# Weak Convergence of Spectral Shift Functions Revisited

Carson Connard  Department of Mathematics, Kansas State University, 1228 N.
Martin Luther King Jr Dr., Manhattan, KS 66506, USA<EMAIL_ADDRESS>,
Benjamin Ingimarson  Department of Mathematical Sciences, Carnegie Mellon
University, 5000 Forbes Avenue, Pittsburgh, PA 15289, USA<EMAIL_ADDRESS>,  Roger Nichols  Department of Mathematics (Dept. 6956), The University of
Tennessee at Chattanooga, 615 McCallie Ave., Chattanooga, TN 37403, USA
<EMAIL_ADDRESS>https://sites.google.com/mocs.utc.edu/rogernicholshomepage/home and  Andrew
Paul  Department of Mathematics; University of California, San Diego; 9500
Gilman Drive, La Jolla, CA 92093-0112, USA<EMAIL_ADDRESS>https://anpaul.weebly.com

###### Abstract.

We study convergence of the spectral shift function for the finite interval
restrictions of a pair of full-line Schrödinger operators to an interval of
the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as
$\ell\to\infty$ in the case when the finite interval restrictions are
relatively prime to those with Dirichlet boundary conditions. Using a Krein-
type resolvent identity we show that the spectral shift function for the
finite interval restrictions converges weakly to that for the pair of full-
line Schrödinger operators as the length of the interval tends to infinity.

###### Key words and phrases:

Spectral shift function, Schrödinger operator, Sturm–Liouville operator,
coupled boundary conditions, weak convergence.

###### 2020 Mathematics Subject Classification:

Primary 34L40, 35J10; Secondary 34B24, 47B25, 47E05.

##  1\. Introduction

It is a classical fact that the one-dimensional self-adjoint Schrödinger
operators $H_{0}=-d^{2}/dx^{2}$ and $H=-d^{2}/dx^{2}+V$, with $V$ real-valued
and Lebesgue integrable on ${\mathbb{R}}$, are resolvent comparable in the
sense that their resolvent difference

$\big{(}H-zI_{L^{2}({\mathbb{R}})}\big{)}^{-1}-\big{(}H_{0}-zI_{L^{2}({\mathbb{R}})}\big{)}^{-1}$
(1.1)

belongs to the trace class for all $z\in{\mathbb{C}}\setminus{\mathbb{R}}$.
Consequently, there is a corresponding unique real-valued spectral shift
function $\xi(\,\cdot\,;H,H_{0})$ defined a.e. on ${\mathbb{R}}$ that vanishes
in a neighborhood of $-\infty$ and satisfies

$\int_{-\infty}^{\infty}\frac{|\xi(\lambda;H,H_{0})|}{1+\lambda^{2}}\,\rm{d}\lambda<\infty.$
(1.2)

Moreover, for a wide class of functions $f$ (see (A.25)–(A.26)), Krein’s trace
formula holds:

$\mbox{tr}_{L^{2}({\mathbb{R}})}{(f(H)-f(H_{0}))}=\int_{-\infty}^{\infty}f^{\prime}(\lambda)\xi(\lambda;H,H_{0})\,\rm{d}\lambda,$
(1.3)

where $\mbox{tr}_{L^{2}({\mathbb{R}})}{(\,\cdot\,)}$ denotes the trace
functional.

When $H_{0}$ and $H$ are restricted to a finite interval, say $(-\ell,\ell)$
with $\ell\in{\mathbb{N}}$, to obtain the finite interval Schrödinger
operators $H_{0,\ell}$ and $H_{\ell}$ (imposing a fixed common choice of self-
adjoint boundary conditions at the endpoints $\pm\ell$, thereby ensuring the
self-adjointness of $H_{0,\ell}$ and $H_{\ell}$), the finite interval
restrictions $H_{0,\ell}$ and $H_{\ell}$ are also resolvent comparable, so a
unique real-valued spectral shift function
$\xi(\,\cdot\,;H_{\ell},H_{\ell,0})$ with properties analogous to those of
$\xi(\,\cdot\,;H,H_{0})$ exists. A problem of interest, dating back at least
to the work of Borovyk and Makarov [2] (see also [1] and the much earlier,
related work of Kirsch [8]) has been to study the modes of convergence of
$\xi(\,\cdot\,;H_{\ell},H_{0,\ell})$ to $\xi(\,\cdot\,;H,H_{0})$ as
$\ell\to\infty$. Since $\xi(\,\cdot\,;H_{\ell},H_{0,\ell})$ is integer-valued,
owing to the discrete nature of the spectra of $H_{\ell}$ and $H_{0,\ell}$,
and $\xi(\,\cdot\,;H,H_{0})$ is continuous and nonconstant on $(0,\infty)$,
pointwise convergence of $\xi(\,\cdot\,;H_{\ell},H_{0,\ell})$ to
$\xi(\,\cdot\,;H,H_{0})$ as $\ell\to\infty$ is immediately dismissed as
impossible. Borovyk and Makarov showed that a certain averaging is needed in
order to obtain convergence. More specifically, in [2], Borovyk and Makarov
proved that when Dirichlet boundary conditions are imposed at the endpoints of
the finite intervals, the sequence $\xi(\,\cdot\,;H_{\ell},H_{0,\ell})$
converges to $\xi(\,\cdot\,;H,H_{0})$ weakly in the sense that

$\lim_{\ell\to\infty}\int_{-\infty}^{\infty}g(\lambda)\xi(\lambda;H_{\ell},H_{0,\ell})\,\textrm{d}\lambda=\int_{-\infty}^{\infty}g(\lambda)\xi(\lambda;H,H_{0})\,\textrm{d}\lambda,\quad
g\in C_{c}({\mathbb{R}}),$ (1.4)

where $C_{c}({\mathbb{R}})$ denotes the set of compactly supported continuous
complex-valued functions on ${\mathbb{R}}$. To be as precise as possible, the
finite intervals considered in [2] are of the form $(0,\ell)$ and the infinite
interval is the half-line $(0,\infty)$, but the arguments given in [2] may be
modified to treat the symmetric intervals $(-\ell,\ell)$ and the line
${\mathbb{R}}$. Using an abstract approach based on modified Fredholm
determinants, (1.4) was extended to all separated self-adjoint boundary
conditions in [4] (the abstract approach is separately developed in [5]).
Actually, in [4], (1.4) was improved to

$\lim_{\ell\to\infty}\int_{-\infty}^{\infty}h(\lambda)\frac{\xi(\lambda;H_{\ell},H_{0,\ell})}{1+\lambda^{2}}\,\textrm{d}\lambda=\int_{-\infty}^{\infty}h(\lambda)\frac{\xi(\lambda;H,H_{0})}{1+\lambda^{2}}\,\textrm{d}\lambda,\quad
h\in C_{b}({\mathbb{R}}),$ (1.5)

where $C_{b}({\mathbb{R}})$ denotes the set of bounded continuous complex-
valued functions on ${\mathbb{R}}$. However, [4] does not discuss coupled
self-adjoint boundary conditions at $\pm\ell$, which are of the form

$\begin{pmatrix}y(\ell)\\\
y^{\prime}(\ell)\end{pmatrix}=e^{i\varphi}R\begin{pmatrix}y(-\ell)\\\
y^{\prime}(-\ell)\end{pmatrix}$ (1.6)

for a fixed parameter $\varphi\in[0,\pi)$ and a fixed matrix
$R=(R_{j,k})_{1\leq j,k\leq 2}\in{\mathbb{R}}^{2\times 2}$ with $\det{(R)}=1$.
In [9], (1.5) was shown to hold for a special subclass of the coupled boundary
conditions (1.6), namely those for which $R_{1,2}=0$, so that $R$ has the
special form

$R=\begin{pmatrix}a&0\\\ b&a^{-1}\end{pmatrix}$ (1.7)

for some $a,b\in{\mathbb{R}}$ with $a\neq 0$. The approach taken in [9] is
based on an analysis of the coefficient in the Krein identity connecting the
resolvent of the finite interval restriction with the boundary conditions
(1.6)–(1.7) to the resolvent of the finite interval restriction with Dirichlet
boundary conditions. In the case of (1.7), the difference of the resolvents of
the coupled and Dirichlet restrictions is a rank one operator, so the Krein
identity contains exactly one coefficient. The work in [9] leaves open the
question of convergence—in particular, the validity of (1.5)—when $R_{1,2}\neq
0$.

For $R_{1,2}\neq 0$, the situation is slightly more delicate as the resolvent
difference is then a rank two operator and the corresponding Krein identity is
more easily understood in terms of four coefficients, see Theorem 2.8 below
and [3, Theorem 3.2 $(i)$]. Our main aim here is to extend (1.5) to the case
$R_{1,2}\neq 0$. To do this, we carefully analyze the $\ell\to\infty$ behavior
of the four coefficients in the Krein identity and use the abstract criteria
for convergence of spectral shift functions developed in [5] to extend (1.5)
to the case $R_{1,2}\neq 0$. When combined with [4] and [9], the results we
obtain here show that (1.5) holds for all self-adjoint boundary conditions.

In Section 2, we rigorously define the full-line Schrödinger operators $H$ and
$H_{0}$ and their finite interval counterparts and recall some of their basic
properties. We also recall Krein’s resolvent identity and separately
investigate the behavior of its coefficients with respect to both the finite
interval length and the spectral parameter. The key results needed to invoke
the abstract criteria from [4] for convergence of spectral shift functions—in
particular, convergence of various Birman–Schwinger-type operators in the
trace and Hilbert–Schmidt classes—are established in Section 3. Finally, in
Section 4, we combine the results from Section 3 with the abstract convergence
criteria from [5] to obtain our main result on convergence of spectral shift
functions corresponding to coupled boundary conditions with $R_{1,2}\neq 0$.
For completeness, Appendix A recalls the convergence criteria from [5] in a
form that is suitably tailored to the application required in Section 4.

Notation: If $A\in{\mathbb{C}}^{m\times n}$ for some $m,n\in{\mathbb{N}}$,
then the $(j,k)$-entry of $A$ is denoted by $A_{j,k}$.
$\text{{SL}}_{2}{({\mathbb{R}})}$ denotes the special linear group of order
$2$ (i.e., the set of all matrices in ${\mathbb{R}}^{2\times 2}$ with
determinant equal to one). If $[a,b]\subseteq{\mathbb{R}}$, then
$\text{{AC}}([a,b])$ denotes the set of all functions
$f\colon[a,b]\to{\mathbb{R}}$ that are absolutely continuous. If
$(a,b)\subseteq{\mathbb{R}}$, then $\text{{AC}}_{\operatorname{loc}}{((a,b))}$
denotes the set of all $f\colon(a,b)\to{\mathbb{C}}$ that are locally
absolutely continuous on $(a,b)$ (i.e., absolutely continuous on every compact
subinterval of $(a,b)$). The symbol “a.e.” abbreviates the phrase “almost
everywhere (with respect to Lebesgue measure on ${\mathbb{R}}$),” and the
symbol “$\coloneqq$” means “defined to be equal to.” If
$f\colon(a,b)\to{\mathbb{R}}$, then $f_{\pm}=(|f|\pm f)/2$ denote the positive
and negative parts of $f$, respectively. If
$T\colon\operatorname{dom}{(T)}\subseteq{\mathcal{H}}\to{\mathcal{H}}$ is a
closable linear operator in the Hilbert space ${\mathcal{H}}$, then
$\overline{T}$ denotes the closure of $T$. If $T$ is a closed linear operator
in a Hilbert space, then $\rho(T)$ and $\sigma(T)$ denote the resolvent set
and spectrum of $T$, respectively. If $J\subseteq{\mathbb{R}}$ is an open
interval, then $H^{k}(J)=W^{k,2}(J)$ denotes the Sobolev space of order
$k\in{\mathbb{N}}$. If ${\mathcal{H}}$ is a separable complex Hilbert space,
then ${\mathcal{B}}({\mathcal{H}})$ and ${\mathcal{B}}_{p}({\mathcal{H}})$,
$p\in[1,\infty)$, denote the set of bounded linear operators and the
$\ell^{p}$-based Schatten–von Neumann trace ideals on ${\mathcal{H}}$,
respectively. $C_{c}({\mathbb{R}})$, $C_{b}({\mathbb{R}})$, and
$C_{\infty}({\mathbb{R}})$ denote the sets of continuous complex-valued
functions on ${\mathbb{R}}$ that are compactly supported, bounded, and
converge to zero at $\pm\infty$, respectively.

##  2\. One-Dimensional Schrödinger Operators

We begin by introducing the following hypothesis which shall be assumed
throughout.

######  Hypothesis 2.1.

$V\in L^{1}({\mathbb{R}})$ is real-valued a.e., $\varphi\in[0,\pi)$, and
$R\in\text{{SL}}_{2}{({\mathbb{R}})}$ with $R_{1,2}\neq 0$.

