# Semiclassical resolvent bounds for compactly supported radial potentials

Kiril Datchev  Department of Mathematics, Purdue University, West Lafayette,
IN, 47907-2067, USA<EMAIL_ADDRESS>,  Jeffrey Galkowski  Department of
Mathematics, University College London, London, WC1H 0AY, UK
<EMAIL_ADDRESS>and  Jacob Shapiro  Department of Mathematics,
University of Dayton, Dayton, OH 45469-2316, USA<EMAIL_ADDRESS>
###### Abstract.

We employ separation of variables to prove weighted resolvent estimates for
the semiclassical Schrödinger operator $-h^{2}\Delta+V(|x|)-E$ in dimension
$n\geq 2$, where $h,\,E>0$, and $V:[0,\infty)\to\mathbb{R}$ is $L^{\infty}$
and compactly supported. The weighted resolvent norm grows no faster than
$\exp(Ch^{-1})$, while an exterior weighted norm grows $\sim h^{-1}$. We
introduce a new method based on the Mellin transform to handle the two-
dimensional case.

##  1\. Introduction and statement of results

Let
$\Delta\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{j=1}^{n}\partial^{2}_{x_{j}}$
be the Laplacian on $\mathbb{R}^{n}$, $n\geq 2$. We consider the semiclassical
Schrödinger operator on $L^{2}(\mathbb{R}^{n})$ given by

$Pu=P(h)u\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-h^{2}\Delta
u+V(|x|)u,$

where $V\colon[0,\infty)\to\mathbb{R}$ is $L^{\infty}$ and compactly
supported, and $h>0$ is a semiclassical parameter. Then $P\colon
H^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$ is self-adjoint, and the
resolvent $(P-z)^{-1}$ is bounded on $L^{2}(\mathbb{R}^{n})$ for all
$z\in\mathbb{C}\setminus\mathbb{R}$. Throughout the article, we let
$r\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|x|$, and
$\langle
x\rangle\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(1+|x|^{2})^{1/2}$.

Our first result is an exponential upper bound on the limiting absorption
resolvent.

######  Theorem 1.

Let $n\geq 2$. Fix $[E_{\emph{min}},E_{\emph{max}}]\subseteq(0,\infty)$ and
$1/2<s\leq 1$. There exist $C,\,h_{0}>0$, such that

$\|\langle x\rangle^{-s}(P-E-i\varepsilon)^{-1}\langle
x\rangle^{-s}\|_{L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})}\leq e^{C/h},$
(1.1)

for all $\varepsilon>0$, $h\in(0,h_{0}]$, and
$E\in[E_{\emph{min}},E_{\emph{max}}]$.

In addition, we prove a ‘non-trapping’ type estimate for the resolvent in the
exterior of a large ball. Let

$R_{0}=R_{0}(V)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\\{r\in[0,\infty)\,:\,r\in\mathop{\rm
ess\,supp}V\\},$

and define

$\begin{split}M_{0}&=M_{0}(V,E)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\\
&\inf\\{m>0\mid V(r)+mr^{-2}-E\geq 0,\text{ for almost all }r\text{ in a
neighborhood of $(0,R_{0}(V)]$}\\}.\end{split}$ (1.2)

For example, if $V$ is the characteristic function of the interval
$(0,R_{0}]$, then $M_{0}=ER^{2}_{0}$. If $V$ is continuous at $R_{0}$,
$M_{0}=\mathop{\rm ess\,sup}_{[0,R_{0}]}r^{2}(E-V)$. Note that always
$M_{0}\geq ER_{0}^{2}$ because we require $M_{0}r^{-2}-E\geq 0$ for some
$r>R_{0}$. Finally, put

$R_{1}=R_{1}(V,E)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{M_{0}(V,E)/E}.$
(1.3)

######  Theorem 2.

Let $n\geq 2$. Fix $[E_{\emph{min}},E_{\emph{max}}]\subseteq(0,\infty)$,
$1/2<s\leq 1$ and $R>\sup_{E\in[E_{\min},E_{\max}]}R_{1}(V,E)$. There exist
$C,\,h_{0}>0$, such that

$\|\langle x\rangle^{-s}\mathbf{1}_{\geq
R}(P-E-i\varepsilon)^{-1}\mathbf{1}_{\geq R}\langle
x\rangle^{-s}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\leq\frac{C}{h},$ (1.4)

for all $\varepsilon>0$, $h\in(0,h_{0}]$, and
$E\in[E_{\emph{min}},E_{\emph{max}}]$, where $\mathbf{1}_{\geq R}$ is the
characteristic function of $\\{x\in\mathbb{R}^{n}\colon|x|\geq R\\}$.

Theorem 2 is optimal in the sense that [DaJi20, Theorem 3] shows that (1.4) is
false in general when $R<\sqrt{M_{0}/E}$. For example, if $V\in
C_{0}^{\infty}([0,1))$, $E<\max(-r^{2}V)$, and $R<\sqrt{M_{0}/E}$, then (by
[DaJi20, (2.9) and (4.10)]) the left hand side of (1.4) is bounded below by
$e^{1/Ch}$ for $h$ tending to zero along a sequence of positive values.

The novelty of Theorems 1 and 2 is that they bound the weighted resolvent for
an arbitrary compactly supported, radial $L^{\infty}$ potential. The
$h$-dependencies on the right sides of (1.4) and (1.1) are sharp in general,
see [DDZ15] and [DaJi20] for exponential lower bounds, and recall that the
free resolvent ($V\equiv 0$) has a $Ch^{-1}$ lower bound (to see this,
consider $u=e^{i\sqrt{E}h^{-1}x_{1}}\chi(x)$ for some $0\not\equiv\chi\in
C_{0}^{\infty}(\mathbb{R}^{n})$).

Vodev’s work [Vo21] shows, for dimension $n\geq 3$, a bound like (1.1) still
holds for radial potentials decaying like $\langle r\rangle^{-\delta}$,
$\delta>2$, except with the right side replaced by $e^{Ch^{-4/3}}$ (bounds
with additional losses hold for $V$ decaying more slowly). For $V\in
L^{\infty}(\mathbb{R}^{n};\mathbb{R})$ not necessarily radial, $n\geq 2$, with
$V=O(\langle r\rangle^{-\delta})$, $\delta>2$, the best known weighted
resolvent upper bound is $e^{Ch^{-4/3}\log(h^{-1})}$ [GaSh20]. In dimension
$n\geq 2$, it is an open problem to determine the optimal $h$-dependence of
the resolvent for $V\in L^{\infty}$. In contrast, when $n=1$, an $e^{Ch^{-1}}$
bound holds even if $V\in L^{1}(\mathbb{R};\mathbb{R})$ [DaSh20]. As far as
the authors are aware, Theorem 2 is the first exterior estimate for any class
of $L^{\infty}$ potentials in dimension higher than one.

Proofs of semiclassical resolvent estimates have a long history and are an
active research topic. Burq [Bu98] was the first to show an $e^{Ch^{-1}}$
bound for smooth perturbations of the Laplacian on $\mathbb{R}^{n}$. Several
extensions followed [Vo00, Bu02, Sj02, CaVo02]. The exterior bound (1.4) was
first proved by Cardoso and Vodev [CaVo02], refining a preliminary estimate of
Burq [Bu02]. More recent works on resolvent estimates in lower regularity
include [Da14, Vo14, RoTa15, KlVo19, Sh19, Vo19, GaSh20, GaSh21, Sh20, Vo20a,
Vo20b, Vo20c, Vo21].

Stronger bounds on the resolvent are known when $V\in
C^{\infty}_{0}(\mathbb{R}^{n};\mathbb{R})$ and conditions are imposed on the
classical flow $\Phi(t)=\exp
t(2\xi\partial_{x}-\partial_{x}V(x)\partial_{\xi})$ (note that $\Phi(t)$ may
be undefined in our case). The key dynamical object is the trapped set
$\mathcal{K}(E)$ at energy $E>0$, defined as the set of $(x,\xi)\in
T^{*}\mathbb{R}^{n}$ such that $|\xi|^{2}+V(x)=E$ and $|\Phi(t)(x,\xi)|$ is
bounded as $|t|\to\infty$. If $\mathcal{K}(E)=\emptyset,$ that is, if $E$ is
nontrapping, Robert and Tamura [RoTa87] showed the weighted resolvent is
bounded by $Ch^{-1}$. We may thus think of (1.4) as a low regularity analog;
it says that applying cutoffs supported sufficiently far from zero removes the
losses from (1.1) due to trapping.

Theorem 1 allows one to obtain, at high frequency, bounds on and resonance
free regions for, the meromorphic continuation of the cutoff resolvent of the
operator $-c^{2}(|x|)\Delta$ on $L^{2}(\mathbb{R}^{n},c^{-2}dx)$. Here $c\in
L^{\infty}([0,\infty);(0,\infty))$ is called the wavespeed and satisfies

$\displaystyle\mathop{\rm supp}(1-c)\text{ is compact},$ (1.5) $\displaystyle
c_{0}<c(r)<c_{1}\text{ for some $c_{0},c_{1}>0$ and for all
$r\in[0,\infty)$.}$ (1.6)

More precisely, one obtains,

######  Theorem 3.

Let $n\geq 2$, and suppose $c\in L^{\infty}([0,\infty);(0,\infty))$ obeys
(1.5) and (1.6). For each $\chi\in C^{\infty}_{0}(\mathbb{R}^{n})$, there
exist constants $C_{1},C_{2},M>0$ such that the cutoff resolvent
$\chi
R(\lambda)\chi\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\chi(-c^{2}\Delta-\lambda^{2})^{-1}\chi$
continues analytically from $\mathop{\rm Im}\lambda>0$ into the set
$\\{\lambda\in\mathbb{C}:|\mathop{\rm Re}\lambda|>M,\,\mathop{\rm
Im}\lambda>-e^{C_{2}|\mathop{\rm Re}\lambda|}\\}$, where it satisfies the
bound

$\|\chi R(\lambda)\chi\|_{L^{2}(\mathbb{R}^{n},\,c^{-2}dx)\to
L^{2}{(\mathbb{R}^{n},\,c^{-2}dx)}}\leq e^{C_{1}|\mathop{\rm Re}\lambda|}.$
(1.7)

The proof of Theorem 3 is the same as the proof of [Sh18, Proposition 5.1],
and is seen by identifying $V=1-c^{-2}$, $h=|\mathop{\rm Re}\lambda|^{-1}$ and
applying (1.1).

Theorem 3 implies logarithmic local energy decay for the wave equation

$\begin{cases}(\partial_{t}^{2}-c^{2}(x)\Delta)u(x,t)=0,&(x,t)\in\mathbb{R}^{n}\times(0,\infty),\,n\geq
2,\\\ u(x,0)=u_{0}(x)\in H^{2}(\mathbb{R}^{n}),\\\
\partial_{t}u(x,0)=u_{1}(x)\in H^{1}(\mathbb{R}^{n}),\end{cases}$ (1.8)

where the initial data are compactly supported. Such a decay rate was first
proved by Burq [Bu98] for $c$ smooth, and allowing for a smooth Dirichlet
obstacle.

The assumption that $c=1$ outside of a compact set is necessary not only to
establish (1.7), but also to study the low-frequency behavior of the cutoff
resolvent. Under this assumption, one can see that this behavior is exactly
the same as the case $c\equiv 1$ [Sh18, Proposition 4.1], which in turn is
well-known (see, e.g., [Vo01, Section 1.1]). With both the low and high
frequency behavior of the $\chi R(\lambda)\chi$ illuminated, a logarithmic
local energy decay rate for the solution of (1.8) follows exactly as in [Sh18,
Sections 6 and 7].

###  1.1. Ideas from the proofs

When $V:\mathbb{R}^{n}\to\mathbb{R}$ has limited regularity, proofs of
resolvent estimates typically proceed by a modified positive commutator
strategy, see e.g., [GaSh20, Section 2]. The potential is treated as a
perturbation because it cannot be differentiated in some or all directions.
The issue one needs to overcome is that, in the positive commutator scheme,
the potential appears in a term without an apparent sign (even though $V$ is
real-valued), which in turn is difficult to control as $h\to 0^{+}$.
Controlling this term results in an estimate from above by
$e^{Ch^{-4/3}\log(h^{-1})}$.

Due to the work of Meshkov [Me92] on the Landis conjecture, one knows that the
exponent $h^{-4/3}$ is optimal for compactly supported complex valued
potentials (see Appendix C for an explanation). Therefore, any improvement
upon the exponent $h^{-4/3}$ in the resolvent estimate for an $L^{\infty}$
compactly supported potential must involve additional assumptions on the
potential $V$ (e.g. reality/radiality).

