded for all $\lambda\in\mathbb{R}$, to obtain

$\begin{split}\int\|W_{1}(t)f\|^{2}\,dt&=C\int\|\chi(G+\lambda-i\varepsilon)^{-1}\chi(\varphi^{\prime}Uf)\check{~{}}(\lambda-i\varepsilon)\|^{2}\,d\lambda\\\
&\leq
C_{\varepsilon}\int\|(\varphi^{\prime}Uf)\check{~{}}(\lambda-i\varepsilon)\|^{2}\,d\lambda\\\
&=C_{\varepsilon}\int e^{2\varepsilon
t}\|\varphi^{\prime}(t)U(t)f\|^{2}\,dt\leq
C_{\varepsilon}\|f\|^{2}.\end{split}$ (7.7)

Next, let $\tilde{\chi}\in C^{\infty}_{0}(\mathbb{R};[0,1])$ with
$\tilde{\chi}=1$ on $\mathop{\rm supp}\chi$. Observe that

$G\chi(G+\lambda)^{-1}\chi=[G,\chi]\tilde{\chi}(G+\lambda+i\varepsilon)^{-1}\tilde{\chi}\chi-\lambda\chi(G+\lambda)^{-1}\chi+\chi^{2}$
(7.8)

holds initially for $\mathop{\rm Im}\lambda\gg 1$ by (6.13), and continues
meromorphically to $\mathbb{C}$ by (6.14). In particular, decreasing
$\varepsilon$ if necessary, (6.15) implies that (7.8) holds for everywhere in
$|\mathop{\rm Im}\lambda|<2\varepsilon$, except possibly at $\lambda=0$.
Therefore, setting,

$\widetilde{W}_{1}(t)f\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{-\infty}^{\infty}e^{-it\lambda}\tilde{\chi}(G+\lambda-i\varepsilon)^{-1}\tilde{\chi}(i\varphi^{\prime}Uf)\check{~{}}(\lambda-i\varepsilon)\,d\lambda,$

we have

$(\partial_{t}-iG)W_{1}(t)f=-i[G,\chi]\widetilde{W}_{1}(t)f+\varepsilon
W_{1}(t)f-i\int
e^{-it\lambda}(i\varphi^{\prime}Uf)\check{~{}}(\lambda-i\varepsilon)\,d\lambda=\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}W_{2}(t)f.$

Integrating both sides of

$\begin{split}\partial_{s}(U(t-s)W_{1}(s)f)&=-iGU(t-s)W_{1}(s)f+U(t-s)\big{(}iGW_{1}(s)f+W_{2}(s)f\big{)}\\\
&=U(t-s)W_{2}(s)f\end{split}$

from $s=0$ to $s=t$ gives

$W_{1}(t)f=U(t)W_{1}(0)f+U(t)\int_{0}^{t}U(-s)W_{2}(s)f\,ds.$

Thus

$\|W_{1}(t)f\|\leq C\big{(}\|f\|+\int_{0}^{t}\|W_{2}(s)f\|ds\big{)}\leq
C\big{(}\|f\|+t^{1/2}\Big{(}\int_{0}^{t}\|W_{2}(s)f\|^{2}ds\Big{)}^{1/2}\big{)}.$

Now check that, since $\|[G,\chi]\widetilde{W}_{1}(t)f\|\leq
C\|\widetilde{W}_{1}(t)f\|$, calculating as in (7.7), we obtain
$\int\|W_{2}(s)f\|^{2}\,ds\leq C\|f\|^{2}$, and hence

$\|W_{1}(t)f\|\leq C(1+t^{1/2})\|f\|$

as desired.
∎

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