In turn, the full-line Schrödinger operator with potential $V$ is defined by

$Hf\coloneqq-f^{\prime\prime}+Vf,\quad
f\in\operatorname{dom}{(H)}\coloneqq\big{\\{}g\in
L^{2}({\mathbb{R}})\,\big{|}\,g,g^{\prime}\in\text{{AC}}_{\operatorname{loc}}{({\mathbb{R}})},\,-g^{\prime\prime}+Vg\in
L^{2}({\mathbb{R}})\big{\\}}.$ (2.1)

Since the condition $V\in L^{1}({\mathbb{R}})$ implies that the differential
expression $\tau\coloneqq-d^{2}/dx^{2}+V(x)$ is in the limit point case at
$\pm\infty$, the operator $H$ is self-adjoint in the Hilbert space
$L^{2}({\mathbb{R}})$ equipped with the inner product

$\langle
f,g\rangle_{L^{2}({\mathbb{R}})}\coloneqq\int_{-\infty}^{\infty}{\overline{f(x)}g(x)\,\mathrm{d}x},\quad
f,g\in L^{2}({\mathbb{R}}).$ (2.2)

Moreover, the sesquilinear form uniquely associated to $H$ is

$\mathfrak{Q}[f,g]\coloneqq\int_{-\infty}^{\infty}\Big{[}\overline{f^{\prime}(x)}g^{\prime}(x)+\overline{f(x)}V(x)g(x)\Big{]}\,\mathrm{d}x,\quad
f,g\in\operatorname{dom}{(\mathfrak{Q})}\coloneqq H^{1}(\mathbb{R}).$ (2.3)

######  Remark 2.2.

In the case when $V=0$ a.e. on ${\mathbb{R}}$, the corresponding “free”
Schrödinger operator and its sesquilinear form are denoted by $H^{{(0)}}$ and
$\mathfrak{Q}^{{(0)}}$, respectively. $\diamond$

The following fixed quantities will play an important role:

$M_{V}\coloneqq\int_{-\infty}^{\infty}V_{-}(x)\,\mathrm{d}x,\quad
N_{R}\coloneqq\frac{|R_{1,1}|+|R_{2,2}|+2}{|R_{1,2}|}.$ (2.4)

In fact, standard estimates imply that $H$ is lower semibounded with a lower
bound expressed in terms of the quantity $M_{V}$.

######  Lemma 2.3.

If Hypothesis 2.1 holds, then

$\mathfrak{Q}[f,f]\geq-M_{V}(1+M_{V})\|f\|_{L^{2}({\mathbb{R}})}^{2},\quad
f\in H^{1}({\mathbb{R}}).$ (2.5)

In particular, if $V_{-}=0$ a.e. on ${\mathbb{R}}$, then $H$ is nonnegative.

###### Proof.

Let $f\in H^{1}({\mathbb{R}})$. Writing $V=V_{+}-V_{-}$, one obtains

$\displaystyle\mathfrak{Q}[f,f]$
$\displaystyle=\int_{-\infty}^{\infty}\Big{[}|f^{\prime}(x)|^{2}+V(x)|f(x)|^{2}\Big{]}\,\mathrm{d}x\geq\|f^{\prime}\|_{L^{2}({\mathbb{R}})}^{2}-\int_{-\infty}^{\infty}V_{-}(x)|f(x)|^{2}\,\mathrm{d}x.$
(2.6)

If $M_{V}=0$, then $V_{-}=0$ a.e. on ${\mathbb{R}}$ and (2.6) implies
$\mathfrak{Q}[f,f]\geq 0$, which is (2.5) for $M_{V}=0$. Thus, $H$ is
nonnegative in this case.

If $M_{V}>0$, then one bounds the second integral in (2.6) from above in terms
of $\|f^{\prime}\|_{L^{2}({\mathbb{R}})}^{2}$ as follows. By [12, Lemma 9.32],
for every $\varepsilon>0$ and every $n\in{\mathbb{Z}}$,

$\sup_{x\in[n,n+1]}|f(x)|^{2}\leq\varepsilon\int_{n}^{n+1}|f^{\prime}(x)|^{2}\,\mathrm{d}x+\left(1+\frac{1}{\varepsilon}\right)\int_{n}^{n+1}|f(x)|^{2}\,\mathrm{d}x\eqqcolon
B_{f}(\varepsilon,n),$ (2.7)

and the scalars $B_{f}(\varepsilon,n)$ satisfy

$\displaystyle\sum_{n\in\mathbb{Z}}B_{f}(\varepsilon,n)$
$\displaystyle=\varepsilon\|f^{\prime}\|_{L^{2}({\mathbb{R}})}^{2}+\left(1+\frac{1}{\varepsilon}\right)\|f\|_{L^{2}({\mathbb{R}})}^{2},\quad\varepsilon>0.$
(2.8)

Therefore, by (2.7) and (2.8),

$\displaystyle\int_{-\infty}^{\infty}V_{-}(x)|f(x)|^{2}\,\mathrm{d}x$
$\displaystyle=\sum_{n\in\mathbb{Z}}\int_{n}^{n+1}V_{-}(x)|f(x)|^{2}\,\mathrm{d}x$
$\displaystyle\leq\sum_{n\in\mathbb{Z}}B_{f}(\varepsilon,n)\int_{n}^{n+1}V_{-}(x)\,\mathrm{d}x$
$\displaystyle\leq\varepsilon
M_{V}\|f^{\prime}\|_{L^{2}({\mathbb{R}})}^{2}+M_{V}\bigg{(}1+\frac{1}{\varepsilon}\bigg{)}\|f\|_{L^{2}({\mathbb{R}})}^{2},\quad\varepsilon>0.$
(2.9)

Choosing $\varepsilon=1/M_{V}$ in (2.9) yields

$\int_{-\infty}^{\infty}V_{-}(x)|f(x)|^{2}\,\mathrm{d}x\leq\|f^{\prime}\|_{L^{2}({\mathbb{R}})}^{2}+M_{V}\left(1+M_{V}\right)\|f\|_{L^{2}({\mathbb{R}})}^{2}.$
(2.10)

Finally, applying (2.10) in (2.6) yields (2.5). ∎

Factoring the potential coefficient $V$ according to

$V=uv,\quad u\coloneqq\operatorname{sgn}{(V)}|V|^{1/2},\quad
v\coloneqq|V|^{1/2}\,\text{ a.e.~{}on ${\mathbb{R}}$},$ (2.11)

the following well-known trace ideal properties of the resolvent of
$H^{{(0)}}$ when multiplied by the factors $u$ and $v$ hold (see, e.g., [5,
(4.19) and (4.20)] and [9, (2.49) and (2.50)]).

######  Lemma 2.4.

If Hypothesis 2.1 holds, then

$\begin{split}u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1},\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\in{\mathcal{B}}_{2}\big{(}L^{2}({\mathbb{R}})\big{)},&\\\
\overline{u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\in{\mathcal{B}}_{1}\big{(}L^{2}({\mathbb{R}})\big{)},&\quad
z\in{\mathbb{C}}\setminus[0,\infty).\end{split}$ (2.12)

In addition,

$\lim_{z\to-\infty}\Big{\|}\overline{u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\Big{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0.$
(2.13)

######  Remark 2.5.

Lemma 2.4 with $V$ replaced by $|V|$ yields

$\begin{split}&|V|^{1/2}\big{(}H^{{(0)}}+I_{L^{2}({\mathbb{R}})}\big{)}^{-1/2}\overline{\big{(}H^{{(0)}}+I_{L^{2}({\mathbb{R}})}\big{)}^{-1/2}|V|^{1/2}}\\\
&\quad=\overline{|V|^{1/2}\big{(}H^{{(0)}}+I_{L^{2}({\mathbb{R}})}\big{)}^{-1}|V|^{1/2}}\in{\mathcal{B}}_{1}\big{(}L^{2}({\mathbb{R}})\big{)},\end{split}$
(2.14)

which implies
$|V|^{1/2}\big{(}H^{{(0)}}+I_{L^{2}({\mathbb{R}})}\big{)}^{-1/2}\in{\mathcal{B}}_{2}\big{(}L^{2}({\mathbb{R}})\big{)}$.
Therefore, $|V|$ is relatively form compact, hence infinitesimally form
bounded, with respect to $H^{{(0)}}$. In particular, the positive and negative
parts of $V$ are infinitesimally form bounded with respect to $H^{{(0)}}$.
$\diamond$

For each $\ell\in{\mathbb{N}}$, let

$V_{\ell}\coloneqq V|_{(-\ell,\ell)}$ (2.15)

denote the restriction of $V$ to $(-\ell,\ell)$ and introduce the finite-
interval Schrödinger operator $H_{\ell,\varphi,R}$ with coupled boundary
conditions at the endpoints of $(-\ell,\ell)$ by

$\displaystyle H_{\ell,\varphi,R}f\coloneqq-f^{\prime\prime}+V_{\ell}f,$
(2.16) $\displaystyle
f\in\operatorname{dom}{(H_{\ell,\varphi,R})}\coloneqq\bigg{\\{}g\in
L^{2}((-\ell,\ell))\,\bigg{|}\,g,g^{\prime}\in\text{{AC}}([-\ell,\ell]),\,\begin{pmatrix}g(\ell)\\\
g^{\prime}(\ell)\end{pmatrix}=e^{i\varphi}R\begin{pmatrix}g(-\ell)\\\
g^{\prime}(-\ell)\end{pmatrix},$ $\displaystyle\hskip
284.52756pt-g^{\prime\prime}+V_{\ell}g\in L^{2}((-\ell,\ell))\bigg{\\}}.$

The operator $H_{\ell,\varphi,R}$ is self-adjoint (see, e.g., [3, Theorem
2.5]) in the Hilbert space $L^{2}((-\ell,\ell))$ equipped with the inner
product

$\langle
f,g\rangle_{L^{2}((-\ell,\ell))}\coloneqq\int_{-\ell}^{\ell}\overline{f(x)}g(x)\,\mathrm{d}x,\quad
f,g\in L^{2}((-\ell,\ell)).$ (2.17)

The sesquilinear form uniquely associated to $H_{\ell,\varphi,R}$ is (see,
e.g., [6, Section 3.3 and (3.3.143)])

$\displaystyle\mathfrak{Q}_{\ell,\varphi,R}[f,g]\coloneqq\int_{-\ell}^{\ell}\Big{[}\overline{f^{\prime}(x)}g^{\prime}(x)+\overline{f(x)}V_{\ell}(x)g(x)\Big{]}\,\mathrm{d}x$
(2.18) $\displaystyle\hskip
65.44142pt-\frac{1}{R_{1,2}}\Big{[}R_{1,1}\overline{f(-\ell)}g(-\ell)-e^{-i\varphi}\overline{f(-\ell)}g(\ell)-e^{i\varphi}\overline{f(\ell)}g(-\ell)+R_{2,2}\overline{f(\ell)}g(\ell)\Big{]},$
$\displaystyle
f,g\in\operatorname{dom}{(\mathfrak{Q}_{\ell,\varphi,R})}\coloneqq
H^{1}((-\ell,\ell)).$

Since the differential expression
$\tau_{\ell}\coloneqq-d^{2}/dx^{2}+V_{\ell}(x)$ is regular on $(-\ell,\ell)$,
the self-adjoint operator $H_{\ell,\varphi,R}$ is lower semibounded. A careful
analysis of the sesquilinear form $\mathfrak{Q}_{\ell,\varphi,R}$ shows that
$H_{\ell,\varphi,R}$ is lower semibounded uniformly with respect to
$\ell\in{\mathbb{N}}$.

######  Lemma 2.6.

If Hypothesis 2.1 holds, then for every $\ell\in{\mathbb{N}}$,

$\mathfrak{Q}_{\ell,\varphi,R}[f,f]\geq-(1+M_{V}+N_{R})\|f\|_{L^{2}((-\ell,\ell))}^{2},\quad
f\in H^{1}((-\ell,\ell)).$ (2.19)

###### Proof.

Let $\ell\in{\mathbb{N}}$ and $f\in H^{1}((-\ell,\ell))$. By (2.18),

$\displaystyle\mathfrak{Q}_{\ell,\varphi,R}[f,f]$
$\displaystyle=\underbrace{\int_{-\ell}^{\ell}\Big{[}|f^{\prime}(x)|^{2}+V_{\ell}(x)|f(x)|^{2}\Big{]}\,\mathrm{d}x}_{\eqqcolon
I_{\ell}[f]}$ (2.20)
$\displaystyle\quad-\underbrace{\frac{1}{R_{1,2}}\Big{[}R_{1,1}f(-\ell)\overline{f(-\ell)}-e^{-i\varphi}\overline{f(-\ell)}f(\ell)-e^{i\varphi}\overline{f(\ell)}f(-\ell)+R_{2,2}f(\ell)\overline{f(\ell)}\Big{]}}_{\eqqcolon
J_{\ell,\varphi,R}[f]}.$

The integral $I_{\ell}[f]$ is estimated from below as follows:

$\displaystyle I_{\ell}[f]$
$\displaystyle=\int_{-\ell}^{\ell}|f^{\prime}(x)|^{2}\,\mathrm{d}x+\int_{-\ell}^{\ell}[V_{+}(x)-V_{-}(x)]|f(x)|^{2}\,\mathrm{d}x$
$\displaystyle\geq\|f^{\prime}\|_{L^{2}((-\ell,\ell))}^{2}-\int_{-\ell}^{\ell}V_{-}(x)|f(x)|^{2}\,\mathrm{d}x$
$\displaystyle\geq\|f^{\prime}\|_{L^{2}((-\ell,\ell))}^{2}-M_{V}\sup_{x\in[-\ell,\ell]}{|f(x)|^{2}}.$
(2.21)

In addition, the boundary terms $J_{\ell,\varphi,R}[f]$ satisfy:

$\displaystyle J_{\ell,\varphi,R}[f]$
$\displaystyle=\frac{1}{R_{1,2}}\Big{[}R_{1,1}|f(-\ell)|^{2}+R_{2,2}|f(\ell)|^{2}\Big{]}-\frac{1}{R_{1,2}}\Big{[}e^{-i\varphi}\overline{f(-\ell)}f(\ell)+e^{i\varphi}\overline{f(\ell)}f(-\ell)\Big{]}$
$\displaystyle\leq\frac{1}{|R_{1,2}|}\Big{[}|R_{1,1}||f(-\ell)|^{2}+|R_{2,2}||f(\ell)|^{2}+2|f(\ell)||f(-\ell)|\Big{]}$
$\displaystyle\leq\frac{1}{|R_{1,2}|}\Big{[}|R_{1,1}||f(-\ell)|^{2}+|R_{2,2}||f(\ell)|^{2}+|f(-\ell)|^{2}+|f(\ell)|^{2}\Big{]}$
$\displaystyle\leq N_{R}\sup_{x\in[-\ell,\ell]}{|f(x)|^{2}}.$ (2.22)