The radial symmetry of the potential is used in two ways. First, it allows us
to study the high angular momenta separately from the low angular momenta. In
particular, we use a spherical energy type estimate to obtain resolvent
estimates for low frequencies. Second, decomposition by angular momentum
enables us to take advantage of reality of $V$ using ODE techniques. In
particular, for large enough angular momenta $m_{j}$, reality of $V$ yields
useful monotonicity properties of certain solutions $u_{0}$ and $u_{1}$ to
$(-h^{2}\partial^{2}_{r}+V+m_{j}r^{-2}-E)u=0$ (see their construction in
Section 2). Control of $u_{0}$ and $u_{1}$ gives a bound on the integral
kernel of $(-h^{2}\partial^{2}_{r}+V+mr^{-2}-E-i0)^{-1}$, which together with
WKB and Bessel function asymptotics, yields the sharp semiclassical resolvent
estimates.

We remark that, in the setting of Theorem 1, our ODE methods do not yield
estimates on $u_{0}$ and $u_{1}$ for small $m_{j}$. However, we are able to
prove good enough resolvent estimates for these frequencies. In doing so, we
prove the following proposition.

######  Proposition 1.1.

Fix $[E_{\min},E_{\max}]\subseteq(0,\infty)$ and $1/2<s\leq 1$. There exist
$C,\,h_{0}>0$ such that for $0\leq\varepsilon\leq 1$, $h\in(0,h_{0}]$,
$m\geq-h^{2}/4$, and $E\in[E_{\min},E_{\max}]$,

$\|\langle
r\rangle^{-s}(-h^{2}\partial^{2}_{r}+V+mr^{-2}-E-i\varepsilon)^{-1}\langle
r\rangle^{-s}\|_{L^{2}(0,\infty)\to L^{2}(0,\infty)}\leq
e^{C(1+|m|^{1/2})/h}.$ (1.9)

Proposition 1.1 is motivated by [Vo21, Proposition 3.1] where a similar
estimate is proved for $m\geq 0$. However, in order to handle the case of
$n=2$, we need to extend these resolvent estimates to $m\geq-h^{2}/4$. The
proof of Proposition 1.1 utilizes methods inspired by the b-calculus from
microlocal analysis to estimate $u$ by $(-h^{2}\partial_{r}^{2}+V+mr^{-2}-E)u$
near $0$, and then employs a spherical energy method to handle the region away
from zero.

We expect Theorems 1 and 2 still hold for potentials $V$ which are radial,
real and non-compactly supported, with sufficient decay toward infinity. A
difficulty with treating this case is finding a suitable replacement for the
WKB and Bessel function asymptotics we use in Section 3. A development in this
direction is that, since we posted this article as a preprint, Vodev [Vo22]
designed a new Carleman estimate for $n\geq 3$, large angular momenta, and
radial $V=O(\langle r\rangle)^{-\delta}$, $\delta>4$; it yields improved
resolvent bounds compared to [Vo21], and an alternative proof of (1.1).

The organization of the paper is as follows. In Section 2 we give an overview
of the plan to prove Theorems 1 and 2 via separation of variables. In Section
3 we use WKB and Bessel function asymptotics to prove Theorem 2. In Section 4
we use the Mellin transform and energy estimates to prove an exponential
estimate which is optimal for low angular momenta but not for high ones. In
Section 5 we use ODE analysis to remove the losses for high angular momenta
and complete the proof of Theorem 1.

Acknowledgements: We thank the anonymous referee, whose helpful comments
improved the exposition. The authors gratefully acknowledge from the following
sources: KD from NSF DMS-1708511, J.G. from EPSRC Early Career Fellowship
EP/V001760/1, and J.S. from NSF DMS-2204322 and a University of Dayton
Research Council Seed Grant. On behalf of all authors, the corresponding
author states that there is no conflict of interest.

Data Availability Statement: Data sharing not applicable to this article as no
datasets were generated or analyzed during the current study.

##  2\. Reduction to a family of one dimensional resovlent estimates

In this section, we use separation of variables to reduce the proofs of
Theorems 1 and 2 to a family of one dimensional resolvent estimates. We begin
by recalling the conjugation

$r^{(n-1)/2}(-\Delta)r^{-(n-1)/2}=-\partial^{2}_{r}-\frac{\Delta_{\mathbb{S}^{n-1}}}{r^{2}}+\frac{(n-1)(n-3)}{4r^{2}}.$

We then put

$m_{j}=h^{2}(\sigma_{j}+4^{-1}(n-1)(n-3)),$

where $0=\sigma_{0}<\sigma_{1}=\sigma_{2}\leq\sigma_{3}\leq\cdots$ are the
eigenvalues of the nonnegative Laplace-Beltrami operator on the unit sphere
$\mathbb{S}^{n-1}\subseteq\mathbb{R}^{n}$, repeated according to multiplicity.
(Recall that the eigenvalues of the unit sphere are $k^{2}+(n-2)k$,
$k=0,\,1,\,\dots$.) Denote by $\mathbf{Y}_{0},\ \mathbf{Y}_{1},\dots$ a
corresponding sequence of orthonormal real eigenfunctions. Also define

$P_{m}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-h^{2}\partial^{2}_{r}+V(r)+mr^{-2},\qquad
m\geq-\frac{h^{2}}{4}.$

The operator $P_{m}$, acting on $L^{2}(\mathbb{R}_{+})$,
$\mathbb{R}_{+}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(0,\infty)$,
with domain $C^{\infty}_{0}(\mathbb{R}_{+})$, is symmetric. As $P_{m}$
commutes with complex conjugation, by von Neumann’s theorem it has equal
defect indices, and thus has self adjoint extensions. Elements in the domain
$\mathcal{D}(P^{*}_{m})$ are precisely those functions $u\in
L^{2}(\mathbb{R}_{+})$ having $u$ and $\partial_{r}u$ absolutely continuous
along with $P_{m}u\in L^{2}(\mathbb{R}_{+})$. Moreover, since
$\|r^{-1}\varphi\|_{L^{2}(\mathbb{R}_{+})}\leq
2\|\partial_{r}\varphi\|_{L^{2}(\mathbb{R}_{+})}$ for any $\varphi\in
C^{\infty}_{0}(\mathbb{R}_{+})$, $P_{m}$ is a semibounded operator, thus a
particular self-adjoint extension of $(P_{m},C^{\infty}_{0}(\mathbb{R}_{+}))$
is its Friedrichs extension (see [ReSi, Theorem X.23]), which we denote by
$(P_{m},\mathcal{D}_{m})$; this is the self-adjoint extension of
$(P_{m},C^{\infty}_{0}(\mathbb{R}_{+}))$ that we work with throughout the
paper.

We start by studying the resolvent kernel for $P_{m}$. To do this, we
construct two convenient linearly independent solutions, $u_{0}$ and $u_{1}$,
which are not in $\mathcal{D}_{m}$, to

$P_{m}u_{j}=(E+i\varepsilon)u_{j},\qquad j\in\\{0,1\\}.$ (2.1)

To define these solutions, let

$\begin{gathered}\varphi_{J}(r)=r^{1/2}J_{\nu}(\lambda
r),\qquad\varphi_{Y}(r)=r^{1/2}Y_{\nu}(\lambda r),\\\
m\geq-\frac{h^{2}}{4},\quad\nu=h^{-1}\big{(}m+\frac{h^{2}}{4}\big{)}^{1/2}\geq
0,\quad\lambda=\frac{\sqrt{E+i\varepsilon}}{h},\quad
E\in[E_{\min},E_{\max}],\quad\varepsilon\geq 0,\end{gathered}$

where $J_{\nu}(\cdot)$ and $Y_{\nu}(\cdot)$ are the Bessel functions of the
first and second kind, respectively [DLMF, Section 10.2]. Note that

$(-h^{2}\partial_{r}+mr^{-2}-E-i\varepsilon)\varphi_{J/Y}=0.$

Next, define $\varphi_{0}=\varphi_{J}$ and, inductively,

$\varphi_{n+1}(r)=\frac{\pi}{2h^{2}}\int_{0}^{r}(\varphi_{Y}(r)\varphi_{J}(r^{\prime})-\varphi_{J}(r)\varphi_{Y}(r^{\prime}))V(r^{\prime})\varphi_{n}(r^{\prime})dr^{\prime},\qquad
n\geq 0.$

Finally, put

$u_{0}(r):=\sum_{n=0}^{\infty}\varphi_{n}(r).$ (2.2)

In Appendix B, we prove

######  Lemma 2.1.

The series (2.2) and its first derivative converge uniformly for
$(r,E,\varepsilon)$ in compact subsets of
$\mathbb{R}_{+}\times[E_{\min},E_{\max}]\times[0,\infty)$. In addition,
$u_{0}(r)>0$ near $r=0$,

$\displaystyle
u_{0}(r)=\varphi_{0}(r)+\frac{\pi}{2h^{2}}\int_{0}^{r}(\varphi_{Y}(r)\varphi_{J}(r^{\prime})-\varphi_{J}(r)\varphi_{Y}(r^{\prime}))V(r^{\prime})u_{0}(r^{\prime})dr^{\prime},$
(2.3)
$\displaystyle(-h^{2}\partial_{r}^{2}+V(r)+mr^{-2}-E-i\varepsilon)u_{0}=0,$
(2.4) $\displaystyle\lim_{r\to
0^{+}}\big{(}\tfrac{n-1}{2r}u_{0}(r)-u_{0}^{\prime}(r)\big{)}r^{(n-1)/2}=0,$
(2.5)

and for all $r^{*}>0,\nu\geq 0$, there is $C_{\nu,r^{*}}$ such that

$|u_{0}(r)|\leq C_{\nu,r^{*}}r^{\nu+\frac{1}{2}},\qquad r\in(0,r^{*}],$ (2.6)

Next, put

$u_{1}(r)=\varphi_{J}(r)+i\varphi_{Y}(r),\qquad r>R_{0},$ (2.7)

and extend $u_{1}$ by requiring that it solve (2.1). By (2.7) and [Ol, Theorem
2.1 in Section 5.2.1], $u_{1}$ depends continuously on $(r,E,\varepsilon)$
varying in
$\mathbb{R}_{+}\times[E_{\text{min}},E_{\text{max}}]\times[0,\infty)$. Also,
from [DLMF, 10.17.5], for all $r^{*}\geq 1$, $\nu\geq 0$, there is
$C_{r^{*},\nu}>0$ such that

$|u_{1}(r)|\leq C_{r^{*},\nu}e^{-\mathop{\rm Im}\lambda r},\qquad r\geq
r^{*}.$ (2.8)

######  Lemma 2.2.

The functions $u_{0}$ in (2.2) and $u_{1}$ in (2.7) are linearly independent
for all $\varepsilon\geq 0$.

###### Proof.

First consider $\varepsilon>0$ and suppose $u_{1}$ and $u_{0}$ are linearly
dependent. Then, since
$u_{0}\in L^{2}((0,1])$ by (2.6) and $u_{1}\in L^{2}([1,\infty))$ by (2.8), we
would have $u_{1}\in L^{2}([0,\infty))$. In particular, $u_{1}$ would be an
$L^{2}$-solution of $(P_{m}-E-i\varepsilon)u=0$ and thus must vanish
identically. Next, when $\varepsilon=0$, then by the Bessel function equation
(A.1), $u_{0}(r)=A\varphi_{J}(\lambda r)+B\varphi_{Y}(\lambda r)$ when
$r>R_{0}$ and for some real constants $A$ and $B$. Comparing with (2.7) and
using the linear independence of $J_{\nu}(z)$ and $Y_{\nu}(z)$, $z>0$, we
conclude that $u_{0}$ and $u_{1}$ are linearly independent when
$\varepsilon=0$, too.

∎

We can now define the resolvent kernel for $P_{m}$,

$\begin{gathered}K(r,r^{\prime})=K(r,r^{\prime};m,E,\varepsilon,h)=-\frac{u_{0}(r)u_{1}(r^{\prime})}{h^{2}W},\\\
r\leq
r^{\prime},\,m\geq-\frac{h^{2}}{4},\,E\in[E_{\text{min}},E_{\text{max}}],\,\varepsilon\geq
0,\end{gathered}$ (2.9)

and for $r^{\prime}<r$, $K(r,r^{\prime})=K(r^{\prime},r)$, where
$W=u_{0}u^{\prime}_{1}-u^{\prime}_{0}u_{1}$ is the Wronskian of $u_{0}$ and
$u_{1}$.

######  Lemma 2.3.

For $E,\varepsilon>0$, $u\in L^{2}(\mathbb{R}^{n})$ and $v\in
L^{2}(\mathbb{R}_{+})$,

$\displaystyle\begin{split}(P_{m}-E-i\varepsilon)^{-1}v&=\int^{\infty}_{0}K(r,r^{\prime};m,E,\varepsilon,h)v(r^{\prime})dr^{\prime}.\\\
&=-h^{-2}W^{-1}\big{(}u_{1}(r)\int^{r}_{0}u_{0}(r^{\prime})v(r^{\prime})dr^{\prime}+u_{0}(r)\int_{r}^{\infty}u_{1}(r^{\prime})v(r^{\prime})dr^{\prime}\big{)},\end{split}$
(2.10)
$\displaystyle(P-E-i\varepsilon)^{-1}u=\sum_{j=0}^{\infty}\mathbf{Y}_{j}\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}r^{-(n-1)/2}K(r,r^{\prime};m_{j},E,\varepsilon,h)(r^{\prime})^{(n-1)/2}u(r^{\prime},\theta)\mathbf{Y}_{j}(\theta)d\theta
dr^{\prime}.$ (2.11)

###### Proof.

To prove (2.10), it suffices to work with $v\in
C^{\infty}_{0}(\mathbb{R}_{+})$. We check that the right side of (2.10)
belongs to $\mathcal{D}_{m}$ and, that applying $P_{m}-E-i\varepsilon$ yields
$v$. The latter is a direct computation, while the former follows from (2.6),
(2.8), and the fact that a characterization of $\mathcal{D}_{m}$ is

$\mathcal{D}_{m}=\begin{cases}\big{\\{}f\in L^{2}(\mathbb{R}_{+}):P_{m}f\in
L^{2}(\mathbb{R}_{+}),\,r^{-\nu-\tfrac{1}{2}}f\in L^{\infty}\text{ near
$r=0$}\big{\\}}&\tfrac{-h^{2}}{4}\leq m<\tfrac{3h^{2}}{4},\\\ \big{\\{}f\in
L^{2}(\mathbb{R}_{+}):P_{m}f\in
L^{2}(\mathbb{R}_{+})\big{\\}}&m\geq\tfrac{3h^{2}}{4}.\end{cases}$ (2.12)

See [NiZe92, Section 6, Example I].

We need only verify (2.11) for $u=\mathbf{Y}_{j}v$ with $v\in
C_{0}^{\infty}(\mathbb{R}_{+})$, as such functions have dense linear span in
$L^{2}(\mathbb{R}^{n})$. In this case, the right side of (2.11) reduces to

$\mathbf{Y}_{j}\int_{0}^{\infty}r^{-(n-1)/2}K(r,r^{\prime};m_{j},E,\varepsilon,h)(r^{\prime})^{(n-1)/2}v(r^{\prime})dr^{\prime}.$
(2.13)

To show (2.13) and $(P-E-i\varepsilon)^{-1}\mathbf{Y}_{j}v$ coincide, we check
that applying $P-E-i\varepsilon$, respectively $\Delta$, to (2.13) in the
sense of distributions, results in $\mathbf{Y}_{j}v$, respectively some
function in $L^{2}(\mathbb{R}^{n})$. Both computations are handled by
integrating by parts in polar coordinates, on domains of the form
$\\{x\in\mathbb{R}^{n}:|x|>\delta>0\\}$, and sending $\delta\to 0$. All
boundary terms that appear in the calculation vanish as $\delta\to 0$, thanks
to (2.5) and (2.6). We leave the remaining details to the reader.
∎

We next consider the limit as $\varepsilon\to 0^{+}$. Recall that [Ag75,
Theorem 4.2] for any $s>1/2$, and $E\in[E_{\min},E_{\max}]$ , the limit

$\langle x\rangle^{-s}(P-E-i0)^{-1}\langle
x\rangle^{-s}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{\varepsilon\to
0^{+}}\langle x\rangle^{-s}(P-E-i\varepsilon)^{-1}\langle x\rangle^{-s}$
(2.14)

exists in the uniform topology $L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})$. By (2.11), (2.10), and because
$K(r,r^{\prime};m_{j},E,\varepsilon,h)$ depends continuously on
$(r,r^{\prime},E,\varepsilon)$ varying in compact subsets of
$\mathbb{R}_{+}\times\mathbb{R}_{+}\times[E_{\text{min}},E_{\text{max}}]\times[0,\infty)$,
for each $j$ and $u,v\in C^{\infty}_{0}(\mathbb{R}_{+})$,

$\begin{split}\langle\mathbf{Y}_{j}u,\langle&x\rangle^{-s}(P-E-i0)^{-1}\langle
x\rangle^{-s}\mathbf{Y}_{j}v\rangle_{L^{2}}\\\ &=\lim_{\varepsilon\to
0^{+}}\langle r^{(n-1)/2}u,\langle
r\rangle^{-s}\int_{0}^{\infty}K(r,r^{\prime};m_{j},E,\varepsilon,h)\langle
r^{\prime}\rangle^{-s}(r^{\prime})^{(n-1)/2}v(r^{\prime})dr^{\prime}\rangle_{L^{2}(\mathbb{R}_{+})}\\\
&=\lim_{\varepsilon\to 0^{+}}\langle r^{(n-1)/2}u,\langle
r\rangle^{-s}(P_{m_{j}}-E-i\varepsilon)^{-1}\langle
r^{\prime}\rangle^{-s}(r^{\prime})^{(n-1)/2}v(r^{\prime})\rangle_{L^{2}(\mathbb{R}_{+})}\\\
&=\langle r^{(n-1)/2}u,\langle
r\rangle^{-s}\int_{0}^{\infty}K(r,r^{\prime};m_{j},E,0,h)\langle
r^{\prime}\rangle^{-s}(r^{\prime})^{(n-1)/2}v(r^{\prime})dr^{\prime}\rangle_{L^{2}(\mathbb{R}_{+})}.\end{split}$

Thus, as functions $\\{\mathbf{Y}_{j}v:j\in\mathbb{N}_{0},\,v\in
C^{\infty}_{0}(\mathbb{R}_{+})\\}$ have dense linear span in
$L^{2}(\mathbb{R}^{n})$, to bound the norm of (2.14) on
$L^{2}(\mathbb{R}^{n})$, it suffices to bound, uniformly in $j$, the kernel
$K(r,r^{\prime};m_{j},E,0,h)$, or alternatively, $\|\langle
r\rangle^{-s}(P_{m_{j}}-E-i\varepsilon)^{-1}\langle
r\rangle^{-s}\|_{L^{2}(\mathbb{R}_{+})\to L^{2}(\mathbb{R}_{+})}$ for
$\varepsilon>0$ small.

If the bounds (1.1) and (1.4) are established for $\varepsilon=0$, it is well-
known that one can use resolvent identities to show they hold for
$\varepsilon>0$ as well. We omit the proof of this but refer the reader to the
relevant results, see [BrPe00, Proposition 3] and [Vo14, Theorem 1.5].

##  3\. Exterior estimates

Theorem 2 follows from the stronger statement that, for any
$R>\sup_{E\in[E_{\min},E_{\max}]}R_{1}(V,E)$ (see (1.3)), we have

$|K(r,r^{\prime})|=|K(r,r^{\prime},m,E,0,h)|\leq\frac{C}{h},$ (3.1)

uniformly for $r$ and $r^{\prime}$ in $[R,\infty)$ obeying $r\leq r^{\prime}$,
$m\geq-h^{2}/4$, $h$ small, and $E\in[E_{\max},E_{\min}]$. We prove (3.1) in
this section, over the course of two lemmas. The first lemma establishes (3.1)
for $m\leq M_{+}$, where

$M_{+}=M_{+}(E)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=M_{0}+\frac{E(R^{2}-R^{2}_{1})}{2}.$
(3.2)

In the second lemma, we prove (3.1) for $m>M_{+}$.

######  Lemma 3.1.

There are $C$ and $h_{0}$ such that (3.1) holds for all $r$ and $r^{\prime}$
in $[R,\infty)$ obeying $r\leq r^{\prime}$, $h\in(0,h_{0}]$,
$E\in[E_{\min},E_{\max}]$ and for all $m\in[-h^{2}/4,M_{+}]$.

###### Proof.

The proof is essentially the same as the one in [DaJi20, Lemma 1]. It is based
on WKB or Liouville-Green asymptotics for solutions to an ODE with real
coefficients in a region with no turning point. First, observe that if $r\geq
R$ and $m\leq M_{+}$, then

$mr^{-2}\leq\frac{2M_{0}+E(R^{2}-R^{2}_{1})}{2r^{2}}\leq\frac{2ER^{2}_{1}+E(R^{2}-R^{2}_{1})}{2R^{2}}=E-\frac{E(R^{2}-R^{2}_{1})}{2R^{2}}.$

Thus $E-mr^{-2}$ is bounded below by a positive constant uniformly in $E$, $r$
and $m$.

By [Ol, Section 6.2.4], there are real numbers $A=A(h)$ and $B=B(h)$ such
that, for $r\geq R$ we have

$u_{0}(r)=\frac{1}{\sqrt[4]{E-mr^{-2}}}\Big{(}\sum_{\pm}(A\pm
iB)\exp\Big{(}\pm\frac{i}{h}\int_{R}^{r}\sqrt{E-ms^{-2}}ds\Big{)}(1+\varepsilon_{\pm}(r))\Big{)},$

where $\varepsilon_{+}$ and $\varepsilon_{-}$ satisfy

$|\varepsilon_{\pm}(r)|+h|\varepsilon_{\pm}^{\prime}(r)|\lesssim\frac{h}{r^{3}}.$

By [Ol, Section 6.2.4], there is a constant $D=D(h)$ so that for $r\geq R$,

$u_{1}(r)=\frac{D}{\sqrt[4]{E-mr^{-2}}}\exp\Big{(}\frac{i}{h}\int_{R}^{r}\sqrt{E-ms^{-2}}ds\Big{)}(1+\varepsilon_{+}(r)),$

where we rule out the presence of an
$\exp\Big{(}-\frac{i}{h}\int_{R}^{r}\sqrt{E-ms^{-2}}ds\Big{)}$ term by using
large-argument Bessel function asymptotics [DLMF, 10.17.5 and 10.17.11] to
show that

$\partial_{r}u_{1}-i\frac{\sqrt{E}}{h}u_{1}\to 0,\qquad\text{as
$r\to\infty$}.$

Next we compute the Wronskian

$W=\frac{D}{\sqrt{E-mr^{-2}}}\Big{(}A-iB\Big{)}\frac{2i}{h}\sqrt{E-mr^{-2}}\Big{(}1+O(hr^{-3})\Big{)}=\frac{2D(B+iA)}{h},$

where we dropped the remainder because $W$ is independent of $r$. Plugging
these formulas for $u_{0}$, $u_{1}$, and $W$ into the formula (2.9) for $K$
gives the conclusion.
∎

######  Lemma 3.2.

There are $C$ and $h_{0}$ such that (3.1) holds for all $r$ and $r^{\prime}$
in $[R_{0},\infty)$ obeying $r\leq r^{\prime}$, $h\in(0,h_{0}]$,
$E\in[E_{\min},E_{\max}]$, and for all $m>M_{+}$.

###### Proof.

By the Bessel function differential equation (A.1), there are real numbers $A$
and $B$ such that, for $r\geq R_{0}$,

$u_{0}(r)=r^{1/2}(AJ_{\nu}(\nu z)+BY_{\nu}(\nu z)),$

where

$\nu=h^{-1}(m+\tfrac{1}{4}h^{2})^{1/2},\qquad
z=(m+\tfrac{1}{4}h^{2})^{-1/2}E^{1/2}r.$

(Note that the constants $A$ and $B$ here are analogous to but different from
the ones from Lemma 3.1.) Recall from (2.7) that

$u_{1}(r)=r^{1/2}(J_{\nu}(\nu z)+iY_{\nu}(\nu z)).$

By the Bessel function Wronskian formula (A.2),

$W=2\pi^{-1}(iA-B).$

To bound $B$ in terms of $A$ we use the fact that
$u_{0}(R_{0}),\,u^{\prime}_{0}(R_{0})\geq 0$ which follows from
$V+mr^{-2}-E\geq 0$ on $(0,R_{0}]$; see Lemma B.1. By the Bessel function
bounds (A.7), for $h$ small enough, we have

$B\leq\frac{J_{\nu}(\nu z_{0})}{-Y_{\nu}(\nu z_{0})}A\lesssim
e^{-2\nu\xi_{0}}A,\qquad-B\leq\frac{J_{\nu}(\nu
z_{0})+2R_{0}J^{\prime}_{\nu}(\nu z_{0})}{Y_{\nu}(\nu
z_{0})+2R_{0}Y^{\prime}_{\nu}(\nu z_{0})}A\lesssim e^{-2\nu\xi_{0}}A$

where

$\xi_{0}=\int_{z_{0}}^{1}t^{-1}(1-t^{2})^{1/2}dt,\qquad
z_{0}=(m+\tfrac{1}{4}h^{2})^{-1/2}E^{1/2}R_{0}.$

Note that $z_{0}<1$ because

$z_{0}\leq
M_{+}^{-1/2}E^{1/2}R_{0}=\left(\frac{M_{0}}{ER^{2}_{0}}+\frac{R^{2}-R^{2}_{0}}{2R^{2}_{0}}\right)^{-1/2}\leq\left(1+\frac{R^{2}-R^{2}_{0}}{2R^{2}_{0}}\right)^{-1/2}.$

Then, when $R_{0}\leq r\leq r^{\prime}$, letting
$z^{\prime}=(m+\tfrac{1}{4}h^{2})^{-1/2}E^{1/2}r^{\prime}$ and inserting
$|B|\lesssim e^{-2\nu\xi_{0}}A$ and the Bessel function bound (A.10) into
(2.9) gives

$\begin{split}|K(r,r^{\prime})|&\lesssim h^{-2}(rr^{\prime})^{1/2}(J_{\nu}(\nu
z)+e^{-2\nu\xi_{0}}|Y_{\nu}(\nu z)|)|J_{\nu}(\nu z^{\prime})+iY_{\nu}(\nu
z^{\prime})|\\\ &\lesssim h^{-2}\nu^{-1}(rr^{\prime})^{1/2}\langle
z\rangle^{-1/2}\langle z^{\prime}\rangle^{-1/2}\lesssim h^{-1},\end{split}$
(3.3)

as desired.
∎

##  4\. One dimensional resolvent estimates for fixed angular momentum

We now study the equation

$(P_{m}-E-i\varepsilon)u=(-h^{2}\partial_{r}^{2}+V(r)+mr^{-2}-E-i\varepsilon)u=f,\qquad\quad
m\geq-\frac{h^{2}}{4},$

on
$L^{2}=L^{2}(\mathbb{R}_{+})\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{2}(0,\infty)$.
Recall that the self-adjoint extension of
$(P_{m},C^{\infty}_{0}(\mathbb{R}_{+}))$ we employ is its Friedrichs extension
$(P_{m},\mathcal{D}_{m})$. Put
$\mathcal{D}_{\text{comp},m}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{u\in\mathcal{D}_{m}:\mathop{\rm
supp}u\text{ is compact in $[0,\infty)$}\\}$.