Applying (2) and (2.22) in (2.20), one obtains:

$\mathfrak{Q}_{\ell,\varphi,R}[f,f]=I_{\ell}[f]-J_{\ell,\varphi,R}[f]\geq\|f^{\prime}\|_{L^{2}((-\ell,\ell))}^{2}-(M_{V}+N_{R})\sup_{x\in[-\ell,\ell]}{|f(x)|^{2}}.$
(2.23)

Note that $\sup_{x\in[-\ell,\ell]}|f(x)|^{2}=\sup_{x\in[n,n+1]}|f(x)|^{2}$ for
some $n=n(f)\in[-\ell,\ell-1]\cap\mathbb{N}$. By [12, Lemma 9.32],

$\displaystyle\sup_{x\in[-\ell,\ell]}|f(x)|^{2}$
$\displaystyle=\sup_{x\in[n,n+1]}|f(x)|^{2}$
$\displaystyle\leq\varepsilon\int_{n}^{n+1}|f^{\prime}(x)|^{2}\,\mathrm{d}x+\left(1+\frac{1}{\varepsilon}\right)\int_{n}^{n+1}|f(x)|^{2}\,\mathrm{d}x$
$\displaystyle\leq\varepsilon\|f^{\prime}\|_{L^{2}((-\ell,\ell))}^{2}+\left(1+\frac{1}{\varepsilon}\right)\|f\|_{L^{2}((-\ell,\ell))}^{2},\quad\varepsilon>0.$
(2.24)

Finally, taking $\varepsilon=(M_{V}+N_{R})^{-1}>0$ in (2.24) and applying the
resulting estimate in (2.23) yields (2.19). Neither $M_{V}$ nor $N_{R}$ depend
on $\ell\in{\mathbb{N}}$, so (2.19) implies that $H_{\ell,\varphi,R}$ is lower
semibounded uniformly with respect to $\ell\in{\mathbb{N}}$. ∎

For each $\ell\in{\mathbb{N}}$, the finite-interval Schrödinger operator with
Dirichlet boundary conditions at the endpoints of $(-\ell,\ell)$ is defined by

$\displaystyle H_{\ell,D}f\coloneqq-f^{\prime\prime}+V_{\ell}f,$ (2.25)
$\displaystyle f\in\operatorname{dom}{(H_{\ell,D})}\coloneqq\big{\\{}g\in
L^{2}((-\ell,\ell))\,\big{|}\,g,g^{\prime}\in\text{{AC}}([-\ell,\ell]),\,g(-\ell)=g(\ell)=0,$
$\displaystyle\hskip 230.46748pt-g^{\prime\prime}+V_{\ell}g\in
L^{2}((-\ell,\ell))\big{\\}}.$

The operator $H_{\ell,D}$ is self-adjoint in $L^{2}((-\ell,\ell))$ for each
$\ell\ \in{\mathbb{N}}$, and its corresponding sesquilinear form is

$\displaystyle\mathfrak{Q}_{\ell,D}[f,g]\coloneqq\int_{-\ell}^{\ell}\Big{[}\overline{f^{\prime}(x)}g^{\prime}(x)+\overline{f(x)}V_{\ell}(x)g(x)\Big{]}\,\mathrm{d}x,$
(2.26) $\displaystyle
f,g\in\operatorname{dom}{(\mathfrak{Q}_{\ell,D})}\coloneqq\big{\\{}h\in
L^{2}((-\ell,\ell))\,\big{|}\,h\in\text{{AC}}([-\ell,\ell]),\,h(-\ell)=h(\ell)=0,\,h^{\prime}\in
L^{2}((-\ell,\ell))\big{\\}}$ $\displaystyle\hskip
83.65117pt=H_{0}^{1}((-\ell,\ell)).$

######  Remark 2.7.

$(i)$ The condition $R_{1,2}\neq 0$ in Hypothesis 2.1 implies that
$H_{\ell,D}$ and $H_{\ell,\varphi,R}$ are relatively prime with respect to
their underlying minimal Sturm–Liouville operator:

$\displaystyle H_{\ell,\min}f\coloneqq-f^{\prime\prime}+V_{\ell}f,$ (2.27)
$\displaystyle f\in\operatorname{dom}{(H_{\ell,\min})}\coloneqq\big{\\{}g\in
L^{2}((-\ell,\ell))\,\big{|}\,g,g^{\prime}\in\text{{AC}}([-\ell,\ell]),\,g(-\ell)=g^{\prime}(-\ell)=g(\ell)=g^{\prime}(\ell)=0,$
$\displaystyle\hskip 317.24805pt-g^{\prime\prime}+V_{\ell}g\in
L^{2}((-\ell,\ell))\big{\\}},$

in the sense that
$\operatorname{dom}{\big{(}H_{\ell,D}\big{)}}\cap\operatorname{dom}{\big{(}H_{\ell,\varphi,R}\big{)}}=\operatorname{dom}{\big{(}H_{\ell,\min}\big{)}}$.
$(ii)$ In the case when $V=0$ a.e. on ${\mathbb{R}}$, the corresponding “free”
finite-interval Schrödinger operators and their sesquilinear forms will be
denoted by $H^{{(0)}}_{\ell,\varphi,R}$, $H^{{(0)}}_{\ell,D}$,
$\mathfrak{Q}^{{(0)}}_{\ell,\varphi,R}$, and $\mathfrak{Q}^{{(0)}}_{\ell,D}$,
respectively, for each $\ell\in{\mathbb{N}}$. In particular, one infers that
$H^{{(0)}}_{\ell,D}$ is nonnegative for every $\ell\in{\mathbb{N}}$.
$\diamond$

The resolvent operators of $H_{\ell,\varphi,R}$ and $H_{\ell,D}$ are related
for each $\ell\in{\mathbb{N}}$ via a Krein-type resolvent identity. To wit,
for each $z\in\rho(H_{\ell,D})$, let
$\\{\psi_{\ell,m}(z,\,\cdot\,)\\}_{m=1,2}$ denote solutions to the Schrödinger
differential equation $-\psi^{\prime\prime}+V_{\ell}\psi=z\psi$ on
$(-\ell,\ell)$ that satisfy the boundary conditions

$\begin{split}&\psi_{\ell,1}(z,-\ell)=0,\quad\psi_{\ell,1}(z,\ell)=1,\\\
&\psi_{\ell,2}(z,-\ell)=1,\quad\psi_{\ell,2}(z,\ell)=0.\end{split}$ (2.28)

Krein’s resolvent identity relates the resolvent operators of
$H_{\ell,\varphi,R}$ and $H_{\ell,D}$ via a rank-two operator constructed from
the solutions $\\{\psi_{\ell,m}(z,\,\cdot\,)\\}_{m=1,2}$.

######  Theorem 2.8 ([3, Theorem 3.2]).

Assume Hypothesis 2.1. If $\ell\in{\mathbb{N}}$ and
$z\in\rho(H_{\ell,\varphi,R})\cap\rho(H_{\ell,D})$, then the matrix

$K_{\ell,\varphi,R}(z)\coloneqq\begin{pmatrix}\frac{R_{2,2}}{R_{1,2}}-\psi_{\ell,1}^{\prime}(z,\ell)&\frac{-1}{e^{-i\varphi}R_{1,2}}-\psi_{\ell,2}^{\prime}(z,\ell)\\\
\frac{-1}{e^{i\varphi}R_{1,2}}+\psi_{\ell,1}^{\prime}(z,-\ell)&\frac{R_{1,1}}{R_{1,2}}+\psi_{\ell,2}^{\prime}(z,-\ell)\end{pmatrix}$
(2.29)

is invertible and

$\big{(}H_{\ell,\varphi,R}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}=\big{(}H_{\ell,D}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}+P_{\ell,\varphi,R}(z),$
(2.30)

where the rank-two operator $P_{\ell,\varphi,R}(z)$ is defined by

$P_{\ell,\varphi,R}(z)\coloneqq-\sum_{m,n=1}^{2}\big{[}K_{\ell,\varphi,R}(z)^{-1}\big{]}_{m,n}\langle\psi_{\ell,n}(\overline{z},\,\cdot\,),\,\cdot\,\rangle_{L^{2}((-\ell,\ell))}\psi_{\ell,m}(z,\,\cdot\,).$
(2.31)

######  Remark 2.9.

In the case when $V=0$ a.e. on ${\mathbb{R}}$, the solutions corresponding to
(2.28), the matrix (2.29), and the rank-two operator (2.31) will be denoted by
$\psi_{\ell,m}^{{(0)}}(z,\,\cdot\,)$, $m\in\\{1,2\\}$,
$K_{\ell,\varphi,R}^{{(0)}}(z)$, and $P_{\ell,\varphi,R}^{{(0)}}(z)$,
respectively. $\diamond$

Introducing for each $\ell\in{\mathbb{N}}$ the factorization of the potential
coefficient $V_{\ell}$,

$V_{\ell}=u_{\ell}v_{\ell},\quad
u_{\ell}\coloneqq\operatorname{sgn}{(V_{\ell})}|V_{\ell}|^{1/2},\quad
v_{\ell}\coloneqq|V_{\ell}|^{1/2}\,\text{ a.e.~{}on $(-\ell,\ell)$},$ (2.32)

the following analogue of Lemma 2.4 for the free Dirichlet operator
$H_{\ell,D}^{{(0)}}$ holds (see, e.g., [4, (2.69), (3.12), and (3.14)]).

######  Lemma 2.10.

If Hypothesis 2.1 holds and $\ell\in{\mathbb{N}}$, then

$\begin{split}u_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1},\overline{\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\in{\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)},&\\\
\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\in{\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)},&\quad
z\in\rho\big{(}H_{\ell,D}^{{(0)}}\big{)}.\end{split}$ (2.33)

The analogue of Lemma 2.10 for $H_{\ell,\varphi,R}^{{(0)}}$ is obtained as a
consequence of Theorem 2.8.

######  Lemma 2.11.

If Hypothesis 2.1 holds and $\ell\in{\mathbb{N}}$, then

$\displaystyle
u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1},\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\in{\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)},$
(2.34)
$\displaystyle\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\in{\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)},$
$\displaystyle\quad z\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}.$ (2.35)

###### Proof.

Let $\ell\in{\mathbb{N}}$. Taking $z=i$ (and $V_{\ell}\equiv 0$), (2.30)
implies

$\begin{split}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}&=u_{\ell}\big{(}H_{\ell,D}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\\\
&\quad-\sum_{m,n=1}^{2}\Big{[}K_{\ell,\varphi,R}^{{(0)}}(i)^{-1}\Big{]}_{m,n}\big{\langle}\psi_{\ell,n}^{{(0)}}(-i,\,\cdot\,),\,\cdot\,\big{\rangle}_{L^{2}((-\ell,\ell))}u_{\ell}\psi_{\ell,m}^{{(0)}}(i,\,\cdot\,).\end{split}$
(2.36)

Since $u_{\ell}\psi_{\ell,m}^{{(0)}}(i,\,\cdot\,)\in L^{2}((-\ell,\ell))$,
$m\in\\{1,2\\}$, the sum on the right-hand side in (2.36) is a rank-two
operator in $L^{2}((-\ell,\ell))$. Thus, the right-hand side of (2.36) belongs
to ${\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)}$ by Lemma 2.10. If
$z\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}$, the first resolvent
identity implies

$\begin{split}&u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\\\
&\quad=u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{[}I_{L^{2}((-\ell,\ell))}+(z-i)\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{]},\end{split}$
(2.37)

which is a product of operators in
${\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)}$ and
${\mathcal{B}}\big{(}L^{2}((-\ell,\ell))\big{)}$, respectively, and therefore
belongs to ${\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)}$. The first
containment in (2.34) follows. The same argument mutatis mutandis shows

$v_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\in{\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)},\quad
z\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}.$ (2.38)

In consequence, for each $z\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}$,

$\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}=\Big{[}v_{\ell}\big{(}H_{\ell,\varphi,R}^{{(0)}}-\overline{z}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{]}^{*}\in{\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)},$
(2.39)

which is the second containment in (2.34).