######  Lemma 4.1.

Let $[E_{\min},E_{\max}]\subseteq(0,\infty)$ and $1/2<s\leq 1$. Then there is
$C,\,h_{0}>0$ such that for all $E\in[E_{\min},E_{\max}]$, $h\in(0,h_{0}]$,
$m\geq-h^{2}/4$, $0\leq\varepsilon\leq 1$, and
$u\in\mathcal{D}_{\emph{comp},m}$,

$\|\langle r\rangle^{-s}u\|_{L^{2}(\mathbb{R}_{+})}\leq
e^{C(1+|m|^{1/2})/h}\|\langle
r\rangle^{s}(P_{m}-E-i\varepsilon)u\|_{L^{2}(\mathbb{R}_{+})}.$

To prove Lemma 4.1, we start by studying the operator

$Q_{m}\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-\partial_{r}^{2}+mh^{-2}r^{-2},\qquad
m\geq-\frac{h^{2}}{4},$

on functions compactly supported in $[0,\infty)$. We recall the Mellin
transform and its inverse:

$\mathcal{M}(u)(\sigma):=\int_{0}^{\infty}r^{i\sigma}u(r)\frac{dr}{r},\qquad\mathcal{M}_{t}^{-1}(v)(r):=\frac{1}{2\pi}\int_{\mathbb{R}}r^{-i\sigma}v(\sigma)d\tau,\qquad\sigma=\tau+it,$

where the definitions hold initially, e.g., for $u\in
C_{0}^{\infty}(\mathbb{R}_{+})$ and $v((\cdot)+it)\in
L^{1}_{\tau}(\mathbb{R})\cap L^{2}_{\tau}(\mathbb{R})$, and then extend by
density to bounded operators $\mathcal{M}:L^{2}(0,\infty;r^{-2t-1}dr)\to
L^{2}_{\tau}(\mathbb{R})$,
$\mathcal{M}_{t}^{-1}:L^{2}_{\tau}(\mathbb{R})\to
L^{2}(0,\infty;r^{-2t-1}dr)$. Moreover, since
$\mathcal{M}(u)(\tau+it)=2\pi\mathcal{F}^{-1}(e^{-tx}u(e^{x}))(\tau)$,
$x\in\mathbb{R}$, and $\mathcal{M}^{-1}_{t}(v)(r)=r^{t}\mathcal{F}(v)(\log
r)/2\pi$, where $\mathcal{F}$ denotes Fourier transform,

$\begin{gathered}\|\mathcal{M}(u)(\tau+it)\|_{L^{2}_{\tau}(\mathbb{R})}=(2\pi)^{1/2}\|r^{-t-1/2}u\|_{L^{2}(\mathbb{R}_{+})},\\\
\|r^{-t-1/2}\mathcal{M}_{t}^{-1}(v)(r)\|_{L^{2}(\mathbb{R}_{+})}=(2\pi)^{-1/2}\|v\|_{L^{2}_{\tau}(\mathbb{R})}.\end{gathered}$
(4.1)

Let

$t_{\pm}=t_{\pm}(m):=\frac{1\pm\sqrt{1+4mh^{-2}}}{2},\qquad\Lambda(t,m):=|t^{2}-t-h^{-2}m|^{-1},\qquad
t\neq t_{\pm}.$

######  Lemma 4.2.

There is $C>0$ such that for $m\geq-h^{2}/4$, $N\in\mathbb{R}$,
$t_{0}\in\mathbb{R}\setminus\\{t_{+}(m),t_{-}(m)\\}$, and $u\in
r^{-N}L^{2}_{\operatorname{comp}}[0,\infty)$ with $Q_{m}u\in
r^{t_{0}-\frac{3}{2}}L^{2}$, we have

$u=\Pi_{t_{0}}(r^{2}Q_{m}u)+E_{t_{0}}(r^{2}Q_{m}u),$ (4.2)

where

$\|r^{-t_{0}-\frac{1}{2}}E_{t_{0}}v\|_{L^{2}}\leq
C\Lambda(t_{0},m)\|r^{-{t_{0}}-\frac{1}{2}}v\|_{L^{2}(\mathbb{R}_{+})},$ (4.3)

and

$\Pi_{t_{0}}v=\begin{cases}0&t_{0}<t_{-},\\\ r^{\frac{1}{2}}\log
r\mathcal{M}(v)(\tfrac{i}{2})-r^{\frac{1}{2}}\mathcal{M}(\log
rv)(\tfrac{i}{2})&t_{-}<t_{0},\,m=-\tfrac{h^{2}}{4},\\\
\frac{r^{t_{-}}\mathcal{M}(v)(it_{-})}{t_{-}-t_{+}}&t_{-}<t_{0}<t_{+},\,m>-\tfrac{h^{2}}{4},\\\
\frac{r^{t_{-}}\mathcal{M}(v)(it_{-})}{t_{-}-t_{+}}+\frac{r^{t_{+}}\mathcal{M}(v)(it_{+})}{t_{+}-t_{-}}&t_{+}<t_{0},\,m>-\tfrac{h^{2}}{4}.\end{cases}$

Remark: Applying $(r\partial_{r})^{j}$, $j=1,\,2$, to the expression for
$E_{t_{0}}v$, see (4.6) below, yields a strengthening of (4.3), namely

$\|r^{-t_{0}-\frac{1}{2}}(r\partial_{r})^{j}E_{t_{0}}v\|_{L^{2}}\leq\Lambda_{j}(t_{0},m)\|r^{-{t_{0}}-\frac{1}{2}}v\|_{L^{2}(\mathbb{R}_{+})},\qquad
j=0,\,1,\,2,$

However, we omit the proof since in the sequel we do not need the estimates
for $j=1,\,2$.

###### Proof.

Without loss of generality, we take $N$ positive and large enough so that
$-N<t_{0},\,t_{+},\,t_{-}$. Since $u\in r^{-N}L_{\operatorname{comp}}^{2}$ and
$r^{2}Q_{m}u\in r^{t_{0}+\frac{1}{2}}L^{2}_{\operatorname{comp}}$,
$\mathcal{M}(u)(\sigma)$ is holomorphic in $\operatorname{Im}\sigma<-N-1/2$,
while $\mathcal{M}(r^{2}Q_{m}u)(\sigma)$, is holomorphic in
$\operatorname{Im}\sigma<t_{0}$ and extends continuously (as an $L^{2}$
function) to $\operatorname{Im}\sigma=t_{0}$. In addition,

$\mathcal{M}(r^{2}Q_{m}u)(\sigma)=(\sigma^{2}-i\sigma+mh^{-2})\mathcal{M}(u)(\sigma),\qquad\operatorname{Im}\sigma<-N-\frac{1}{2}.$
(4.4)

In particular, (4.4) implies $\mathcal{M}{u}(\sigma)\in
L^{1}_{\tau}(\mathbb{R})\cap L^{2}_{\tau}(\mathbb{R})$ for
$\operatorname{Im}\sigma<-N-1/2$, so by Fourier inversion,

$u(r)=\frac{1}{2\pi}\int_{\operatorname{Im}\sigma=-N-1}\frac{r^{-i\sigma}\mathcal{M}(r^{2}Q_{m}u)(\sigma)}{\sigma^{2}-i\sigma+mh^{-2}}d\sigma.$
(4.5)

We now deform the contour to $\operatorname{Im}\sigma=t_{0}-\epsilon$ and the
send $\epsilon\to 0$. From (4.5),

$u(r)=\frac{1}{2\pi}\lim_{R\to\infty}\int_{\gamma_{R,-N-1}}\frac{r^{-i\sigma}\mathcal{M}(r^{2}Q_{m}u)(\sigma)}{\sigma^{2}-i\sigma+mh^{-2}}d\sigma,\qquad\gamma_{R,t}=\\{\tau+it\,:\,\tau\in[-R,R]\\}.$

Next, observe that since, in $\operatorname{Im}\sigma<t_{0}$,
$\|\mathcal{M}(r^{2}Q_{m}u)(\sigma)\|_{L_{\tau}^{\infty}(\mathbb{R})}$ is
independent of $\operatorname{Re}\sigma$,

$\Big{|}\int_{\gamma_{\pm,R,-N,\epsilon}}\frac{r^{-i\sigma}\mathcal{M}(r^{2}Q_{m}u)(\sigma)}{\sigma^{2}-i\sigma+mh^{-2}}d\sigma\Big{|}\leq\frac{C_{\epsilon}}{R^{2}}\to
0,\qquad\text{as $R\to\infty$.}$

where

$\gamma_{\pm,R,-N,\epsilon}:=\\{\pm R+it\,:\,t\in[-N-1,t_{0}-\epsilon]\\}.$

In particular, using $t_{0}\neq t_{\pm}(m)$ and that
$\mathcal{M}(r^{2}Q_{m}u)(\tau+it)$ varies continuously in
$L^{2}_{\tau}(\mathbb{R})$ for $t\leq t_{0}$, we send $\epsilon\to 0$ to
obtain

$\begin{split}u(r)&=\frac{1}{2\pi}\int_{\operatorname{Im}\sigma=t_{0}}\frac{r^{-i\sigma}\mathcal{M}(r^{2}Q_{m}u)(\sigma)}{\sigma^{2}-i\sigma+mh^{-2}}d\sigma+i\sum_{\begin{subarray}{c}\sigma^{2}-i\sigma+mh^{-2}=0\\\
\operatorname{Im}\sigma<t_{0}\end{subarray}}\operatorname{Res}\Big{(}r^{-i\sigma}\frac{\mathcal{M}(r^{2}Q_{m}u)(\sigma)}{\sigma^{2}-i\sigma+mh^{-2}}\Big{)}\\\
&=:E_{t_{0}}(r^{2}Q_{m}u)+\Pi_{t_{0}}(r^{2}Q_{m}u).\end{split}$ (4.6)

The formula for $\Pi_{t_{0}}$ follows from calculating residues at
$t_{\pm}(m)$, while the bound on $E_{t_{0}}$ follows from minimizing the
modulus

$|(\tau+it_{0})^{2}-i(\tau+it_{0})+mh^{-2}|^{2}=(-t_{0}^{2}+t_{0}+mh^{-2}+\tau^{2})^{2}+\tau^{2}(2t_{0}-1)^{2},$

with respect to $\tau$.
∎

######  Lemma 4.3.

Suppose $m>0$ and $u\in\mathcal{D}_{m}$. Then $r^{-1}u\in
L^{2}(\mathbb{R}_{+})$.