Multiplying both sides of (2.36) by $v_{\ell}$ from the right and taking
closures yields

$\displaystyle\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}$
$\displaystyle=\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}$
(2.40)
$\displaystyle\quad-\sum_{m,n=1}^{2}\Big{[}K_{\ell,\varphi,R}^{{(0)}}(i)^{-1}\Big{]}_{m,n}\big{\langle}v_{\ell}\psi_{\ell,n}^{{(0)}}(-i,\,\cdot\,),\,\cdot\,\big{\rangle}_{L^{2}((-\ell,\ell))}u_{\ell}\psi_{\ell,m}^{{(0)}}(i,\,\cdot\,).$

Since
$v_{\ell}\psi_{\ell,m}^{{(0)}}(-i,\,\cdot\,),u_{\ell}\psi_{\ell,m}^{{(0)}}(i,\,\cdot\,)\in
L^{2}((-\ell,\ell))$, $j\in\\{1,2\\}$, the sum on the right-hand side in
(2.40) is a rank-two operator in $L^{2}((-\ell,\ell))$. Thus, the right-hand
side of (2.40) belongs to ${\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)}$
by Lemma 2.10. If $z\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}$, the
first resolvent identity implies

$\displaystyle\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}$
(2.41)
$\displaystyle\quad=\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}+(z-i)u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-iI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}.$

The first term on the right-hand side in (2.41) was shown above to belong to
${\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)}$. The second term on the
right-hand side in (2.41) belongs to
${\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)}$ since, by (2.34), it is a
product of two operators in
${\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell)\big{)}$. The containment in (2.35)
follows. ∎

######  Remark 2.12.

If $\ell\in{\mathbb{N}}$, and $k>(1+N_{R})^{1/2}$, then upon replacing
$V_{\ell}$ by $|V_{\ell}|$, Lemma 2.11 implies

$\begin{split}&|V_{\ell}|^{1/2}\big{(}H_{\ell,\varphi,R}^{{(0)}}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1/2}\overline{\big{(}H_{\ell,\varphi,R}^{{(0)}}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1/2}|V_{\ell}|^{1/2}}\\\
&\quad=\overline{|V_{\ell}|^{1/2}\big{(}H_{\ell,\varphi,R}^{{(0)}}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}|V_{\ell}|^{1/2}}\in{\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)},\end{split}$
(2.42)

which implies
$|V_{\ell}|^{1/2}\big{(}H_{\ell,\varphi,R}^{{(0)}}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1/2}\in{\mathcal{B}}_{2}\big{(}L^{2}((-\ell,\ell))\big{)}$.
Therefore, $|V_{\ell}|$ is relatively form compact, hence infinitesimally form
bounded, with respect to $H_{\ell,\varphi,R}^{{(0)}}$. In particular, the
positive and negative parts of $V_{\ell}$ are infinitesimally form bounded
with respect to $H_{\ell,\varphi,R}^{{(0)}}$ for $\ell\in{\mathbb{N}}$.
$\diamond$

In the applications below, the results of Theorem 2.8 are most often needed in
the free case $V=0$ a.e. on ${\mathbb{R}}$ and for $z=-k^{2}$ with $k>0$ taken
sufficiently large. In the free case and for negative values of $z$, the
solutions determined by (2.28) have the form

$\psi_{\ell,m}^{{(0)}}(-k^{2},x)=\frac{1}{2}\bigg{[}\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\bigg{]},\quad
x\in[-\ell,\ell],\,m\in\\{1,2\\},\,k\in(0,\infty),$ (2.43)

and the matrix (2.29) reduces to

$\displaystyle
K_{\ell,\varphi,R}^{{(0)}}(-k^{2})=\begin{pmatrix}\frac{R_{2,2}}{R_{1,2}}-\frac{k}{2}[\coth{(k\ell)}+\tanh{(k\ell)}]&\frac{-1}{e^{-i\varphi}R_{1,2}}-\frac{k}{2}[\tanh{(k\ell)}-\coth{(k\ell)}]\\\
\frac{-1}{e^{i\varphi}R_{1,2}}-\frac{k}{2}[\tanh{(k\ell)}-\coth{(k\ell)}]&\frac{R_{1,1}}{R_{1,2}}-\frac{k}{2}[\coth{(k\ell)}+\tanh{(k\ell)}]\end{pmatrix},$
(2.44) $\displaystyle k\in\big{(}(1+N_{R})^{1/2},\infty\big{)}.$

Lemma 2.6 ensures that
$z=-k^{2}\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}$ for all
$\ell\in{\mathbb{N}}$ when $k>(1+N_{R})^{1/2}$. Therefore, since
$H_{\ell,D}^{{(0)}}$ is nonnegative for all $\ell\in{\mathbb{N}}$, the
condition on $k$ in (2.44) ensures that
$z=-k^{2}\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}\cap\rho\big{(}H_{\ell,D}^{{(0)}}\big{)}$
for all $\ell\in{\mathbb{N}}$.

The limiting behaviors of the entries of the inverse matrix
$K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}$ as $k\to\infty$ with
$\ell\in{\mathbb{N}}$ held fixed and as $\ell\to\infty$ with $k$ taken
sufficiently large and held fixed will also play an important role. Upon
inspection, (2.44), implies

$\det\Big{(}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})\Big{)}\underset{k\to\infty}{\sim}k^{2}\,\text{
for each fixed $\ell\in{\mathbb{N}}$}.$ (2.45)

Taking the inverse of the matrix in (2.44) and applying the relation in (2.45)
yields an asymptotic estimate for the entries of
$K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}$:

$\Big{[}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}\Big{]}_{m,n}\underset{k\to\infty}{=}O(1/k)\,\text{
for each fixed $\ell\in{\mathbb{N}}$}.$ (2.46)

In addition, for each fixed $k\in\big{(}(1+N_{R})^{1/2},\infty\big{)}$, (2.44)
and $\det{(R)}=1$ imply

$\displaystyle\lim_{\ell\to\infty}\det\Big{(}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})\Big{)}=k^{2}-\frac{R_{1,1}+R_{2,2}}{R_{1,2}}k+\frac{R_{2,1}}{R_{1,2}}\eqqcolon
p_{R}^{{(0)}}(k).$ (2.47)

Elementary algebraic manipulations, again using $\det{(R)}=1$, reveal that the
discriminant of the polynomial $p_{R}^{{(0)}}$ is positive. If $k_{R}^{{(0)}}$
denotes the largest root of $p_{R}^{{(0)}}$, then

$\lim_{\ell\to\infty}\det{\left(K_{\ell,\varphi,R}^{{(0)}}(-k^{2})\right)}\neq
0\,\text{ if $k>\max{\left\\{k_{R}^{{(0)}},(1+N_{R})^{1/2}\right\\}}$}.$
(2.48)

For fixed $k$, each entry of the matrix in (2.44) has a finite limit as
$\ell\to\infty$. Consequently, (2.48) implies that each entry of
$K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}$ has a finite limit as
$\ell\to\infty$. In particular,

$\Big{[}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}\Big{]}_{m,n}\underset{\ell\to\infty}{=}O(1)\,\text{
for $m,n\in\\{1,2\\}$ and each fixed
$k>\max{\left\\{k_{R}^{{(0)}},(1+N_{R})^{1/2}\right\\}}$}.$ (2.49)

######  Example 2.13.

In general, the discriminant of $p_{R}^{{(0)}}$ is
$\big{[}(R_{1,1}-R_{2,2})^{2}+4\big{]}/R_{1,2}^{2}$, so the largest root of
the polynomial $p_{R}^{{(0)}}$ is

$k_{R}^{{(0)}}=\frac{1}{2}\bigg{\\{}\frac{R_{1,1}+R_{2,2}}{R_{1,2}}+\frac{[(R_{1,1}-R_{2,2})^{2}+4]^{1/2}}{|R_{1,2}|}\bigg{\\}}.$
(2.50)

Choosing

$R=R_{\alpha}\coloneqq\begin{pmatrix}0&\alpha\\\ -\alpha^{-1}&0\end{pmatrix},$
(2.51)

where $\alpha\in{\mathbb{R}}\setminus\\{0\\}$ is a parameter, results in

$k_{R_{\alpha}}^{{(0)}}=\frac{1}{|\alpha|},\quad(1+N_{R_{\alpha}})^{1/2}=\bigg{(}1+\frac{2}{|\alpha|}\bigg{)}^{1/2}.$
(2.52)

The function

$\frac{1}{|\alpha|}-\bigg{(}1+\frac{2}{|\alpha|}\bigg{)}^{1/2},\quad\alpha\in{\mathbb{R}}\setminus\\{0\\},$
(2.53)

exhibits a sign change at $\alpha=\sqrt{2}-1$. Therefore, depending on $R$,
$\max\Big{\\{}k_{R}^{{(0)}},(1+N_{R})^{1/2}\Big{\\}}$ can be either of
$k_{R}^{{(0)}}$ or $(1+N_{R})^{1/2}$.

##  3\. Convergence Results for Resolvent Operators

The Hilbert spaces $L^{2}((-\ell,\ell))$ vary with $\ell\in{\mathbb{N}}$.
Therefore, we introduce the following $\ell$-dependent direct sum to move to
the fixed $\ell$-independent Hilbert space $L^{2}({\mathbb{R}})$. If
$\ell\in{\mathbb{N}}$, $f\in L^{2}((-\ell,\ell))$, and $g\in
L^{2}({\mathbb{R}}\setminus(-\ell,\ell))$, then we define $f\oplus_{\ell}g$ by

$(f\oplus_{\ell}g)(x)\coloneqq\begin{cases}f(x)&\text{for
a.e.~{}$x\in(-\ell,\ell)$},\\\ g(x)&\text{for
a.e.~{}$x\in{\mathbb{R}}\setminus(-\ell,\ell)$}.\end{cases}$ (3.1)

By additivity of the Lebesgue integral, $f\oplus_{\ell}g\in
L^{2}({\mathbb{R}})$ and

$\|f\oplus_{\ell}g\|_{L^{2}({\mathbb{R}})}^{2}=\|f\|_{L^{2}((-\ell,\ell))}^{2}+\|g\|_{L^{2}({\mathbb{R}}\setminus(-\ell,\ell))}^{2}.$
(3.2)

Conversely, if $h\in L^{2}({\mathbb{R}})$, then $h$ can be expressed as

$h=f\oplus_{\ell}g,\,\text{ where $f=h|_{(-\ell,\ell)}$ and
$g=h|_{{\mathbb{R}}\setminus(-\ell,\ell)}$}.$ (3.3)

In this way, for each $\ell\in{\mathbb{N}}$,

$L^{2}({\mathbb{R}})=L^{2}((-\ell,\ell))\oplus_{\ell}L^{2}({\mathbb{R}}\setminus(-\ell,\ell)).$
(3.4)

By additivity of the Lebesgue integral if $h_{1},h_{2}\in L^{2}({\mathbb{R}})$
with

$h_{j}=f_{j}\oplus_{\ell}g_{j},\quad f_{j}\in L^{2}((-\ell,\ell)),\quad
g_{j}\in L^{2}({\mathbb{R}}\setminus(-\ell,\ell)),\quad j\in\\{1,2\\},$ (3.5)

then

$\langle h_{1},h_{2}\rangle_{L^{2}({\mathbb{R}})}=\langle
f_{1},f_{2}\rangle_{L^{2}((-\ell,\ell))}+\langle
g_{1},g_{2}\rangle_{L^{2}({\mathbb{R}}\setminus(-\ell,\ell))}.$ (3.6)

Finally, if

$S\colon\operatorname{dom}{(S)}\subseteq L^{2}((-\ell,\ell))\to
L^{2}((-\ell,\ell))$ (3.7)

and

$T\colon\operatorname{dom}{(T)}\subseteq
L^{2}({\mathbb{R}}\setminus(-\ell,\ell))\to
L^{2}({\mathbb{R}}\setminus(-\ell,\ell))$ (3.8)

are linear operators, then their direct sum $S\oplus_{\ell}T$ with respect to
the decomposition (3.4) is defined in the standard way by

$(S\oplus_{\ell}T)h\coloneqq(Sf)\oplus_{\ell}(Tg),\quad
h=f\oplus_{\ell}g\in\operatorname{dom}{(S\oplus_{\ell}T)}\coloneqq\operatorname{dom}{(S)}\oplus_{\ell}\operatorname{dom}{(T)}.$
(3.9)

Under this direct sum formalism, it is known that the sequence
$\left\\{H_{\ell,D}^{(0)}\oplus_{\ell}0\right\\}_{\ell=1}^{\infty}$ converges
to the free Schrödinger operator $H^{{(0)}}$ in the strong resolvent sense as
$\ell\to\infty$.

######  Lemma 3.1 (Lemma 3.1 in [4]).

If Hypothesis 2.1 holds, then for each fixed
$z\in\mathbb{C}\setminus[0,\infty)$,

$\operatorname*{s-lim}_{\ell\to\infty}{\left(\left[H_{\ell,D}^{(0)}\oplus_{\ell}0\right]-zI_{L^{2}(\mathbb{R})}\right)^{-1}}=\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}.$
(3.10)

In particular, the sequence
$\left\\{H_{\ell,D}^{(0)}\oplus_{\ell}0\right\\}_{\ell=1}^{\infty}$ converges
to $H^{(0)}$ in the strong resolvent sense.

One can apply Theorem 2.8 to establish an analogue of Lemma 3.1 for the
operators $H_{\ell,\varphi,R}^{{(0)}}$. Our proof employs the following
elementary abstract fact, the proof of which is an exercise, for computing the
operator or trace ideal norm of a rank one operator.

######  Proposition 3.2.

Let ${\mathcal{H}}$ denote a Hilbert space with inner product
$\langle\,\cdot\,,\,\cdot\,\rangle_{{\mathcal{H}}}$. If
$\psi,\phi\in{\mathcal{H}}$ and the rank one operator
$A\colon{\mathcal{H}}\to{\mathcal{H}}$ is given by
$A=\langle\psi,\,\cdot\,\rangle_{{\mathcal{H}}}\phi$, then for each
$p\in[1,\infty)$,

$\|A\|_{{\mathcal{B}}_{p}({\mathcal{H}})}=\|\psi\|_{{\mathcal{H}}}\|\phi\|_{{\mathcal{H}}}=\|A\|_{{\mathcal{B}}({\mathcal{H}})}.$
(3.11)

The analogue of Lemma 3.1 for the operators $H_{\ell,\varphi,R}^{{(0)}}$ is
then stated as follows:

######  Lemma 3.3.

If Hypothesis 2.1 holds and
$k_{R,\max}\coloneqq\max\big{\\{}k_{R}^{{(0)}},(1+N_{R})^{1/2}\big{\\}}$, then
for each fixed $z\in\mathbb{C}\setminus[-k_{R,\max}^{2},\infty)$,

$\operatorname*{s-lim}_{\ell\to\infty}{\left(\left[H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\right]-zI_{L^{2}({\mathbb{R}})}\right)^{-1}}=\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}.$
(3.12)

In particular, the sequence
$\left\\{H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\right\\}_{\ell=1}^{\infty}$
converges to $H^{(0)}$ in the strong resolvent sense.

###### Proof.

It suffices to show

$\operatorname*{s-lim}_{\ell\to\infty}{\left(\left[H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\right]+k^{2}I_{L^{2}({\mathbb{R}})}\right)^{-1}}=\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}$
(3.13)

for some $k>k_{R,\max}$, as the statement in (3.12) then follows from an
application of the first resolvent identity. Let $k>k_{R,\max}$, so that
$-k^{2}\in\rho\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}\cap\rho\big{(}H_{\ell,D}^{{(0)}}\big{)}\cap\rho\big{(}H^{{(0)}}\big{)}$
for all $\ell\in{\mathbb{N}}$. By Theorem 2.8,

$\begin{split}\Big{(}\Big{[}H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\Big{]}+k^{2}I_{L^{2}(\mathbb{R})}\Big{)}^{-1}=\Big{(}\Big{[}H_{\ell,D}^{(0)}\oplus_{\ell}0\Big{]}+k^{2}I_{L^{2}({\mathbb{R}})}\Big{)}^{-1}+\Big{[}P_{\ell,\varphi,R}^{(0)}(-k^{2})\oplus_{\ell}0\Big{]},&\\\
\ell\in{\mathbb{N}}.&\end{split}$ (3.14)

In light of (3.10) and (3.14), in order to show (3.13), it suffices to prove

$\operatorname*{s-lim}_{\ell\to\infty}{\Big{[}P_{\ell,\varphi,R}^{(0)}(-k^{2})\oplus_{\ell}0\Big{]}}=0.$
(3.15)

To this end, note that the sequence of operators on the left-hand side in
(3.15) is uniformly bounded with respect to $\ell\in{\mathbb{N}}$. Indeed, by
the triangle inequality and (2.49),

$\displaystyle\Big{\|}P_{\ell,\varphi,R}^{(0)}(-k^{2})\oplus_{\ell}0\Big{\|}_{\mathcal{B}(L^{2}(\mathbb{R}))}$
$\displaystyle\quad=\Big{\|}P_{\ell,\varphi,R}^{(0)}(-k^{2})\Big{\|}_{\mathcal{B}(L^{2}((-\ell,\ell)))}$
$\displaystyle\quad\leq\sum_{m,n=1}^{2}{\Big{|}\Big{[}K_{\ell,\varphi,R}(-k^{2})^{-1}\Big{]}_{m,n}\Big{|}\
\big{\|}\big{\langle}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,),\,\cdot\,\big{\rangle}\psi_{\ell,n}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{\mathcal{B}(L^{2}((-\ell,\ell)))}}$
$\displaystyle\,\;\underset{\ell\to\infty}{\leq}O(1)\sum_{m,n=1}^{2}{\big{\|}\big{\langle}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,),\,\cdot\,\big{\rangle}\psi_{\ell,n}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{\mathcal{B}(L^{2}((-\ell,\ell)))}}$
$\displaystyle\,\;\underset{\ell\to\infty}{=}O(1)\sum_{m,n=1}^{2}{\big{\|}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}\big{\|}\psi_{\ell,n}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}},$
(3.16)

where the final equality follows from Proposition 3.2. One infers that

$\displaystyle\big{\|}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}^{2}$
$\displaystyle=\frac{1}{4}\int_{-\ell}^{\ell}{\bigg{[}\frac{\cosh{(kx)}}{\cosh{(k\ell)}}\pm\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\bigg{]}^{2}\,\mathrm{d}x}$
$\displaystyle=\frac{1}{4}\int_{-\ell}^{\ell}\bigg{[}\frac{\cosh^{2}{(kx)}}{\cosh^{2}{(k\ell)}}+\frac{\sinh^{2}{(kx)}}{\sinh^{2}{(k\ell)}}\bigg{]}\,\mathrm{d}x,\quad
m\in\\{1,2\\},\,\ell\in{\mathbb{N}},$ (3.17)

where the third term that arises from the expansion of the square in the first
integral vanishes as it is an odd function being integrated over a finite
symmetric interval. The last integral in (3.17) is elementary and one obtains:

$\begin{split}\big{\|}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}^{2}=\frac{\tanh{(k\ell)}+\coth{(k\ell)}}{4k}+\frac{\ell}{4}\big{[}\operatorname{sech}^{2}{(k\ell)}-\operatorname{csch}^{2}{(k\ell)}\big{]},&\\\
m\in\\{1,2\\},\,\ell\in{\mathbb{N}}.&\end{split}$ (3.18)

In particular, (3.18) implies

$\big{\|}\psi_{\ell,m}^{(0)}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}\underset{\ell\to\infty}{=}O(1),\quad
m\in\\{1,2\\}.$ (3.19)

Combining (3.16) and (3.19), one obtains

$\Big{\|}P_{\ell,\varphi,R}^{(0)}(-k^{2})\oplus_{\ell}0\Big{\|}_{\mathcal{B}(L^{2}(\mathbb{R}))}\underset{\ell\to\infty}{=}O(1),$
(3.20)

so that
$\Big{\\{}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{\\}}_{\ell=1}^{\infty}$
is a bounded sequence in ${\mathcal{B}}\big{(}L^{2}({\mathbb{R}})\big{)}$.
Therefore, by [11, Exercise 4.28], in order to show (3.15), it suffices to
prove

$\lim_{\ell\to\infty}\Big{[}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{]}f=0,\quad
f\in L^{1}({\mathbb{R}})\cap L^{2}({\mathbb{R}}),$ (3.21)

since $L^{1}({\mathbb{R}})\cap L^{2}({\mathbb{R}})$ is dense in
$L^{2}({\mathbb{R}})$. If $f\in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$, then
one obtains

$\displaystyle\big{|}\chi_{(-\ell,\ell)}(x)\psi_{\ell,m}^{{(0)}}(-k^{2},x)f(x)\big{|}$
$\displaystyle=\frac{1}{2}\bigg{|}\chi_{(-\ell,\ell)}(x)\bigg{[}\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\bigg{]}f(x)\bigg{|}$
$\displaystyle\leq|f(x)|\,\text{ for
a.e.~{}$x\in{\mathbb{R}}$},\quad\ell\in{\mathbb{N}},\,m\in\\{1,2\\}.$ (3.22)

Since $\lim_{\ell\to\infty}{\psi_{\ell,m}^{{(0)}}(-k^{2},x)}=0$ for every
$x\in\mathbb{R}$, it follows that

$\lim_{\ell\to\infty}{\chi_{(-\ell,\ell)}(x)\psi_{\ell,m}^{{(0)}}(-k^{2},x)f(x)}=0\,\text{
for a.e.~{}$x\in{\mathbb{R}}$}.$ (3.23)

Hence, by the dominated convergence theorem,

$\begin{split}\lim_{\ell\to\infty}\big{\langle}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,),f\big{|}_{(-\ell,\ell)}\big{\rangle}_{L^{2}((-\ell,\ell))}=\lim_{\ell\to\infty}{\int_{-\infty}^{\infty}{\chi_{(-\ell,\ell)}\psi_{\ell,m}^{{(0)}}(-k^{2},x)f(x)\,\mathrm{d}x}}=0,&\\\
m\in\\{1,2\\}.&\end{split}$ (3.24)

Finally, applying (2.31), (2.49), (3.19), and (3.24), one obtains

$\displaystyle\Big{\|}\Big{[}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{]}f\,\Big{\|}_{L^{2}({\mathbb{R}})}$
(3.25)
$\displaystyle\quad=\Big{\|}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})f\big{|}_{(-\ell,\ell)}\,\Big{\|}_{L^{2}((-\ell,\ell))}$
$\displaystyle\quad\leq\sum_{m,n=1}^{2}\Big{|}\Big{[}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}\Big{]}_{m,n}\Big{|}\Big{|}\big{\langle}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,),f\big{|}_{(-\ell,\ell)}\big{\rangle}_{L^{2}((-\ell,\ell))}\Big{|}\big{\|}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}$
$\displaystyle\,\;\underset{\ell\to\infty}{=}o(1),$

and (3.21) follows. ∎

The following result, which studies the $z\to-\infty$ behavior of the extended
Birman–Schwinger operator of $H_{\ell,D}^{{(0)}}$ with $\ell\in{\mathbb{N}}$
fixed, is an immediate consequence of [4, Lemma 3.8].

######  Lemma 3.4.

If Hypothesis 2.1 holds, then

$\lim_{z\to-\infty}\left\|\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\right\|_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0,\quad\ell\in{\mathbb{N}}.$
(3.26)

Using the Krein identity (2.30), (2.31), and the large parameter asymptotics
(2.46), one establishes the following analogue of Lemma 3.4 for the operator
$H_{\ell,\varphi,R}^{{(0)}}$.

######  Lemma 3.5.

If Hypothesis 2.1 holds, then

$\lim_{z\to-\infty}\left\|\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\right\|_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0,\quad\ell\in{\mathbb{N}}.$
(3.27)

###### Proof.

Let $\ell\in{\mathbb{N}}$. It suffices to show

$\lim_{k\to\infty}\left\|\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}=0.$
(3.28)

By Theorem 2.8, for $k>k_{R,\max}$,

$\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}=\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}+\overline{u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}},$
(3.29)

and one verifies that

$\overline{u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}}=\sum_{m,n=1}^{2}\Big{[}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}\Big{]}_{m,n}\big{\langle}v_{\ell}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,),\,\cdot\,\big{\rangle}_{L^{2}((-\ell,\ell))}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,).$
(3.30)

To prove (3.30) one observes that the operator on the right-hand side in
(3.30) is bounded on $L^{2}((-\ell,\ell))$ and that
$u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}$ coincides with this
operator on the dense subspace $\operatorname{dom}{(v_{\ell})}$. For
$k>\max{\Big{\\{}k_{R}^{{(0)}},(1+N_{R})^{1/2}\Big{\\}}}$ one estimates:

$\displaystyle\left\|\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}$
$\displaystyle\quad\leq\left\|\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}+\left\|\overline{u_{\ell}P_{\ell,\varphi,R}^{(0)}(-k^{2})v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}.$
(3.31)

The first term on the right-hand side of the inequality in (3.31) converges to
zero as $k\to\infty$ by Lemma 3.4. Therefore, to establish (3.28), it suffices
to show that

$\lim_{k\to\infty}\left\|\overline{u_{\ell}P_{\ell,\varphi,R}^{(0)}(-k^{2})v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}=0.$
(3.32)

To this end, (3.30), (2.46), and Proposition 3.2 imply

$\begin{split}&\left\|\overline{u_{\ell}P_{\ell,\varphi,R}^{(0)}(-k^{2})v_{\ell}}\right\|_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}\\\
&\,\;\underset{k\to\infty}{\leq}O(1/k)\sum_{m,n=1}^{2}\big{\|}v_{\ell}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}.\end{split}$
(3.33)

One infers that for any $k>0$,

$\displaystyle\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}^{2}$
(3.34)
$\displaystyle\quad=\big{\|}v_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}^{2}=\int_{-\ell}^{\ell}\frac{1}{4}\left|\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\right|^{2}|V_{\ell}(x)|\,\mathrm{d}x,\quad
m\in\\{1,2\\}.$

The integrand on the right-hand side in (3.34) is bounded a.e. on
$(-\ell,\ell)$ by $|V_{\ell}|\in L^{1}((-\ell,\ell))$ and since

$\lim_{k\to\infty}\frac{1}{4}\left|\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\right|^{2}=0,\quad
x\in(-\ell,\ell),\,m\in\\{1,2\\},$ (3.35)

the dominated convergence theorem implies

$\lim_{k\to\infty}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}=\lim_{k\to\infty}\big{\|}v_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}=0,\quad
m\in\\{1,2\\}.$ (3.36)

The convergence statement in (3.32) follows from (3.33) and (3.36). ∎

We recall the following convergence results from [4] for the operators
$H_{\ell,D}^{{(0)}}$, $\ell\in{\mathbb{N}}$.

######  Lemma 3.6 (Lemmata 3.1 and 3.2 in [4]).

Assume Hypothesis 2.1. For each fixed $z\in{\mathbb{C}}\setminus[0,\infty)$,
the following convergence results hold in
${\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))$:

$\displaystyle\lim_{\ell\to\infty}\Big{\|}\Big{[}u_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0,$ (3.37)
$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0,$ (3.38)

and the following convergence result holds in
${\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))$:

$\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0.$
(3.39)

######  Remark 3.7.

Statements analogous to (3.37) and (3.38) with the roles of $u_{\ell}$,
$v_{\ell}$ and $u$, $v$ interchanged also hold. That is, under the same
assumptions in Lemma 3.6, one has

$\displaystyle\lim_{\ell\to\infty}\Big{\|}\Big{[}v_{\ell}\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-v\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0,$ (3.40)
$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}u_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}u}\bigg{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0.$ (3.41)

$\diamond$

Using Theorem 2.8 and Lemma 3.6, one establishes analogues of (3.37)–(3.39)
for the operators $H_{\ell,\varphi,R}^{{(0)}}$ with coupled boundary
conditions. Our proof of the analogue of (3.37) makes use of Grümm’s theorem,
which we recall for completeness.