###### Proof.

Observe that there exists $C>0$ so that,

$m\|r^{-1}u\|_{L^{2}(\mathbb{R}_{+})}^{2}\leq
C\|u\|_{L^{2}(\mathbb{R}_{+})}^{2}+|\langle
P_{m}u,u\rangle_{L^{2}(\mathbb{R}_{+})}|,\qquad u\in
C^{\infty}_{0}(\mathbb{R}_{+}).$

As $(P_{m},\mathcal{D}_{m})$ is the Friedrichs extension of
$(P_{m},C_{0}^{\infty}(0,\infty))$, by [ReSi, Theorem X.23], any
$u\in\mathcal{D}_{m}$ has a sequence $\\{\psi_{k}\\}\subseteq
C^{\infty}_{0}(\mathbb{R}_{+})$ such that $\psi_{k}\to u$ in
$L^{2}(\mathbb{R}_{+})$ and $\langle
P_{m}\psi_{k},\psi_{k}\rangle_{L^{2}}\to\langle
P_{m}u,u\rangle_{L^{2}(\mathbb{R}_{+})}$. We also get pointwise almost
everywhere convergence by extracting a subsequence of the $\psi_{k}$, which we
again just call $\psi_{k}$. Sending $k\to\infty$ and using Fatou’s lemma gives
$m\|r^{-1}u\|_{L^{2}(\mathbb{R}_{+})}^{2}\leq
C\|u\|_{L^{2}(\mathbb{R}_{+})}^{2}+\|P_{m}u\|_{L^{2}(\mathbb{R}_{+})}\|u\|_{L^{2}(\mathbb{R}_{+})}<\infty$
for general $u\in\mathcal{D}_{m}$ as desired.
∎

Lemma 4.2 will allow us to control the behavior of solutions, $u$ to
$(P_{m}-E-i\varepsilon)u=f$ near $0$.

###### Proof of Lemma 4.1.

To shorten notation, we set $P_{m,E,\varepsilon}=P_{m}-E-i\varepsilon$. Since
$u\in\mathcal{D}_{\operatorname{comp},m}$ and $u\in L^{2}$, for any compact
$K\subseteq(0,\infty)$, there exists $C_{K,h}>0$ so that

$\|\partial_{r}^{2}u\|_{L^{2}(K)}\leq
C_{K,h}(\|u\|_{L^{2}(K)}+\|P_{m}u\|_{L^{2}(K)}).$

In particular, $u\in H^{2}_{\operatorname{loc}}(0,\infty)$.

Next, set

$\delta=\delta(m):=\delta_{0}h\langle
h^{-2}m\rangle^{1/2},\qquad\delta_{1}=\delta_{1}(m)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\max(\delta(m),\tfrac{1}{2}),$
(4.7)

for some $0<\delta_{0}\ll 1$ independent of $h$, $m$ and
$E\in[E_{\min},E_{\max}]$, to be chosen. Let $\chi\in C_{0}^{\infty}[0,2)$
with $\chi\equiv 1$ near $[0,1]$. Set
$\chi_{\delta_{1}}=\chi(\delta^{-1}_{1}r)$. Then

$\begin{split}Q_{m}\chi_{\delta_{1}}u&=h^{-2}P_{m,E,\varepsilon}\chi_{\delta_{1}}u-h^{-2}(V-E-i\varepsilon)\chi_{\delta_{1}}u\\\
&=h^{-2}\chi_{\delta_{1}}P_{m}u+h^{-2}[P_{m},\chi_{\delta_{1}}]u-h^{-2}\chi_{\delta_{1}}(V-E-i\varepsilon)u.\end{split}$
(4.8)

In particular, since $u\in H^{2}_{\operatorname{loc}}(0,\infty)$, $P_{m}u\in
L^{2}$, and $\chi_{\delta_{1}}$ is constant near zero,
$r^{2}Q_{m}\chi_{\delta_{1}}u\in r^{2}L^{2}$. Setting

$t_{0}=t_{0}(m)\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{cases}-\frac{1}{2}&-\frac{h^{2}}{4}\leq
m\leq\frac{h^{2}}{4},\\\ \ 1&\frac{h^{2}}{4}<m,\end{cases}$

observe that

$\Lambda(t_{0},m)=|t^{2}_{0}-t_{0}-h^{-2}m|^{-1}\leq c\langle
h^{-2}m\rangle^{-1}$ (4.9)

for some $c>0$ independent of $h$ and $m$. Note also that with this definition
of $t_{0}(m)$, $t_{0}(m)<t_{-}(m)$ for $-h^{2}/4\leq m\leq h^{2}/4$, and
$t_{-}(m)<t_{0}(m)<t_{+}(m)$ for $h^{2}/4<m$. Therefore, by Lemma 4.2, since
$u\in L^{2}$, $r^{2}Q_{m}\chi_{\delta_{1}}u\in r^{2}L^{2}$, and $t_{0}(m)\leq
3/2$,

$\chi_{\delta_{1}}u=E_{t_{0}}(r^{2}Q_{m}\chi_{\delta_{1}}u)+\Pi_{t_{0}}(r^{2}Q_{m}\chi_{\delta_{1}}u),$

where

$\Pi_{t_{0}}(r^{2}Q_{m}\chi_{\delta_{1}}u)=\begin{cases}0&-\frac{h^{2}}{4}\leq
m\leq\frac{h^{2}}{4},\\\
\frac{r^{t_{-}}}{t_{-}-t_{+}}\mathcal{M}(r^{2}Q_{m}\chi_{\delta_{1}}u)(it_{-})&\frac{h^{2}}{4}<m,\end{cases}$

and

$\|r^{-t_{0}-\frac{1}{2}}E_{t_{0}}v\|_{L^{2}}\leq
C\Lambda(t_{0},m)\|r^{-t_{0}-\frac{1}{2}}v\|_{L^{2}}.$

Next, by Lemma 4.3, for $m>0$, $r^{-1}u\in L^{2}$. Thus, both
$\mathcal{M}(\chi_{\delta_{1}}u)(\sigma)$ and
$\mathcal{M}(r^{2}Q_{m}\chi_{\delta_{1}}u)(\sigma)$ are holomorphic in
$\mathop{\rm Im}\sigma<1/2$. As $t_{-}<1/2$ when $m>0$, by (4.4),

$\mathcal{M}(r^{2}Q_{m}\chi_{\delta_{1}}u)(it_{-})=((it_{-})^{2}-i(it_{-})-h^{-2}m)\mathcal{M}(\chi_{\delta_{1}}u)(it_{-})=0.$

In particular, for any $m$ and $u\in\mathcal{D}_{\operatorname{comp},m}$,
$\Pi_{t_{0}}(r^{2}Q_{m}\chi_{\delta_{1}}u)=0$. Thus, by (4.8) and
$\delta_{1}\geq 1/2$,

$\displaystyle\|r^{-t_{0}-\frac{1}{2}}u\|_{L^{2}(0,\delta_{1})}$
$\displaystyle\leq\|r^{-t_{0}-\frac{1}{2}}\chi_{\delta_{1}}u\|_{L^{2}(0,\delta_{1})}\leq
C\Lambda(t_{0},m)\|r^{\frac{3}{2}-t_{0}}Q_{m}\chi_{\delta_{1}}u\|_{L^{2}}$
(4.10) $\displaystyle\leq
C\Lambda(t_{0},m)\Big{(}h^{-2}\|r^{\frac{3}{2}-t_{0}}P_{m,E,\varepsilon}u\|_{L^{2}(0,2\delta_{1})}$
$\displaystyle+h^{-1}\|r^{\frac{3}{2}-t_{0}}hu^{\prime}\|_{L^{2}(\delta_{1},2\delta_{1})}+h^{-2}\|r^{\frac{3}{2}-t_{0}}u\|_{L^{2}(0,2\delta_{1})}\Big{)}.$

We now estimate part of the last term on the right side of (4.10), using (4.7)
and (4.9):

$\begin{split}C\Lambda(t_{0},m)h^{-2}\|r^{\frac{3}{2}-t_{0}}u\|_{L^{2}(0,\delta)}&\leq
C\Lambda(t_{0},m)h^{-2}\delta^{2}\|r^{-t_{0}-\frac{1}{2}}u\|_{L^{2}(0,\delta)}\\\
&\leq Cc\delta_{0}\|r^{-t_{0}-\frac{1}{2}}u\|_{L^{2}(0,\delta)}.\end{split}$

Choosing $\delta_{0}$ small enough, this term may be absorbed into the left
side of (4.10), so we find

$\displaystyle\|r^{-t_{0}-\frac{1}{2}}u\|_{L^{2}(0,\delta_{1})}$
$\displaystyle\leq Ch^{-2}\langle
h^{-2}m\rangle^{-1}\Big{(}\|r^{\frac{3}{2}-t_{0}}P_{m,E,\varepsilon}u\|_{L^{2}(0,2\delta_{1})}+$
(4.11) $\displaystyle
h\|r^{\frac{3}{2}-t_{0}}hu^{\prime}\|_{L^{2}(\delta_{1},2\delta_{1})}+\|r^{\frac{3}{2}-t_{0}}u\|_{L^{2}(\delta,2\delta_{1})}\Big{)}.$

We now employ the energy method to study the region $[\delta(m),\infty)$. Let
$s>1/2$, $\eta=\min\\{1,E\\}/2$, and $\phi_{j}\in
C_{0}^{\infty}([0,1);[0,1])$, $j=0,1,2$, with $\phi_{0}\equiv 1$ on $[0,1/2]$
and $\phi_{j}\equiv 1$ on $\mathop{\rm supp}\phi_{j-1}$. Let

$F(r):=|hu^{\prime}(r)|^{2}+E|u(r)|^{2},\qquad
w(r):=\int_{0}^{r}(1-\phi_{1}(r^{\prime}/\delta))\phi_{2}(r^{\prime}/\delta)dr^{\prime}e^{\psi(r)/h},$

with

$\psi\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\int_{0}^{r}\eta^{-1}\big{(}|V(r^{\prime})|+2|m|(1-\phi_{0}(r^{\prime}/\delta))(r^{\prime})^{-2}\big{)}+\langle
r^{\prime}\rangle^{-2s}dr^{\prime}\\\
\leq\eta^{-1}\|V\|_{L^{1}(\mathbb{R}_{+})}+C(1+\eta^{-1}|m|^{1/2}).$

Here, we have used

$\begin{split}|m|\int^{r}_{\tfrac{1}{2}\delta(m)}(r^{\prime})^{-2}dr^{\prime}=|m|\Big{(}(\tfrac{1}{2}\delta(m))^{-1}-r^{-1}\Big{)}\lesssim|m|^{1/2}.\end{split}$

Then,

$\displaystyle(wF)^{\prime}(r)$ $\displaystyle=-2\operatorname{Re}w\langle
P_{m,E,\varepsilon}u,u^{\prime}\rangle+2\varepsilon w\operatorname{Im}\langle
u,u^{\prime}\rangle+w^{\prime}(|hu^{\prime}|^{2}+E|u|^{2})+2w\operatorname{Re}\langle(V+mr^{-2})u,u^{\prime}\rangle,$

where $\langle
z,z_{1}\rangle\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=z\overline{z_{1}}$,
$z,z_{1}\in\mathbb{C}$. Since $\phi_{0}(r/\delta)>0$ implies $w(r)=0$, we have

$\displaystyle w^{\prime}$
$\displaystyle=\frac{\psi^{\prime}}{h}w+(1-\phi_{1}(r/\delta))\phi_{2}(r/\delta))e^{\psi/h}\geq\frac{\psi^{\prime}}{h}w\geq
h^{-1}w[\eta^{-1}|V|+\langle r\rangle^{-2s}+\eta^{-1}|m|r^{-2}].$ (4.12)

So,

$\displaystyle(wF)^{\prime}(r)$ $\displaystyle\geq-2\operatorname{Re}w\langle
P_{m,E,\varepsilon}u,u^{\prime}\rangle+2\varepsilon w\operatorname{Im}\langle
u,u^{\prime}\rangle+w^{\prime}(|hu^{\prime}|^{2}+E|u|^{2})+2w|\operatorname{Re}\langle(V+mr^{-2})u,u^{\prime}\rangle|$
$\displaystyle\geq-h^{-1}\eta^{-1}\langle
r\rangle^{2s}w|P_{m,E,\varepsilon}u|^{2}-h^{-1}\eta w\langle
r\rangle^{-2s}|hu^{\prime}|^{2}$
$\displaystyle-h^{-1}(|V|+\varepsilon+|m|r^{-2})w(|u|^{2}+|hu^{\prime}|^{2})+\min\\{1,E\\}w^{\prime}(|hu^{\prime}|^{2}+|u|^{2})$
$\displaystyle\geq-h^{-1}\eta^{-1}\langle
r\rangle^{2s}w|P_{m,E,\varepsilon}u|^{2}-\varepsilon
h^{-1}w(|u|^{2}+|hu^{\prime}|^{2})+\frac{\min\\{1,E\\}}{2}w^{\prime}(|hu^{\prime}|^{2}+E|u|^{2}).$

We integrate from $0$ to $\infty$, and use the facts that $w(r)=0$ near zero
and $u\in H^{2}_{\operatorname{loc}}((0,\infty))$ is compactly supported in
$[0,\infty)$:

$\displaystyle\|(w^{\prime})^{1/2}(hu^{\prime})\|^{2}_{L^{2}(0,\infty)}$
$\displaystyle+\|(w^{\prime})^{1/2}u\|^{2}_{L^{2}(0,\infty)}$ (4.13)
$\displaystyle\leq Ch^{-1}\int\langle
r\rangle^{2s}w(r)|P_{m,E,\varepsilon}u(r)|^{2}dr+\varepsilon h^{-1}\int
w(|u(r)|^{2}+|hu^{\prime}(r)|^{2})dr.$

Next, observe that

$\operatorname{Re}\int wP_{m,E,\varepsilon}u\bar{u}dr=\int
w|hu^{\prime}|^{2}dr+\operatorname{Re}\int
hw^{\prime}hu^{\prime}\bar{u}dr+\int w(V-E+mr^{-2})|u|^{2}dr.$

So, using (4.12)

$\begin{split}\|(w)^{1/2}hu^{\prime}\|_{L^{2}}&\leq\frac{\gamma
h}{2}\|(w^{\prime})^{1/2}hu^{\prime}\|^{2}_{L^{2}}+\frac{h\gamma^{-1}}{2}\|(w^{\prime})^{1/2}u\|^{2}_{L^{2}}+C\|(w)^{1/2}u\|_{L^{2}}^{2}\\\
&+C\|w^{1/2}|m|^{1/2}r^{-1}u\|_{L^{2}}^{2}+\frac{1}{2}\|w^{1/2}\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}}+\frac{1}{2}\|w^{1/2}\langle
r\rangle^{-s}u\|_{L^{2}}\\\ &\leq\frac{\gamma
h}{2}\|(w^{\prime})^{1/2}hu^{\prime}\|^{2}_{L^{2}}+C(1+\gamma^{-1})h\|(w^{\prime})^{1/2}u\|^{2}_{L^{2}}\\\
&+\frac{1}{2}\|w^{1/2}\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}}+C\|(w)^{1/2}u\|_{L^{2}}.\end{split}$

Plugging this into the right side of (4.13), taking $\gamma$ small enough,
subtracting the corresponding term to the left-hand side, and using,

$\begin{gathered}c\delta(m)\langle r\rangle^{-2s}/h\leq w^{\prime}(r),\qquad
r\geq\delta(m),\\\ |w(r)|,|w^{\prime}(r)|\leq e^{C(1+|m|^{1/2})/h},\qquad
r\geq 0,\\\ \mathop{\rm
supp}w\subseteq(\tfrac{1}{2}\delta(m),\infty),\end{gathered}$

and $\varepsilon\leq 1$, we have

$\displaystyle\|\langle
r\rangle^{-s}(hu^{\prime})\|_{L^{2}(\delta,\infty)}+\|\langle
r\rangle^{-s}u\|_{L^{2}(\delta,\infty)}$ (4.14) $\displaystyle\leq
e^{C(1+|m|^{1/2})/h}\|\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}(0,\infty)}+\varepsilon^{1/2}e^{C(1+|m|^{1/2})/h}\|u\|_{L^{2}(\frac{1}{2}\delta,\infty)}.$

Next, observe that $-1/2\leq t_{0}(m)\leq 3/2$ implies

$\displaystyle\|r^{\frac{3}{2}-t_{0}}v\|_{L^{2}(0,2\delta_{1})}\leq
C\delta_{1}^{t_{0}-\frac{3}{2}}\|v\|_{L^{2}(0,2\delta_{1})},$
$\displaystyle\|v\|_{L^{2}(0,\delta)}\leq
C\delta^{t_{0}+\frac{1}{2}}\|r^{-t_{0}-\frac{1}{2}}v\|_{L^{2}(0,\delta)},$
$\displaystyle\|r^{\frac{3}{2}-t_{0}}v\|_{L^{2}(\delta_{1},2\delta_{1})}\leq
C\delta_{1}^{t_{0}-\frac{3}{2}+s}\|\langle
r\rangle^{-s}v\|_{L^{2}(\delta_{1},2\delta_{1})}.$

Combining this with (4.11) and (4.14), and that $\delta_{1}\leq C\langle
m\rangle^{1/2}$, we have

$\displaystyle\|\langle r\rangle^{-s}u\|_{L^{2}(0,\infty)}\leq
e^{C(1+|m|^{1/2})/h}$ $\displaystyle\|\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}(0,\infty)}+\varepsilon^{1/2}e^{C(1+|m|^{1/2})/h}\|u\|_{L^{2}(\frac{1}{2}\delta,\infty)}.$

Finally, observe that, since $u\in\mathcal{D}_{m}$, for any $\gamma>0$,

$\varepsilon\|u\|^{2}_{L^{2}}=|\operatorname{Im}\langle
P_{m,E,\varepsilon}u,u\rangle|\leq\|\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}}\|\langle
r\rangle^{-s}u\|_{L^{2}}\leq\tfrac{1}{2\gamma}\|\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|^{2}_{L^{2}}+\tfrac{\gamma}{2}\|\langle
r\rangle^{-s}u\|^{2}_{L^{2}},$

and hence

$\displaystyle\|\langle r\rangle^{-s}u\|_{L^{2}(0,\infty)}\leq
e^{C(1+|m|^{1/2})/h}$ $\displaystyle\|\langle
r\rangle^{s}P_{m,E,\varepsilon}u\|_{L^{2}(0,\infty)}.$

This completes the proof of Lemma 4.1.
∎

###### Proof of Proposition 1.1.

Let $1/2<s<1$, and $W_{s}\in C^{\infty}$ with $W_{s}\equiv 1$ near $0$ and
$c\langle r\rangle^{s}\leq W_{s}(r)\leq C\langle r\rangle^{s}$, and
$W_{s}=C\langle r\rangle^{s}$ on $r\geq 2$.

First, recall that for any $\chi\in C_{0}^{\infty}[0,\infty)$ with $\chi\equiv
1$ near $0$,

$\|u\|_{\mathcal{D}_{m}}\sim\|\chi
u\|_{\mathcal{D}_{m}}+\|(1-\chi)u\|_{H^{2}}.$

Thus, since $[W_{s},P_{m}](W_{s})^{-1}:H^{2}\to H^{1}$ is supported away from
zero, for any $v\in\mathcal{D}_{m}$ such that $W_{s}v\in\mathcal{D}_{m}$,

$\|W_{s}(P_{m}-E-i\varepsilon)v\|_{L^{2}}\leq\|[W_{s},P_{m}](W_{s})^{-1}W_{s}v\|_{L^{2}}+\|(P_{m}-E-i\varepsilon)W_{s}v\|_{L^{2}}\leq
C_{\varepsilon,h}\|W_{s}v\|_{\mathcal{D}_{m}}.$ (4.15)

Next, we claim that for $\varepsilon>0$ and $f\in L^{2}(\mathbb{R}_{+})$,

$W_{s}(P_{m}-E-i\varepsilon)^{-1}\langle r\rangle^{-s}f\in\mathcal{D}_{m}.$

Put

$v\mathrel{\vbox{\par\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(P_{m}-E-i\varepsilon)^{-1}\langle
r\rangle^{-s}f\in\mathcal{D}_{m}.$

We shall show $W_{s}v\in\mathcal{D}_{m}$ by demonstrating that

$W_{s}v=(P_{m}-E-i\varepsilon)^{-1}\langle r\rangle^{-s}(W_{s}\langle
r\rangle^{-s}f+g)\ $ (4.16)

for suitable $g\in L^{2}(\mathbb{R}_{+})$.

Let $\chi\in C_{0}^{\infty}([0,\infty);[0,1])$ with $\chi\equiv 1$, and put
$\chi_{k}=\chi(k^{-1}r)$, $k\in\mathbb{N}$. Almost everywhere on
$\mathbb{R}_{+}$,

$(P_{m}-E-i\varepsilon)\chi_{k}W_{s}v=\chi_{k}W_{s}\langle
r\rangle^{-s}f+[P_{m},\chi_{k}W_{s}]v.$

So in particular $P_{m}\chi_{k}W_{s}v\in L^{2}(\mathbb{R}_{+})$, hence
$\chi_{k}W_{s}v\in\mathcal{D}(P^{*}_{m})$. This implies that, for each $k$,

$\chi_{k}W_{s}v=(P_{m}-E-i\varepsilon)^{-1}(\chi_{k}W_{s}\langle
r\rangle^{-s}f+[P_{m},\chi_{k}W_{s}]v).$ (4.17)

We complete the proof of (4.16) by sending $k\to\infty$ in (4.17) and noting
that the $[P_{m},\chi_{k}W_{s}]$, $[P_{m},W_{s}]$ are zero on a fixed
neighborhood of $r=0$ (independent of $k$) and that they are uniformly bounded
$H^{2}(\mathbb{R}_{+})\to L^{2}(\mathbb{R}_{+})$ because $s\leq 1$.

Now, let $v_{k}=\chi(k^{-1}r)W_{s}v\in\mathcal{D}_{\operatorname{comp},m}$,
$k\in\mathbb{N}$, and note that $v_{k}\to W_{s}v$ in $\mathcal{D}_{m}$.
Observe that

$\|\langle
r\rangle^{-s}((W_{s})^{-1}v_{k}-v)\|_{L^{2}}\leq\|(v_{k}-W_{s}v)\|_{L^{2}}\to
0,$ (4.18)

and by (4.15),

$\displaystyle\|\langle
r\rangle^{s}(P-E-i\varepsilon)((W_{s})^{-1}v_{k}-v)\|_{L^{2}}$
$\displaystyle\leq
C\|W_{s}(P_{m}-E-i\varepsilon)((W_{s})^{-1}v_{k}-v)\|_{L^{2}}$ (4.19)
$\displaystyle\leq C_{\varepsilon,h}\|v_{k}-W_{s}v\|_{\mathcal{D}_{m}}\to 0.$

Finally, applying Lemma 4.1,

$\|\langle r\rangle^{-s}(W_{s})^{-1}v_{k}\|_{L^{2}}\leq
Ce^{C(1+|m|^{1/2})/h}\|\langle
r\rangle^{s}(P_{m}-E-i\varepsilon)(W_{s})^{-1}v_{k}\|_{L^{2}}.$

Sending $k\to\infty$ and using (4.18), (4.19), and the definition of $v$, this
implies

$\|\langle r\rangle^{-s}(P_{m}-E-i\varepsilon)^{-1}\langle
r\rangle^{-s}f\|_{L^{2}}\leq Ce^{C(1+|m|^{1/2})/h}\|f\|_{L^{2}},$

completing the proof of the proposition.
∎

##  5\. Exponential estimates for the resolvent

In this section we prove Theorem 1. As before, we use separation of variables
to reduce estimating the resolvent of $P$ to estimating the resolvent of each
$P_{m}$. By Proposition 1.1, for every $m\geq-h^{2}/4$ and
$0\leq\varepsilon\leq 1$

$\|\langle r\rangle^{-s}(P_{m}-E-i\varepsilon)^{-1}\langle
r\rangle^{-s}\|_{L^{2}(\mathbb{R}_{+})\to L^{2}(\mathbb{R}_{+})}\leq
e^{C(|m|^{1/2}+1)/h}.$

Thus it is enough to prove the following lemma.

######  Lemma 5.1.

There are $C$ and $h_{0}$ such that

$|K(r,r^{\prime})|=|K(r,r^{\prime},m,E,0,h)|\leq\frac{C}{h^{2}}$

holds whenever $0<r\leq r^{\prime}$, $m\geq M_{+}$, and $0<h<h_{0}$.

###### Proof.

We use the fact that $u_{0}(0)=0$ and $u_{0}^{\prime}(r)\geq 0$ when $r\leq
R_{0}$, and consider separately three cases.

  1. (1)

Suppose that $R_{0}\leq r\leq r^{\prime}$. Then the result follows from the
stronger estimate proved in Lemma 3.2.

  2. (2)

Suppose that $r\leq R_{0}\leq r^{\prime}$. Then

$|K(r,r^{\prime})|=u_{0}(r)\left|\frac{u_{1}(r^{\prime})}{h^{2}W}\right|\leq
u(R_{0})\frac{|u_{1}(r^{\prime})|}{h^{2}|W|}=|K(R_{0},r^{\prime})|\lesssim
h^{-1},$

where we used $u_{0}^{\prime}\geq 0$ on $(0,R_{0})$ and then (3.3).

  3. (3)

Suppose that $r\leq r^{\prime}\leq R_{0}$. Dividing the definition of the
Wronskian, $u_{0}u_{1}^{\prime}-u_{0}^{\prime}u_{1}=W$, by $u_{0}^{2}$ and
integrating both sides gives

$u_{1}(r)=u_{0}(r)\Big{(}\frac{u_{1}(R_{0})}{u_{0}(R_{0})}+\int_{R_{0}}^{r}\frac{W}{u_{0}(s)^{2}}ds\Big{)},$

thus, using again $u_{0}^{\prime}\geq 0$, on $(0,R_{0})$

$|u_{0}(r)u_{1}(r^{\prime})|\leq
u_{0}(R_{0})|u_{1}(R_{0})|+(R_{0}-r^{\prime})|W|.$

Plugging into (2.9), using (3.3) on the first term, and estimating the second
term directly gives the result.

∎

##  Appendix A Bessel functions

In this paper we use the Bessel functions $J_{\nu}(z)$ and $Y_{\nu}(z)$ [DLMF,
Section 10.2], where $\nu\geq 0$ and $z>0$, and below we review some standard
facts about them. By [DLMF, Equation 10.13.1], the differential equation

$\partial_{r}^{2}w+\left(\lambda^{2}-r^{-2}(\nu^{2}-\tfrac{1}{4})\right)w=0,$
(A.1)

is solved by $w_{1}=r^{1/2}J_{\nu}(\lambda r)$ and
$w_{2}=r^{1/2}Y_{\nu}(\lambda r)$, for any $\lambda>0$. By [DLMF, Equation
10.5.2], we have the Wronskian formula

$w_{1}\partial_{r}w_{2}-w_{2}\partial_{r}w_{1}=2\pi^{-1}.$ (A.2)

We use upper and lower bounds for $J$ and $Y$ derived from Olver’s uniform
asymptotics for large values of $\nu$. To state them, we use the notation
$a\asymp b$ to mean $a\lesssim b$ and $b\lesssim a$. We define a decreasing
bijection $(0,\infty)\ni z\mapsto\zeta(z)\in\mathbb{R}$ by

$\zeta=\begin{cases}\phantom{-}\frac{3}{2}\big{(}\int_{z}^{1}t^{-1}(1-t^{2})^{1/2}dt\big{)}^{2/3},\qquad&z\leq
1,\\\
-\frac{3}{2}\big{(}\int_{1}^{z}t^{-1}(t^{2}-1)^{1/2}dt\big{)}^{2/3},\qquad&z\geq
1,\end{cases}$ (A.3)

and use the Airy functions

$\begin{split}\textrm{Ai}(x)&=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac{t^{3}}{3}+xt\right)dt,\\\
\textrm{Bi}(x)&=\frac{1}{\pi}\int_{0}^{\infty}\left(e^{-\frac{t^{3}}{3}+xt}+\sin\left(\tfrac{t^{3}}{3}+xt\right)\right)dt.\end{split}$
(A.4)

Then, by [DLMF, Section 10.20] and [AbSt, Sections 9.3.35–46], we have

$\begin{split}J_{\nu}(\nu
z)&\asymp\left(\frac{\zeta}{1-z^{2}}\right)^{1/4}\left(\nu^{-1/3}\textrm{Ai}(\nu^{2/3}\zeta)+\nu^{-5/3}B_{0}(\zeta)\textrm{Ai}^{\prime}(\nu^{2/3}\zeta)\right),\\\
-zJ^{\prime}_{\nu}(\nu
z)&\asymp\left(\frac{1-z^{2}}{\zeta}\right)^{1/4}\left(\nu^{-2/3}\textrm{Ai}^{\prime}(\nu^{2/3}\zeta)+\nu^{-4/3}C_{0}(\zeta)\textrm{Ai}(\nu^{2/3}\zeta)\right),\\\
-Y_{\nu}(\nu
z)&\asymp\left(\frac{\zeta}{1-z^{2}}\right)^{1/4}\left(\nu^{-1/3}\textrm{Bi}(\nu^{2/3}\zeta)+\nu^{-5/3}B_{0}(\zeta)\textrm{Bi}^{\prime}(\nu^{2/3}\zeta)\right),\\\
zY^{\prime}_{\nu}(\nu
z)&\asymp\left(\frac{1-z^{2}}{\zeta}\right)^{1/4}\left(\nu^{-2/3}\textrm{Bi}^{\prime}(\nu^{2/3}\zeta)+\nu^{-4/3}C_{0}(\zeta)\textrm{Bi}(\nu^{2/3}\zeta)\right),\end{split}$
(A.5)

uniformly for $\nu\gg 1$ and $z>0$, where $B_{0}$ and $C_{0}$ are positive
smooth functions of $\zeta$ obeying

$B_{0}(\zeta)\asymp\begin{cases}\zeta^{-1/2},\qquad&\zeta\geq 1,\\\
\zeta^{-2},\qquad&\zeta\leq-1,\end{cases}\qquad
C_{0}(\zeta)\asymp\begin{cases}\zeta^{1/2}\qquad&\zeta\geq 1,\\\
-\zeta^{-1},\qquad&\zeta\leq-1.\end{cases}$

These bounds become simpler when $z$ is small. More specifically, by [DLMF,
Section 9.7], we have

$\begin{array}[]{rclrcl}\textrm{Ai}(x)&\asymp&x^{-1/4}\exp(-\tfrac{2}{3}x^{3/2}),&-\textrm{Ai}^{\prime}(x)&\asymp&x^{1/4}\exp(-\tfrac{2}{3}x^{3/2}),\\\
\textrm{Bi}(x)&\asymp&x^{-1/4}\exp(\tfrac{2}{3}x^{3/2}),&\qquad\textrm{Bi}^{\prime}(x)&\asymp&x^{1/4}\exp(\tfrac{2}{3}x^{3/2}),\end{array}$
(A.6)

for $x\gg 1$. Hence, for any $z_{0}\in(0,1)$, we have

$\begin{array}[]{rclrcl}J_{\nu}(\nu
z)&\asymp&\nu^{-1/2}e^{-\nu\xi},&J^{\prime}_{\nu}(\nu
z)&\asymp&z^{-1}J_{\nu}(\nu z),\\\ -Y_{\nu}(\nu
z)&\asymp&\nu^{-1/2}e^{\nu\xi},&\qquad Y^{\prime}_{\nu}(\nu
z)&\asymp&-z^{-1}Y_{\nu}(\nu z),\end{array}$ (A.7)

uniformly for $z\in(0,z_{0}]$ and $\nu\gg 1$, where

$\xi=\int_{z}^{1}t^{-1}(1-t^{2})^{1/2}dt.$ (A.8)

The bounds also become simpler when $z$ is large. More specifically, by by
[DLMF, Section 9.7], we have

$\begin{split}\textrm{Ai}(-x)&\asymp
x^{-1/4}\left(\cos(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})+\tfrac{5}{48}x^{-3/2}\sin(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})\right),\\\
-\textrm{Ai}^{\prime}(-x)&\asymp
x^{1/4}\left(\sin(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})-\tfrac{7}{48}x^{-3/2}\cos(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})\right),\\\
\textrm{Bi}(-x)&\asymp
x^{-1/4}\left(-\sin(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})+\tfrac{5}{48}x^{-3/2}\cos(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})\right),\\\
-\textrm{Bi}^{\prime}(-x)&\asymp
x^{1/4}\left(\cos(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})+\tfrac{7}{48}x^{-3/2}\sin(\tfrac{2}{3}x^{3/2}-\tfrac{\pi}{4})\right),\end{split}$
(A.9)

for $x\gg 1$. In particular, combining (A.6) and (A.9) gives

$\exp(\tfrac{2}{3}x_{+}^{3/2})|\textrm{Ai}(x)|+\exp(-\tfrac{2}{3}x_{+}^{3/2})|\textrm{Bi}(x)|\lesssim\langle
x\rangle^{-1/4},$
$\exp(\tfrac{2}{3}x_{+}^{3/2})|\textrm{Ai}^{\prime}(x)|+\exp(-\tfrac{2}{3}x_{+}^{3/2})|\textrm{Bi}^{\prime}(x)|\lesssim\langle
x\rangle^{1/4},$

uniformly for $x\in\mathbb{R}$, where $x_{+}=\max(x,0)$. Hence

$\begin{split}|J_{\nu}(\nu
z)|&\lesssim\left|\frac{\zeta}{1-z^{2}}\right|^{1/4}e^{-\nu\xi_{+}}\left(\nu^{-1/2}\langle\zeta\rangle^{-1/4}+\nu^{-3/2}\langle\zeta\rangle^{-1/4}\right)\lesssim\nu^{-1/2}\langle
z\rangle^{-1/2}e^{-\nu\xi_{+}},\\\ |Y_{\nu}(\nu
z)|&\lesssim\left|\frac{\zeta}{1-z^{2}}\right|^{1/4}e^{\nu\xi_{+}}\left(\nu^{-1/2}\langle\zeta\rangle^{-1/4}+\nu^{-3/2}\langle\zeta\rangle^{-1/4}\right)\lesssim\nu^{-1/2}\langle
z\rangle^{-1/2}e^{\nu\xi_{+}},\end{split}$ (A.10)

uniformly for $\nu\gg 1$ and $z>0$, where $\xi_{+}=\xi$ as given by (A.8) when
$z<1$, and $\xi_{+}=0$ when $z\geq 1$.

##  Appendix B Properties of $u_{0}$

In this Appendix, we prove Lemma 2.1, following [Ya10, Chapter 4, Section
1.1]. Recall our notation from Section 2,

$\begin{gathered}\varphi_{J}(r)=r^{1/2}J_{\nu}(\lambda
r),\qquad\varphi_{Y}(r)=r^{1/2}Y_{\nu}(\lambda r),\\\
\nu=h^{-1}\big{(}m+\frac{h^{2}}{4}\big{)}^{1/2},\quad
m\geq-\frac{h^{2}}{4},\quad\lambda=\frac{\sqrt{E+i\varepsilon}}{h},\quad
E\in[E_{\min},E_{\max}],\quad\varepsilon\geq 0.\end{gathered}$

Then put $\varphi_{0}=\varphi_{J}$ and define inductively

$\varphi_{n+1}(r)=\frac{\pi}{2h^{2}}\int_{0}^{r}(\varphi_{Y}(r)\varphi_{J}(r^{\prime})-\varphi_{J}(r)\varphi_{Y}(r^{\prime}))V(r^{\prime})\varphi_{n}(r^{\prime})dr^{\prime}.$
(B.1)

We shall prove suitable estimates on the $\varphi_{n}$ to be able to put

$u_{0}(r)=\sum_{n=0}^{\infty}\varphi_{n}(r),$ (B.2)

with the series and its first derivative converging uniformly as
$(r,E,\varepsilon)$ vary in compact subsets of
$\mathbb{R}_{+}\times[E_{\min},E_{\max}]\times[0,\infty)$.

###### Proof of Lemma 2.1.

To prove the lemma, we recall several estimates on $\varphi_{Y}$ and
$\varphi_{J}$. Given $r^{*}>0$, by [DLMF, Sections, 10.6 (ii), 10.7(i)] there
is $C_{0}$ depending continuously on $r^{*}>0$, $\nu\geq 0$, $\delta>0$, and
$\lambda\neq 0$ such that

$\begin{gathered}|\varphi_{J}(r)|r^{-\nu-\frac{1}{2}}+|\varphi_{Y}(r)|r^{\nu+\delta-\frac{1}{2}}\leq
C_{0}\\\
|\varphi^{\prime}_{J}(r)|r^{-\nu+\frac{1}{2}}+|\varphi^{\prime}_{Y}(r)|r^{\nu+\delta+\frac{1}{2}}\leq
C_{0}\\\
|\varphi^{\prime\prime}_{J}(r)|r^{-\nu+\frac{3}{2}}+|\varphi^{\prime\prime}_{Y}(r)|r^{\nu+\delta+\frac{3}{2}}\leq
C_{0}\end{gathered},\qquad r\in(0,r^{*}].$ (B.3)

Next,

$|\varphi_{0}(r)|\leq
C_{0}r^{\nu+\frac{1}{2}},\qquad|\varphi_{0}^{\prime}(r)|\leq
C_{0}r^{\nu-\frac{1}{2}},\qquad|\varphi_{0}^{\prime\prime}(r)|\leq
C_{0}r^{\nu-\frac{3}{2}},\qquad r\in(0,r^{*}].$ (B.4)

To see that the series (B.2) converges uniformly on compact sets, we claim
that

$|\varphi_{n}(r)|\leq C_{0}\cdots C_{n}r^{\nu+n(2-\delta)+\frac{1}{2}},\qquad
r\in(0,r^{*}].$ (B.5)

where

$C_{n}=C_{0}^{2}\pi\sup|V|\frac{2\nu+(2n-1)(2-\delta)+2}{2h^{2}(2-\delta)n(2\nu+(n-1)(2-\delta)+2)}.$
(B.6)

Once we have (B.5), since $\lim C_{n}=0$, the Weierstrass and ratio tests
shows that the convergence of (B.2) is uniform as $(r,E,\varepsilon)$ vary in
compact subsets of $\mathbb{R}_{+}\times[E_{\min},E_{\max}]\times[0,\infty)$.
Moreover, since $\varphi_{0}\sim r^{\nu+\frac{1}{2}}$ as $r\to 0$, $u_{0}>0$
near zero.