######  Theorem 3.8 (Grümm’s Theorem, [7]).

Let $p\in[1,\infty)$, $A\in\mathcal{B}_{p}(\mathcal{H})$, and suppose that
$\\{A_{\ell}\\}_{\ell=1}^{\infty}\subseteq\mathcal{B}_{p}(\mathcal{H})$ with
$\lim_{\ell\to\infty}{\|A_{\ell}-A\|_{\mathcal{B}_{p}(\mathcal{H})}}=0$. If
$B\in\mathcal{B}(\mathcal{H})$,
$\\{\mathcal{B}_{\ell}\\}_{\ell=1}^{\infty}\subseteq\mathcal{B}(\mathcal{H})$
with
$\sup_{\ell\in\mathbb{N}}{\|B_{\ell}\|_{\mathcal{B}(\mathcal{H})}}<\infty$ and
$\operatorname*{s-lim}_{\ell\to\infty}{B_{\ell}}=B$, then

$\lim_{\ell\to\infty}{\|A_{\ell}B_{\ell}-AB\|_{\mathcal{B}_{p}(\mathcal{H})}}=\lim_{\ell\to\infty}{\|B_{\ell}A_{\ell}-BA\|_{\mathcal{B}_{p}(\mathcal{H})}}=0.$
(3.42)

The analogue of (3.37) and (3.38) for $H_{\ell,\varphi,R}^{{(0)}}$ then reads
as follows:

######  Lemma 3.9.

Assume Hypothesis 2.1 and let
$k_{R,\max}\coloneqq\max\big{\\{}k_{R}^{{(0)}},(1+N_{R})^{1/2}\big{\\}}$. For
each fixed $z\in\mathbb{C}\setminus[-k_{R,\max}^{2},\infty)$, the following
convergence results hold in ${\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))$:

$\displaystyle\lim_{\ell\to\infty}\Big{\|}\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0,$ (3.43)
$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle=0.$ (3.44)

###### Proof.

Let $k>k_{R,\max}$. We first verify (3.43) for $z=-k^{2}$. Using Theorem 2.8
to relate the resolvent operators of $H_{\ell,\varphi,R}^{{(0)}}$ and
$H_{\ell,D}^{{(0)}}$, one obtains:

$\displaystyle\Big{\|}\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
(3.45)
$\displaystyle\quad\leq\Big{\|}\Big{[}u_{\ell}\big{(}H_{\ell,D}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle\qquad+\Big{\|}u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))},\quad\ell\in{\mathbb{N}}.$

The first ${\mathcal{B}}_{2}$-norm on the right-hand side of the inequality in
(3.45) converges to zero as $\ell\to\infty$ by Lemma 3.6. Thus, it suffices to
show

$\lim_{\ell\to\infty}\Big{\|}u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}=0.$
(3.46)

To this end, (2.31), (2.49), and Proposition 3.2 imply

$\displaystyle\Big{\|}u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\oplus_{\ell}0\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle\quad=\Big{\|}u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})\Big{\|}_{{\mathcal{B}}_{2}(L^{2}((-\ell,\ell)))}$
$\displaystyle\quad=\Bigg{\|}\sum_{m,n=1}^{2}\Big{[}K_{\ell,\varphi,R}^{{(0)}}(-k^{2})^{-1}\Big{]}_{m,n}\big{\langle}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,),\,\cdot\,\big{\rangle}_{L^{2}((-\ell,\ell))}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\Bigg{\|}_{{\mathcal{B}}_{2}(L^{2}((-\ell,\ell)))}$
$\displaystyle\,\;\underset{\ell\to\infty}{\leq}O(1)\cdot\sum_{m,n=1}^{2}\big{\|}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}$
$\displaystyle\,\;\underset{\ell\to\infty}{\leq}O(1)\cdot\sum_{m=1}^{2}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))},\quad\ell\in{\mathbb{N}}.$
(3.47)

The first inequality in (3.47) follows from (2.49) while the second inequality
follows from (3.19). The $L^{2}$-norms in the sum after the final inequality
in (3.47) converge to zero as $\ell\to\infty$. Indeed, writing for
$m\in\\{1,2\\}$ and $\ell\in{\mathbb{N}}$,

$\begin{split}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}^{2}=\frac{1}{4}\int_{-\infty}^{\infty}\bigg{|}\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\bigg{|}^{2}\chi_{(-\ell,\ell)}(x)|V(x)|\,\textrm{d}x,\end{split}$
(3.48)

one notes that for $m\in\\{1,2\\}$ and $\ell\in{\mathbb{N}}$,

$\frac{1}{4}\bigg{|}\frac{\cosh{(kx)}}{\cosh{(k\ell)}}+(-1)^{m-1}\frac{\sinh{(kx)}}{\sinh{(k\ell)}}\bigg{|}^{2}\chi_{(-\ell,\ell)}(x)|V(x)|\leq|V(x)|\,\text{
for a.e.~{}$x\in{\mathbb{R}}$}.$ (3.49)

Since $V\in L^{1}({\mathbb{R}})$ and the left-hand side of the inequality in
(3.49) converges to zero pointwise a.e. on ${\mathbb{R}}$, the dominated
convergence theorem implies

$\lim_{\ell\to\infty}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}=0,\quad
m\in\\{1,2\\}.$ (3.50)

Thus, by combining (3.47) and (3.50), one obtains for any $k>k_{R,\max}$,

$\lim_{\ell\to\infty}\Big{\|}\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}=0.$
(3.51)

To prove (3.43) in full generality, let
$z\in{\mathbb{C}}\setminus[-k_{R,\max}^{2},\infty)$. Fix any $k>k_{R,\max}$.
By the first resolvent identity applied to $H_{\ell,\varphi,R}^{{(0)}}$,

$\displaystyle
u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0$
(3.52)
$\displaystyle\quad=\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}$
$\displaystyle\qquad+(z+k^{2})\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}\Big{[}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}-z^{-1}I_{L^{2}({\mathbb{R}}\setminus(-\ell,\ell))}\Big{]}$
$\displaystyle\quad=\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}$
$\displaystyle\qquad+(z+k^{2})\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}\Big{(}\Big{[}H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\Big{]}-zI_{L^{2}({\mathbb{R}})}\Big{)}^{-1},\quad\ell\in{\mathbb{N}}.$

In light of the estimate

$\begin{split}\Big{\|}\Big{(}\Big{[}H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\Big{]}-zI_{L^{2}({\mathbb{R}})}\Big{)}^{-1}\Big{\|}_{{\mathcal{B}}(L^{2}({\mathbb{R}}))}&\leq\Big{\|}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}(L^{2}((-\ell,\ell)))}+|z|^{-1}\\\
&\leq\operatorname{dist}{\big{(}z,[-k_{R,\max}^{2},\infty)\big{)}}+|z|^{-1},\quad\ell\in{\mathbb{N}},\end{split}$
(3.53)

where $\operatorname{dist}{(\zeta,\Omega)}$ denotes the distance between the
point $\zeta\in{\mathbb{C}}$ and the set $\Omega\subseteq{\mathbb{C}}$, one
concludes

$\sup_{\ell\in{\mathbb{N}}}\Big{\|}\Big{(}\Big{[}H_{\ell,\varphi,R}^{(0)}\oplus_{\ell}0\Big{]}-zI_{L^{2}({\mathbb{R}})}\Big{)}^{-1}\Big{\|}_{{\mathcal{B}}(L^{2}({\mathbb{R}}))}<\infty.$
(3.54)

Therefore, by (3.51) and Lemma 3.3 combined with Grümm’s theorem (to treat the
second term after the second equality in (3.52)), the right-hand side in
(3.52) converges in ${\mathcal{B}}_{2}\big{(}L^{2}({\mathbb{R}})\big{)}$ as
$\ell\to\infty$ and

$\displaystyle\lim_{\ell\to\infty}\Big{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}$
(3.55)
$\displaystyle\quad=u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}+(z+k^{2})u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}$
$\displaystyle\quad=u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\,\text{
in ${\mathcal{B}}_{2}\big{(}L^{2}({\mathbb{R}})\big{)}$},$

which establishes (3.43). The final equality in (3.55) is the result of
another application of the first resolvent identity, this time for
$H^{{(0)}}$.

Finally, to prove (3.44), note that the same argument employed above to prove
(3.43) shows mutatis mutandis (using (3.40)) that

$\lim_{\ell\to\infty}\Big{\|}\Big{[}v_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-v\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}=0$
(3.56)

for $z\in{\mathbb{C}}\setminus[-k_{R,\max}^{2},\infty)$. Therefore, for
$z\in{\mathbb{C}}\setminus[-k_{R,\max}^{2},\infty)$, taking adjoints one
obtains

$\displaystyle\bigg{\|}\bigg{[}\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
(3.57)
$\displaystyle\quad=\Big{\|}\Big{[}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}\oplus_{\ell}0\Big{]}^{\ast\ast}-\Big{[}\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v\Big{]}^{\ast\ast}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle\quad=\Big{\|}\Big{[}v_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-\overline{z}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}^{\ast}-\Big{[}v\big{(}H^{(0)}-\overline{z}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{]}^{\ast}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))}$
$\displaystyle\quad=\Big{\|}\Big{[}v_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-\overline{z}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\Big{]}-v\big{(}H^{(0)}-\overline{z}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}(L^{2}({\mathbb{R}}))},\quad\ell\in{\mathbb{N}},$

which converges to zero as $\ell\to\infty$ by (3.56). The claim in (3.44)
follows. ∎

Our proof of the analogue of (3.39) for $H_{\ell,\varphi,R}^{{(0)}}$ relies on
the following classical result for the limit of a sequence that is the term-
wise product of operators belonging to appropriate trace ideals.

######  Lemma 3.10.

Let $p,q,r\in[1,\infty)$ with $p^{-1}+q^{-1}=r^{-1}$. If
$\\{A_{\ell}\\}_{\ell=1}^{\infty}\subset{\mathcal{B}}_{p}({\mathcal{H}})$,
$\\{B_{\ell}\\}_{\ell=1}^{\infty}\subset{\mathcal{B}}_{q}({\mathcal{H}})$,
$A\in{\mathcal{B}}_{p}({\mathcal{H}})$, and
$B\in{\mathcal{B}}_{q}({\mathcal{H}})$ with

$\lim_{\ell\to\infty}\|A_{\ell}-A\|_{{\mathcal{B}}_{p}({\mathcal{H}})}=0\quad\text{and}\quad\lim_{\ell\to\infty}\|B_{\ell}-B\|_{{\mathcal{B}}_{q}({\mathcal{H}})}=0,$
(3.58)

then

$\lim_{\ell\to\infty}\|A_{\ell}B_{\ell}-AB\|_{{\mathcal{B}}_{r}({\mathcal{H}})}=0.$
(3.59)

The proof of Lemma 3.10 is an application of Hölder’s inequality for the trace
ideals (cf., e.g., [10, Theorem 2.8]). Lemma 3.10 then combines with Lemma 3.9
to yield the following analogue of (3.39) for $H_{\ell,\varphi,R}^{{(0)}}$.

######  Lemma 3.11.

Assume Hypothesis 2.1 and let
$k_{R,\max}\coloneqq\max\big{\\{}k_{R}^{{(0)}},(1+N_{R})^{1/2}\big{\\}}$. For
each fixed $z\in\mathbb{C}\setminus[-k_{R,\max}^{2},\infty)$, the following
convergence result holds in ${\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))$:

$\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0.$
(3.60)

###### Proof.

Let $k>k_{R,\max}$. We first verify (3.60) for $z=-k^{2}$. Using Theorem 2.8
to relate the resolvent operators of $H_{\ell,\varphi,R}^{{(0)}}$ and
$H_{\ell,D}^{{(0)}}$, one obtains:

$\displaystyle\bigg{\|}\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}$
(3.61)
$\displaystyle\quad\leq\bigg{\|}\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,D}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v}\oplus_{\ell}0\bigg{]}-\overline{u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}$
$\displaystyle\qquad+\bigg{\|}\overline{u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}}\oplus_{\ell}0\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))},\quad\ell\in{\mathbb{N}}.$

The first ${\mathcal{B}}_{1}$-norm on the right-hand side of the inequality in
(3.61) converges to zero as $\ell\to\infty$ by Lemma 3.6. Invoking (3.30),
(2.49), and Proposition 3.2, one obtains

$\displaystyle\bigg{\|}\overline{u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}}\oplus_{\ell}0\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}$
(3.62)
$\displaystyle\quad=\bigg{\|}\overline{u_{\ell}P_{\ell,\varphi,R}^{{(0)}}(-k^{2})v_{\ell}}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}((-\ell,\ell)))}$
$\displaystyle\,\;\underset{\ell\to\infty}{\leq}O(1)\cdot\sum_{m,n=1}^{2}\big{\|}v_{\ell}\psi_{\ell,n}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))}\big{\|}u_{\ell}\psi_{\ell,m}^{{(0)}}(-k^{2},\,\cdot\,)\big{\|}_{L^{2}((-\ell,\ell))},\quad\ell\in{\mathbb{N}}.$

Since $|u_{\ell}|=|v_{\ell}|$ a.e. on $(-\ell,\ell)$, (3.50) implies that each
factor under the sum in (3.62), and hence the sum itself, converges to zero as
$\ell\to\infty$. Thus, taking $\ell\to\infty$ throughout (3.61), one
concludes:

$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}-\overline{u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\bigg{\|}_{{\mathcal{B}}_{1}(L^{2}({\mathbb{R}}))}=0.$
(3.63)

To prove (3.60) in full generality, let
$z\in{\mathbb{C}}\setminus[-k_{R,\max}^{2},\infty)$. Fix any $k>k_{R,\max}$.
By the first resolvent identity applied to $H_{\ell,\varphi,R}^{{(0)}}$,