To obtain (B.5), we start with $n=1$,

$\begin{split}\frac{2h^{2}}{C_{0}^{3}\pi\sup|V|}&|\varphi_{1}(r)|\\\ &\leq
r^{-\nu-\delta+\frac{1}{2}}\int_{0}^{r}(r^{\prime})^{\nu+\frac{1}{2}}(r^{\prime})^{\nu+\frac{1}{2}}dr^{\prime}+r^{\nu+\frac{1}{2}}\int_{0}^{r}(r^{\prime})^{-\nu-\delta+\frac{1}{2}}(r^{\prime})^{\nu+\frac{1}{2}}dr^{\prime}\\\
&=(2\nu+2)^{-1}r^{\nu+\frac{5}{2}-\delta}+(2-\delta)^{-1}r^{\nu+\frac{5}{2}-\delta}=\frac{2\nu+4-\delta}{(2\nu+2)(2-\delta)}r^{\nu+\frac{5}{2}-\delta},\qquad
r\in(0,r^{*}],\end{split}$

or

$|\varphi_{1}(r)|\leq C_{0}C_{1}r^{\nu+\frac{5}{2}-\delta},\qquad
r\in(0,r^{*}],$

where $C_{1}=C_{0}^{2}\sup|V|(2\nu+4-\delta)/(2\pi h^{2}(2\nu+2)(2-\delta))$.

Now, suppose the claim (B.5) holds with $n=k$ for some $k\geq 1$, with $C_{k}$
given by (B.6). Then,

$\begin{split}&\frac{2h^{2}}{C_{0}C_{1}C_{2}\dots
C_{k}C_{0}^{2}\pi\sup|V|}|\varphi_{k+1}(r)|\\\ &\leq
r^{-\nu+\frac{1}{2}-\delta}\int_{0}^{r}(r^{\prime})^{\nu+\frac{1}{2}}(r^{\prime})^{\nu+k(2-\delta)+\frac{1}{2}}dr^{\prime}+r^{\nu+\frac{1}{2}}\int_{0}^{r}(r^{\prime})^{-\nu+\frac{1}{2}-\delta}(r^{\prime})^{\nu+k(2-\delta)+\frac{1}{2}}dr^{\prime}\\\
&=(2\nu+k(2-\delta)+2)^{-1}r^{\nu+2(k+1)+\frac{1}{2}}+((k+1)(2-\delta))^{-1}r^{\nu+(k+1)(2-\delta)+\frac{1}{2}},\qquad
r\in(0,r^{*}],\end{split}$

or

$|\varphi_{k+1}(r)|\leq C_{0}C_{1}\dots
C_{k+1}r^{\nu+(k+1)(2-\delta)+\frac{1}{2}},\qquad r\in(0,r^{*}],$

where

$C_{k+1}=C_{0}^{2}\pi\sup|V|\frac{2\nu+(2k+1)(2-\delta)+2}{2h^{2}(2-\delta)(k+1)(2\nu+k(2-\delta)+2)}.$

In particular, (B.5) holds by induction.

To see that the first derivative of (B.2) converges uniformly on compact sets,
observe that for $n\geq 0$,

$\begin{split}\frac{|\varphi_{n+1}^{\prime}(r)|}{\sup|V|}&=\frac{\pi}{2\sup|V|h^{2}}\Big{|}\int_{0}^{r}(\varphi_{Y}^{\prime}(r)\varphi_{J}(r^{\prime})-\varphi_{J}^{\prime}(r)\varphi_{Y}(r^{\prime}))V(r^{\prime})\varphi_{n}(r^{\prime})dr^{\prime}\Big{|}\\\
&\leq(2\pi h^{2})^{-1}C_{0}\dots
C_{n}C_{0}^{2}\Big{(}r^{-\nu-\frac{1}{2}-\delta}\int_{0}^{r}(r^{\prime})^{2\nu+n(2-\delta)+1})dr^{\prime}+r^{\nu-\frac{1}{2}}\int_{0}^{r}(r^{\prime})^{n(2-\delta)+1-\delta}dr^{\prime}\Big{)}\\\
&\leq(2\pi h^{2})^{-1}C_{0}\dots
C_{n}C_{0}^{2}r^{\nu+n(2-\delta)+\frac{3}{2}-\delta}\Big{(}\frac{1}{2\nu+n(2-\delta)+2}+\frac{1}{(n+1)(2-\delta)}\Big{)}\\\
&\leq C_{0}\cdots C_{n+1}r^{\nu+(n+1)(2-\delta)-\frac{1}{2}},\qquad
r\in(0,r^{*}].\end{split}$ (B.7)

Now (2.3) and (2.4) follow from (B.5) and (B.7) by summing (B.1) from $n=0$ to
$n=N$ and sending $N\to\infty$.

Next, to see that (2.5) holds, observe that for any $\delta>0$,

$\sum_{n=1}^{\infty}\varphi_{n}(r)\leq
C_{\nu,\delta}r^{\nu+\frac{5}{2}-\delta},\qquad\sum_{n=1}^{\infty}\varphi_{n}^{\prime}(r)\leq
C_{\nu,\delta}r^{\tfrac{3}{2}-\delta}.$

Therefore,

$\displaystyle\lim_{r\to
0^{+}}\big{(}\tfrac{n-1}{2r}u_{0}(r)-u_{0}^{\prime}(r)\big{)}r^{-\nu+1/2}$
$\displaystyle=\lim_{r\to
0^{+}}\big{(}\tfrac{n-1}{2r}\varphi_{0}(r)-\varphi_{0}^{\prime}(r)\big{)}r^{-\nu+1/2}$
$\displaystyle=\lim_{r\to 0^{+}}\big{(}\tfrac{n-2}{2}J_{\nu}(\lambda
r)r^{-1/2}-\lambda J_{\nu}^{\prime}(\lambda r)r^{1/2}\big{)}r^{-\nu+1/2}$
$\displaystyle=\frac{\lambda^{\nu}}{2^{\nu}\Gamma(\nu+1)}\big{(}\frac{n-2}{2}-\nu\big{)}.$

Since $\nu\geq 0$, (2.5) holds.
∎

Now that we have Lemma 2.1, we show that $u_{0}$ has useful monotonicity
properties when $m$ is large enough. Recall the defining property (1.2) of
$M_{0}$ and consider $m\geq M_{0}$.

######  Lemma B.1.

Let $M_{0}$ as in (1.2) and $R_{1}$ as in (1.3). Then for $m\geq M_{0}$,
$u_{0}(r),u_{0}^{\prime}(r)\geq 0$ for $r\in(0,R_{0}]$.

###### Proof.

First, by Lemma 2.1, we have

$u^{\prime\prime}_{0}=-h^{-2}(E-V-mr^{-2})u_{0}.$

Using $m\geq M_{0}$ and the definition (1.2), $E-V-mr^{-2}\leq 0$ on
$(0,R_{0}]$. Hence $u_{0}(r)u_{0}^{\prime\prime}(r)\geq 0$ on $(0,R_{0}]$.
Since also $u_{0}(0)=0$ and $u_{0}(r)>0$ for $r>0$ small enough, the proof is
completed by the following lemma.
∎

######  Lemma B.2.

Suppose $f,f^{\prime}\in AC_{\operatorname{loc}}(a,b)$, $f\in
C([a,b];\mathbb{R})$, $f(a)=0$, $f^{\prime}\in C(a,b]$, and
$f^{\prime\prime}(t)f(t)\geq 0$, for almost every in $t\in(a,b)$ in the sense
of Lebesgue measure. Then $f$ has a fixed sign in $[a,b]$ and $f(b)$,
$f^{\prime}(b)$ have the same sign.

###### Proof.

If $f$ is identically 0 on $[a,b]$, then the claim is trivially true.
Therefore, we will assume $f$ is not identically 0. Assume $f$ attains its
extremum $L\neq 0$ at an interior point of $(a,b)$. Replacing $f$ by $-f$ if
necessary, we may assume $L>0$. Set

$x^{*}=\inf\\{x\in(a,b):f(x)=L\\}.$

Then $f>0$ near $x^{*}$, hence $f^{\prime\prime}\geq 0$ almost everywhere, in
a neighborhood of $x^{*}$. Also $f^{\prime}(x^{*})=0$. Now, for $x$
sufficiently close to but less than $x^{*}$,

$-f^{\prime}(x)=\int^{x^{*}}_{x}f^{\prime\prime}(s)ds\geq 0\implies
f^{\prime}(x)\leq 0.$

But then, by the mean value theorem, for $a<x<x^{*}$ and some
$\overline{x}\in(x,x^{*})$,

$0<f(x^{*})-f(x)=f^{\prime}(\overline{x})(x^{*}-x)\leq 0,$

a contradiction.

So the extrema of $f$ must occur at the endpoints: $f(b)$ is the maximum or
minimum of $f$ on $[a,b]$, and $f(a)=0$ is the minimum or maximum,
respectively. So $f$ has a fixed sign, hence also $f^{\prime\prime}$ has this
sign almost everywhere.

Using the mean value theorem again,

$\frac{f(b)-f(a)}{b-a}=\frac{f(b)}{b-a}=f^{\prime}(\overline{x}),\qquad\text{some
$\overline{x}\in(a,b).$}$

So $f(b)$ and $f^{\prime}(\overline{x})$ have the same sign, so

$f^{\prime}(b)=f^{\prime}(\overline{x})+\int_{\overline{x}}^{b}f^{\prime\prime}(s)ds$

has the same sign as $f(b)$ as well.
∎

##  Appendix C Consequences of Meshkov’s example for resolvent estimates

Recall from Fredholm theory [DyZw19, Theorems 3.8 and equation 6.0.2] that,
for $V\in L^{\infty}_{\text{comp}}(\mathbb{R}^{n};\mathbb{C})$, the cutoff
resolvent $\chi(-h^{2}\Delta+V-z)^{-1}\chi$, $\chi\in
C_{0}^{\infty}(\mathbb{R}^{n})$, is meromorphic in
$(E_{\min}/2,2E_{\max})+i(0,\infty)$ and continues meromorphically to
$(E_{\min}/2,2E_{\max})+i\mathbb{R}$.

To see that the optimal power in a resolvent estimate for $L^{\infty}$,
compactly supported, complex valued potentials is $h^{-4/3}$, recall that
[Me92] constructs a _complex-valued_ potential $V\in
L^{\infty}(\mathbb{R}^{2})$ and a function $u$ such that

$(-\Delta+V)u=0,\quad\text{in }\mathbb{R}^{2},\qquad|u(x)|\leq
C\exp(-C|x|^{4/3}),\qquad\|u\|_{L^{2}(\mathbb{R}^{n})}=1.$

Changing variables, $y=hx$, and putting $w(y)=u(h^{-1}y)$,
$V_{h}(y)=V(h^{-1}y)$, we have
$\|V_{h}\|_{L^{\infty}}\leq C<\infty$,

$(-h^{2}\Delta+V_{h})w=0,\qquad|w(y)|\leq C\exp(-C|y|h^{-4/3}).$

Let $\chi\in C_{0}^{\infty}(B(0,3))$ with $\chi\equiv 1$ near $B(0,2)$ and
$\tilde{\chi}\in C_{0}^{\infty}(B(0,3))$ with $\tilde{\chi}\equiv 1$ on
$\mathop{\rm supp}\chi$. Then, for $E\in[E_{\min},E_{\max}]$,

$\displaystyle\|\tilde{\chi}(-h^{2}\Delta$ $\displaystyle+(V_{h}+E)-E)(\chi
w)\|_{L^{2}(\mathbb{R}^{n})}=\|[-h^{2}\Delta,\chi]w\|_{L^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq Ch\|w\|_{H_{h}^{1}(B(0,3)\setminus B(0,2))}$
$\displaystyle\leq Ch(\|-h^{2}\Delta w\|_{L^{2}(B(0,4)\setminus
B(0,1))}+\|w\|_{L^{2}(B(0,4)\setminus B(0,1))})$ $\displaystyle\leq
Ch(\|V_{h}w\|_{L^{2}(B(0,4)\setminus B(0,1))}+\|w\|_{L^{2}(B(0,4)\setminus
B(0,1))})\leq Che^{-Ch^{-4/3}},$

For $h$ small enough,

$\|\chi w\|_{L^{2}}\geq h-h\|u\|_{L^{2}(\mathbb{R}^{2}\setminus
B(0,h^{-1})}\geq\tfrac{h}{2}.$

Therefore, if the cutoff resolvent
$\tilde{\chi}(-h^{2}\Delta+\tilde{\chi}(V_{h}+E)-E-i0)^{-1}\tilde{\chi}$
exists, it must have

$\|\tilde{\chi}(-h^{2}\Delta+\tilde{\chi}(V_{h}-E)-E-i0)^{-1}\tilde{\chi}\|_{L^{2}(\mathbb{R}^{n})\to
L^{2}(\mathbb{R}^{n})}\geq C\exp(Ch^{-4/3}).$

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