$\displaystyle\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0$
(3.64)
$\displaystyle\quad=\bigg{[}\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]}$
$\displaystyle\qquad+(z+k^{2})\bigg{[}u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}+k^{2}I_{L^{2}((-\ell,\ell))}\big{)}^{-1}\oplus_{\ell}0\bigg{]}\bigg{[}\overline{\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0\bigg{]},\quad\ell\in{\mathbb{N}}.$

Taking $\ell\to\infty$ throughout (3.64) and invoking (3.63) to treat the
first term on the right-hand side in (3.64) and using Lemmata 3.9 and 3.10 to
treat the second term on the right-hand side in (3.64), one obtains

$\displaystyle\lim_{\ell\to\infty}\overline{u_{\ell}\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}v_{\ell}}\oplus_{\ell}0$
(3.65)
$\displaystyle\quad=\overline{u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}v}+(z+k^{2})u\big{(}H^{(0)}+k^{2}I_{L^{2}(\mathbb{R})}\big{)}^{-1}\overline{\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}$
$\displaystyle\quad=\overline{u\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}v}\,\text{
in ${\mathcal{B}}_{1}\big{(}L^{2}({\mathbb{R}})\big{)}$},$

and (3.60) follows. The final equality in (3.65) is the result of another
application of the first resolvent identity, this time for $H^{{(0)}}$. ∎

##  4\. Convergence Results for Spectral Shift Functions

Hypothesis 2.1 guarantees that $H^{{(0)}}$ and $H$ are resolvent comparable;
that is,

$\Big{[}\big{(}H-zI_{L^{2}({\mathbb{R}})}\big{)}^{-1}-\big{(}H^{(0)}-zI_{L^{2}(\mathbb{R})}\big{)}^{-1}\Big{]}\in{\mathcal{B}}_{1}\big{(}L^{2}({\mathbb{R}})\big{)},\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}}.$ (4.1)

In a similar vein, for each $\ell\in{\mathbb{N}}$,

$\Big{[}\big{(}H_{\ell,D}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}-\big{(}H_{\ell,D}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{]}\in{\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)},\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}}.$ (4.2)

Moreover, Theorem 2.8 and (4.2) imply $H_{\ell,\varphi,R}$ and
$H_{\ell,\varphi,R}^{{(0)}}$ are resolvent comparable for every
$\ell\in{\mathbb{N}}$:

$\Big{[}\big{(}H_{\ell,\varphi,R}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}-\big{(}H_{\ell,\varphi,R}^{(0)}-zI_{L^{2}((-\ell,\ell))}\big{)}^{-1}\Big{]}\in{\mathcal{B}}_{1}\big{(}L^{2}((-\ell,\ell))\big{)},\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}}.$ (4.3)

By (4.1) and (4.3) there exist unique real-valued (a.e.) spectral shift
functions $\xi\big{(}\,\cdot\,;H,H^{{(0)}}\big{)}$ and
$\xi\big{(}\,\cdot\,;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}$ for
the pairs $\big{(}H,H^{{(0)}}\big{)}$ and
$\big{(}H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}$,
$\ell\in{\mathbb{N}}$, respectively, that satisfy

$\int_{-\infty}^{\infty}\frac{\big{|}\xi\big{(}\lambda;H,H^{{(0)}}\big{)}\big{|}}{1+\lambda^{2}}\,\textrm{d}\lambda<\infty,\quad\text{and}\quad\xi\big{(}\lambda;H,H^{{(0)}}\big{)}=0,\quad\lambda<\min\sigma(H),$
(4.4)

and

$\begin{split}&\int_{-\infty}^{\infty}\frac{\big{|}\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}\big{|}}{1+\lambda^{2}}\,\textrm{d}\lambda<\infty,\\\
&\xi\big{(}\lambda;H_{\lambda,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}=0,\quad\lambda<\min\big{[}\sigma\big{(}H_{\ell,\varphi,R}\big{)}\cup\sigma\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}\big{]},\quad\ell\in{\mathbb{N}}.\end{split}$
(4.5)

and for which Krein’s trace formulæ hold: For any
$f\in\mathfrak{F}({\mathbb{R}})$ (cf. (A.25)–(A.26)),

$\displaystyle\mbox{tr}_{L^{2}({\mathbb{R}})}\big{(}f(H)-f\big{(}H^{{(0)}}\big{)}\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}f^{\prime}(\lambda)\xi\big{(}\lambda;H,H^{{(0)}}\big{)}\,\textrm{d}\lambda,$
(4.6)
$\displaystyle\mbox{tr}_{L^{2}((-\ell,\ell))}\big{(}f\big{(}H_{\ell,\varphi,R}\big{)}-f\big{(}H_{\ell,\varphi,R}^{{(0)}}\big{)}\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}f^{\prime}(\lambda)\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}\,\textrm{d}\lambda,\quad\ell\in{\mathbb{N}}.$
(4.7)

The convergence properties in Lemmata 3.1, 3.3, 3.5, 3.9, and 3.11 imply the
following weak convergence property of
$\xi\big{(}\,\cdot\,;H_{\ell,\varphi,R}\big{)}$ to
$\xi\big{(}\,\cdot\,;H,H^{{(0)}}\big{)}$ as $\ell\to\infty$.

######  Theorem 4.1.

If Hypothesis 2.1 holds, then

$\lim_{\ell\rightarrow\infty}\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}}{1+\lambda^{2}}\,f(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;H,H^{{(0)}}\big{)}}{1+\lambda^{2}}\,f(\lambda)\,\mathrm{d}\lambda,\quad
f\in C_{b}({\mathbb{R}}).$ (4.8)

###### Proof.

It suffices to verify that conditions $(i)$–$(ix)$ in Hypothesis A.1 hold with
the identifications

$\begin{split}&T^{{(0)}}=H^{{(0)}},\hskip 22.1931ptT=H,\hskip
36.98866ptV_{1}=u,\hskip 20.48601ptV_{2}=v,\\\
&T^{{(0)}}_{\ell}=H_{\ell,\varphi,R}^{{(0)}},\quad
T_{\ell}=H_{\ell,\varphi,R},\quad V_{1,\ell}=u_{\ell},\quad
V_{2,\ell}=v_{\ell},\quad\ell\in{\mathbb{N}}.\end{split}$ (4.9)

The conclusion of the theorem then follows from Theorem A.2. Condition $(i)$
holds by construction and Lemmata 2.3 and 2.6 imply that $(ii)$ holds. Since
$u,v\in L^{2}({\mathbb{R}})$ are real-valued a.e. and functions in
$\operatorname{dom}{(|H^{{(0)}}|^{1/2})}=H^{1}({\mathbb{R}})$ are bounded, one
infers that the operators of multiplication by $u$ and $v$ are self-adjoint,
hence closed, and that (A.2) an (A.4) hold. Similarly, one verifies that (A.3)
and (A.5) hold. Therefore, condition $(iii)$ is satisfied. Condition $(iv)$
holds by Lemmata 2.4, 2.11, and 3.5. Condition $(v)$ holds by Lemma 3.3, and
condition $(vi)$ holds by Lemmata 3.9 and 3.11. Condition $(vii)$ holds since
$V_{1}=u$, $V_{2}=v$, $V_{1,\ell}=u_{\ell}$, and $V_{2,\ell}=v_{\ell}$ are
operators of multiplication by real-valued functions, and condition $(viii)$
holds by Remarks 2.5 and 2.12. Finally, the explicit structures of the
sesquilinear forms $\mathfrak{Q}$ and $\mathfrak{Q}_{\ell,\varphi,R}$ (cf.
(2.3) and (2.18)) imply

$\mathfrak{Q}=\mathfrak{Q}^{{(0)}}+\mathfrak{Q}_{V},\quad\mathfrak{Q}_{\ell,\varphi,R}=\mathfrak{Q}_{\ell,\varphi,R}^{{(0)}}+\mathfrak{Q}_{V_{\ell}},\quad\ell\in{\mathbb{N}},$
(4.10)

where $\mathfrak{Q}_{V}$ and $\mathfrak{Q}_{V_{\ell}}$ denote the sesquilinear
forms uniquely associated to the self-adjoint operators of multiplication by
$V$ and $V_{\ell}$ in $L^{2}({\mathbb{R}})$ and $L^{2}((-\ell,\ell))$,
$\ell\in{\mathbb{N}}$, respectively. In particular, $H$ is the form sum of
$H^{{(0)}}$ with $V$ and $H_{\ell,\varphi,R}$ is the form sum of
$H_{\ell,\varphi,R}^{{(0)}}$ with $V_{\ell}$, $\ell\in{\mathbb{N}}$. Hence,
condition $(ix)$ holds. ∎

Invoking the abstract Corollaries A.3, A.4, and A.5, one obtains the following
additional results for convergence of the spectral shift functions.

######  Corollary 4.2.

If Hypothesis 2.1 holds, then

$\lim_{\ell\to\infty}\int_{-\infty}^{\infty}\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}\,g(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\xi\big{(}\lambda;H,H^{{(0)}}\big{)}\,g(\lambda)\,\mathrm{d}\lambda,\quad
g\in C_{c}({\mathbb{R}}).$ (4.11)

######  Corollary 4.3.

If Hypothesis 2.1 holds, then the convergence in (4.8) holds for any bounded
Borel measurable function that is continuous almost everywhere with respect to
Lebesgue measure on ${\mathbb{R}}$. In particular,

$\lim_{\ell\to\infty}\int_{S}\frac{\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}}{1+\lambda^{2}}\,\mathrm{d}\lambda=\int_{S}\frac{\xi\big{(}\lambda;H,H^{{(0)}}\big{)}}{1+\lambda^{2}}\,\mathrm{d}\lambda$
(4.12)

holds for any set $S\subseteq{\mathbb{R}}$ that is boundaryless with respect
to Lebesgue measure $($i.e., any set $S\subseteq{\mathbb{R}}$ for which the
boundary of $S$ has Lebesgue measure equal to zero$)$.

######  Corollary 4.4.

If Hypothesis A.1 holds, then for every bounded Borel measurable function $g$
that is compactly supported and Lebesgue almost everywhere continuous on
${\mathbb{R}}$,

$\lim_{\ell\rightarrow\infty}\int_{-\infty}^{\infty}\xi\big{(}\lambda;H_{\ell,\varphi,R},H_{\ell,\varphi,R}^{{(0)}}\big{)}\,g(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\xi\big{(}\lambda;H,H^{{(0)}}\big{)}\,g(\lambda)\,\mathrm{d}\lambda.$
(4.13)

##  Appendix A Criteria for Vague and Weak Convergence of Spectral Shift
Functions

We recall the criteria for convergence of sequences of spectral shift
functions introduced in [5] suitably tailored to the context in which they are
applied in this paper; namely, for pairs of self-adjoint operators acting in
the Hilbert spaces $L^{2}((-\ell,\ell))$ and $L^{2}({\mathbb{R}})$.

######  Hypothesis A.1 (Hypothesis 3.1 in [5]).

Set ${\mathcal{H}}\coloneqq L^{2}({\mathbb{R}})$.
$(i)$ For each $\ell\in{\mathbb{N}}$, decompose ${\mathcal{H}}$ according to

$L^{2}({\mathbb{R}})=L^{2}((-\ell,\ell))\oplus_{\ell}L^{2}({\mathbb{R}}\setminus(-\ell,\ell)),$
(A.1)

and define ${\mathcal{H}}_{\ell}:=L^{2}((-\ell,\ell))$ and
${\mathcal{H}}_{\ell}^{c}=L^{2}({\mathbb{R}}\setminus(-\ell,\ell))$.
$(ii)$ Let $T^{(0)}$ be a self-adjoint operator in ${\mathcal{H}}$, and for
each $\ell\in{\mathbb{N}}$, let $T_{\ell}^{(0)}$ be self-adjoint operators in
${\mathcal{H}}_{\ell}$. In addition, suppose that $T^{(0)}$ is lower
semibounded in ${\mathcal{H}}$, and that for each $\ell\in{\mathbb{N}}$,
$T_{\ell}^{(0)}$ is lower semibounded in ${\mathcal{H}}_{\ell}$.
$(iii)$ Suppose that $V_{1}$, and $V_{2}$ are closed operators in
${\mathcal{H}}$, and for each $\ell\in{\mathbb{N}}$, assume that $V_{1,\ell}$,
and $V_{2,\ell}$ are closed operators in ${\mathcal{H}}_{\ell}$ such that

$\displaystyle\operatorname{dom}{(V_{1})}\cap\operatorname{dom}{(V_{2})}\supseteq\operatorname{dom}{\big{(}|T^{(0)}|^{1/2}\big{)}},$
(A.2)
$\displaystyle\operatorname{dom}{(V_{1,\ell})}\cap\operatorname{dom}{(V_{2,\ell})}\supseteq\operatorname{dom}{\big{(}|T_{\ell}^{(0)}|^{1/2}\big{)}},\quad\ell\in{\mathbb{N}},$
(A.3)

where

$V=V_{1}^{*}V_{2}\,\text{ is a self-adjoint operator in ${\mathcal{H}}$},$
(A.4)

and for each $\ell\in{\mathbb{N}}$,

$V_{\ell}=V_{1,\ell}^{*}V_{2,\ell}\,\text{ is a self-adjoint operator in
${\mathcal{H}}_{\ell}$}.$ (A.5)

$(iv)$ Suppose

$\displaystyle\overline{V_{2}\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}V_{1}^{*}},\,\overline{V_{2,\ell}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}V_{1,\ell}^{*}}\oplus_{\ell}0\in{\mathcal{B}}_{1}({\mathcal{H}}),\quad\ell\in{\mathbb{N}},$
(A.6) $\displaystyle
V_{2}\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1},\,V_{2,\ell}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}\oplus_{\ell}0\in{\mathcal{B}}_{2}({\mathcal{H}}),\quad\ell\in{\mathbb{N}},$
(A.7)
$\displaystyle\overline{\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}V_{1}^{*}},\,\overline{\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}V_{1,\ell}^{*}}\oplus_{\ell}0\in{\mathcal{B}}_{2}({\mathcal{H}}),\quad\ell\in{\mathbb{N}},$
(A.8)

for some $($and hence for all $)$ $z\in{\mathbb{C}}\setminus{\mathbb{R}}$. In
addition, assume that

$\displaystyle\begin{split}&\lim_{z\downarrow-\infty}\Big{\|}\,\overline{V_{2}\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}V_{1}^{*}}\,\Big{\|}_{{\mathcal{B}}_{1}({\mathcal{H}})}=0,\\\
&\lim_{z\downarrow-\infty}\Big{\|}\,\overline{V_{2,\ell}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}V_{1,\ell}^{*}}\oplus_{\ell}0\,\Big{\|}_{{\mathcal{B}}_{1}({\mathcal{H}})}=0,\quad\ell\in{\mathbb{N}}.\end{split}$
(A.9)

$(v)$ Assume that for some $($and hence for all $)$
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$,

$\operatorname*{s-lim}_{\ell\to\infty}\bigg{[}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}\oplus_{\ell}\frac{-1}{z}I_{{\mathcal{H}}_{\ell}^{c}}\bigg{]}=\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}.$
(A.10)

$(vi)$ Suppose that for some $($and hence for all $)$
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$,

$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\,\overline{V_{2,\ell}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}V_{1,\ell}^{*}}\oplus_{\ell}0\bigg{]}-\overline{V_{2}\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}V_{1}^{*}}\,\bigg{\|}_{{\mathcal{B}}_{1}({\mathcal{H}})}=0,$
(A.11)
$\displaystyle\lim_{\ell\to\infty}\Big{\|}\Big{[}V_{2,\ell}\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}}\big{)}^{-1}\oplus_{\ell}0\Big{]}-V_{2}\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}\Big{\|}_{{\mathcal{B}}_{2}({\mathcal{H}})}=0,$
(A.12)
$\displaystyle\lim_{\ell\to\infty}\bigg{\|}\bigg{[}\,\overline{\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}_{\ell}})^{-1}V_{1,\ell}^{*}}\oplus_{\ell}0\bigg{]}-\overline{\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}V_{1}^{*}}\,\bigg{\|}_{{\mathcal{B}}_{2}({\mathcal{H}})}=0.$
(A.13)

$(vii)$ Suppose that

$\displaystyle\begin{split}&\langle
V_{2}f,V_{1}g\rangle_{{\mathcal{H}}}=\langle
V_{1}f,V_{2}g\rangle_{{\mathcal{H}}},\quad
f,g\in\operatorname{dom}{(V_{1})}\cap\operatorname{dom}{(V_{2})},\\\ &\langle
V_{2,\ell}f,V_{1,\ell}g\rangle_{{\mathcal{H}}}=\langle
V_{1,\ell}f,V_{2,\ell}g\rangle_{{\mathcal{H}}},\quad
f,g\in\operatorname{dom}{(V_{1,\ell})}\cap\operatorname{dom}{(V_{2,\ell})},\quad\ell\in{\mathbb{N}}.\end{split}$
(A.14)

$(viii)$ Decomposing $V,V_{\ell}$, $\ell\in{\mathbb{N}}$, into their positive
and negative parts,

$V_{\pm}=[|V|\pm V]/2,\quad V_{\ell,\pm}=[|V_{\ell}|\pm
V_{\ell}]/2,\quad\ell\in{\mathbb{N}},$ (A.15)

$V_{\pm}$ are assumed to be infinitesimally form bounded with respect to
$T^{(0)}$, and for each $\ell\in{\mathbb{N}}$, $V_{\ell,\pm}$ are assumed to
be infinitesimally form bounded with respect to $T_{\ell}^{(0)}$.
$(ix)$ Suppose that $T$ and $T_{\ell}$, $\ell\in{\mathbb{N}}$ are the
quadratic forms sums of $T^{{(0)}}$ and $T_{\ell}^{{(0)}}$ with
$V_{1}^{*}V_{2}$ and $V_{1,\ell}^{*}V_{2,\ell}$, respectively; that is,

$T=T^{{(0)}}+_{\mathfrak{q}}V_{1}^{*}V_{2},\quad
T_{\ell}=T_{\ell}^{{(0)}}+_{\mathfrak{q}}V_{1,\ell}^{*}V_{2,\ell},\quad\ell\in{\mathbb{N}}.$
(A.16)

Assuming Hypothesis A.1, the pairs $\big{(}T,T^{(0)}\big{)}$ and
$\big{(}T_{\ell},T_{\ell}^{(0)}\big{)}$, $\ell\in{\mathbb{N}}$, are resolvent
comparable in the sense that

$\big{[}\big{(}T-zI_{{\mathcal{H}}}\big{)}^{-1}-\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}\big{]}\in{\mathcal{B}}_{1}({\mathcal{H}}),\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}},$ (A.17)

and

$\big{[}\big{(}T_{\ell}-zI_{{\mathcal{H}}}\big{)}^{-1}-\big{(}T_{\ell}^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}\big{]}\in{\mathcal{B}}_{1}({\mathcal{H}}),\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}},\,\ell\in{\mathbb{N}}.$ (A.18)

Thus, Hypothesis A.1 guarantees the existence of real-valued spectral shift
functions $\xi\big{(}\,\cdot\,;T,T^{(0)}\big{)}$ and
$\xi\big{(}\,\cdot\,;T_{\ell},T^{(0)}_{\ell}\big{)}$, $\ell\in{\mathbb{N}}$,
which satisfy

$\begin{split}\mbox{tr}_{{\mathcal{H}}}\Big{(}\big{(}T-zI_{{\mathcal{H}}}\big{)}^{-1}-\big{(}T^{(0)}-zI_{{\mathcal{H}}}\big{)}^{-1}\Big{)}=-\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;T,T^{(0)}\big{)}}{(\lambda-z)^{2}}\,d\lambda,&\\\
\quad z\in\rho(T)\cap\rho\big{(}T^{(0)}\big{)},&\end{split}$ (A.19)

and

$\begin{split}\mbox{tr}_{{\mathcal{H}}_{\ell}}\Big{(}\big{(}T_{\ell}-zI_{{\mathcal{H}}}\big{)}^{-1}-\big{(}T^{(0)}_{\ell}-zI_{{\mathcal{H}}}\big{)}^{-1}\Big{)}=-\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;T_{\ell},T^{(0)}_{\ell}\big{)}}{(\lambda-z)^{2}}\,d\lambda,&\\\
z\in\rho(T_{\ell})\cap\rho\big{(}T^{(0)}_{\ell}\big{)},\,\ell\in{\mathbb{N}},&\end{split}$
(A.20)

and are uniquely determined almost everywhere by the conditions

$\int_{-\infty}^{\infty}\frac{|\xi\big{(}\lambda;T,T^{(0)}\big{)}|}{1+\lambda^{2}}\,\mathrm{d}\lambda<\infty\quad\text{and}\quad\xi\big{(}\lambda;T,T^{(0)}\big{)}=0,\quad\lambda<\min\big{[}\sigma(T)\cup\sigma\big{(}T^{(0)}\big{)}\big{]},$
(A.21)

and for each $\ell\in{\mathbb{N}}$,

$\int_{-\infty}^{\infty}\frac{|\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}|}{1+\lambda^{2}}\,\mathrm{d}\lambda<\infty\quad\text{and}\quad\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}=0,\quad\lambda<\min\big{[}\sigma\big{(}T_{\ell}\big{)}\cup\sigma\big{(}T_{\ell}^{(0)}\big{)}\big{]}.$
(A.22)

Moreover, Krein’s trace formulæ hold:

$\displaystyle\mbox{tr}_{{\mathcal{H}}}\big{(}f(T)-f\big{(}T^{(0)}\big{)}\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}f^{\prime}(\lambda)\,\xi\big{(}\lambda;T,T^{(0)}\big{)}\,\mathrm{d}\lambda,$
(A.23)
$\displaystyle\mbox{tr}_{{\mathcal{H}}}\big{(}f(T_{\ell})-f\big{(}T^{(0)}_{\ell}\big{)}\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}f^{\prime}(\lambda)\,\xi\big{(}\lambda;T_{\ell},T^{(0)}_{\ell}\big{)}\,\mathrm{d}\lambda,\quad\ell\in{\mathbb{N}},$
(A.24)

where $\mathfrak{F}({\mathbb{R}})$ denotes the set of all function
$f\colon{\mathbb{R}}\to{\mathbb{C}}$ with two locally bounded derivatives
satisfying

$\big{(}\lambda^{2}f^{\prime}(\lambda)\big{)}^{\prime}=O\big{(}|\lambda|^{-1-\varepsilon}\big{)},\quad|\lambda|\to\infty$
for some $\varepsilon=\varepsilon(f)>0,$ (A.25)

with

$\lim_{|\lambda|\to\infty}f(\lambda)=C\quad\text{and}\quad\lim_{|\lambda|\to\infty}\lambda^{2}f^{\prime}(\lambda)=D,$
(A.26)

for some constants $C=C(f),D=D(f)\in{\mathbb{C}}$. In particular,

$(\lambda-z)^{-n}\in\mathfrak{F}({\mathbb{R}})$,
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$, $n\in{\mathbb{N}}$, (A.27)

and in this case, the trace formula for $\big{(}T,T^{{(0)}}\big{)}$ reads

$\mbox{tr}_{{\mathcal{H}}}\big{(}(T-zI_{{\mathcal{H}}})^{-n}-\big{(}T^{{(0)}}-zI_{{\mathcal{H}}}\big{)}^{-n}\big{)}=-n\int_{{\mathbb{R}}}\frac{\xi\big{(}\lambda;T,T^{{(0)}}\big{)}}{(\lambda-z)^{n+1}}\,d\lambda,\quad
z\in{\mathbb{C}}\setminus{\mathbb{R}},\,n\in{\mathbb{N}},$ (A.28)

and an analogous formula holds for $\big{(}T_{\ell},T_{\ell}^{{(0)}}\big{)}$,
$\ell\in{\mathbb{N}}$. We refer to [13, Chapter 8] for additional information
related to spectral shift functions.

Under the assumptions in Hypothesis A.1, the following convergence results
hold for the sequence of spectral shift functions
$\big{\\{}\xi\big{(}\,\cdot\,;T_{\ell},T_{\ell}^{(0)}\big{)}\big{\\}}_{\ell=1}^{\infty}$.

######  Theorem A.2 (Theorem 3.13 in [5]).

Assume Hypothesis A.1. Then

$\lim_{\ell\rightarrow\infty}\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}}{1+\lambda^{2}}\,f(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\frac{\xi\big{(}\lambda;T,T^{(0)}\big{)}}{1+\lambda^{2}}\,f(\lambda)\,\mathrm{d}\lambda,\quad
f\in C_{b}({\mathbb{R}}).$ (A.29)

The factor $(1+\lambda^{2})^{-1}$ is essential in (A.29), as the integrals
need not exist without it. One consequence of Theorem A.2 is that the sequence
$\big{\\{}\xi\big{(}\,\cdot\,;T_{\ell},T_{\ell}^{{(0)}}\big{)}\big{\\}}_{\ell=1}^{\infty}$
converges vaguley to $\xi\big{(}\,\cdot\,;T,T^{{(0)}}\big{)}$.

######  Corollary A.3 (Corollary 3.11 in [5]).

Assume Hypothesis A.1. Then

$\lim_{\ell\to\infty}\int_{-\infty}^{\infty}\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}\,g(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\xi\big{(}\lambda;T,T^{(0)}\big{)}\,g(\lambda)\,\mathrm{d}\lambda\quad
g\in C_{c}({\mathbb{R}}).$ (A.30)

Finally, the continuity assumption in Theorem A.2 may be slightly relaxed as
follows.

######  Corollary A.4 (Corollary 3.14 in [5]).

Assume Hypothesis A.1. Then the convergence in (A.29) holds for any bounded
Borel measurable function that is continuous almost everywhere with respect to
Lebesgue measure on ${\mathbb{R}}$. In particular,

$\lim_{\ell\to\infty}\int_{S}\frac{\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}}{1+\lambda^{2}}\,\mathrm{d}\lambda=\int_{S}\frac{\xi\big{(}\lambda;T,T^{(0)}\big{)}}{1+\lambda^{2}}\,\mathrm{d}\lambda$
(A.31)

holds for any set $S\subseteq{\mathbb{R}}$ that is boundaryless with respect
to Lebesgue measure $($i.e., any set $S\subseteq{\mathbb{R}}$ for which the
boundary of $S$ has Lebesgue measure equal to zero$)$.

######  Corollary A.5 (Corollary 3.15 in [5]).

Assume Hypothesis A.1. If $g$ is a bounded Borel measurable function that is
compactly supported and Lebesgue almost everywhere continuous on
${\mathbb{R}}$, then

$\lim_{\ell\rightarrow\infty}\int_{-\infty}^{\infty}\xi\big{(}\lambda;T_{\ell},T_{\ell}^{(0)}\big{)}\,g(\lambda)\,\mathrm{d}\lambda=\int_{-\infty}^{\infty}\xi\big{(}\lambda;T,T^{(0)}\big{)}\,g(\lambda)\,\mathrm{d}\lambda.$
(A.32)

Acknowledgments. The research of the authors was supported by the National
Science Foundation under grant DMS–1852288.

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