# Diverging eigenvalues in domain truncations of Schrödinger operators with
complex potentials

Iveta Semorádová  Faculty of Nuclear Science and Physical Engineering, Czech
Technical University in Prague, Czech Republic & Nuclear Physics Institute,
The Czech Academy of Science, Hlavní 130, 25068 Řež, Czech Republic
<EMAIL_ADDRESS>and  Petr Siegl  School of Mathematics and
Physics, Queen’s University Belfast, University Road, BT7 1NN Belfast, UK
<EMAIL_ADDRESS>
(Date: July 20, 2021)

###### Abstract.

Diverging eigenvalues in domain truncations of Schrödinger operators with
complex potentials are analyzed and their asymptotic formulas are obtained.
Our approach also yields asymptotic formulas for diverging eigenvalues in the
strong coupling regime for the imaginary part of the potential.

###### Key words and phrases:

Schrödinger operators, complex potential, domain truncation, spectral
exactness, diverging eigenvalues

###### 2010 Mathematics Subject Classification:

35J10, 47A10, 47A58

The work of I.S. was supported by the Grant Agency of the Czech Technical
University in Prague, grant No. SGS19/183/OHK4/3T/14. It was initiated at the
International Centre for Theoretical Sciences (ICTS), Bengaluru, India, during
a visit of authors in the program - Non-Hermitian Physics - PHHQP XVIII in
2018 (Code: ICTS/nhp2018/06). Further, I.S. would like to thank Queen’s
University Belfast for the hospitality during her Erasmus+ stay there is
September 2019 – July 2020.

##  1\. Introduction

Approximating spectra of non-self-adjoint partial differential operators is a
major challenge in spectral analysis even in the case of purely discrete
spectra. In this paper, we focus on the spectral convergence of domain
truncations for multidimensional Schrödinger operators $-\Delta+Q$ in
$L^{2}(\Omega)$ with $\Omega\subset\mathbb{R}^{d}$ and a _complex_ potential
$Q:\Omega\to{\mathbb{C}}$, the study of which was initiated in [12]. As an
example, which we use here to indicated the questions addressed in this paper
and our new results, consider the imaginary oscillator

$T_{\infty}=-\Delta_{\rm D}+{\rm i}|x|^{2}$ (1.1)

in $L^{2}(\Omega)$ with $\Omega={\mathbb{R}}^{d}\setminus\overline{B_{1}(0)}$
and the Dirichlet boundary condition imposed at $\partial\Omega$. A possible
sequence of truncations are $T_{n}=-\Delta_{\rm D}+Q$ in $L^{2}(\Omega_{n})$
with $\Omega_{n}:=B_{s_{n}}(0)\cap\Omega$, $s_{n}\nearrow+\infty$, and
Dirichlet boundary conditions at $\partial\Omega_{n}$, $n\in\mathbb{N}$. The
general goal is to determine the relation of spectra of $\\{T_{n}\\}$ and
$T_{\infty}$.

It was established in [11] that for potentials $Q$ with
$\operatorname{Re}Q\geq 0$, $|Q(x)|\to+\infty$ as $|x|\to\infty$ and
satisfying suitable regularity conditions (hence in particular for (1.1)), the
truncations $T_{n}$ converge to $T_{\infty}$ as $n\to\infty$ in the norm
resolvent sense. Consequently the approximation is _spectrally exact_ , _i.e._
all eigenvalues of $T_{\infty}$ are approximated by eigenvalues of $T_{n}$ and
there is no pollution (there are no finite accumulation points of eigenvalues
of $\\{T_{n}\\}$ which are not eigenvalues of $T_{\infty}$), see  _e.g._ [12].
The results in [11] include more general cases with different boundary
conditions and with potentials having negative real part controlled by
$\operatorname{Im}Q$, moreover, the convergence rates of eigenvalues were
related to the decay of eigenfunctions of $T_{\infty}$. (For further works on
spectral approximations using limiting essential spectra and essential
numerical ranges see [8, 9, 10].)

It is however crucial to notice that the spectral exactness does not exclude
eigenvalues of the truncations escaping to infinity as $n\to\infty$, which is
in our example (1.1) illustrated in Figure 1.1. In fact, many other examples,
in particular the one-dimensional imaginary cubic oscillator ($Q(x)={\rm
i}x^{3}$) examined in [11, 24], suggest that diverging eigenvalues are rather
typical and exhibit quite regular patterns.

Figure 1.1. Trajectory in ${\mathbb{C}}$ of eigenvalues of truncations $T_{n}$
of $T_{\infty}$ from (1.1) for $d=3$, $s_{n}=0.1n$, $n=15,16,\dots,115$;
eigenvalues for $l=1$ (red), $l=2$ (pink), $l=3$ (green), $l=4$ (purple),
$l=5$ (brown) only are plotted; see (1.3) and Section 6.2 for details. The
numerics illustrates the spectral exactness (the clusters of eigenvalues at
the ray ${\rm e}^{{\rm i}\pi/4}{\mathbb{R}}_{+}$) as well as the eigenvalues
escaping to infinity along the blue curves, see (1.3) for asymptotic formulas.

The extreme case are the truncations of the imaginary Airy operator
$T_{\infty}=-\partial_{x}^{2}+{\rm i}x$ in $L^{2}({\mathbb{R}})$ to $T_{n}$ in
$L^{2}((-s_{n},s_{n}))$ where, due to the established spectral exactness and
the fact the spectrum of $T_{\infty}$ is empty, _all_ eigenvalues of $T_{n}$
escape to infinity (see Example 5.1 for details, more results can be found in
[6, Thm. 3.1]).

The main goal of this paper is to analyze the diverging eigenvalues employing
an operator convergence and a localization strategy inspired by [6, Thm. 3.1].
This approach yields also improvements of the spectral exactness results in
[11], moreover, it is applicable in some problems with strongly coupled
$\operatorname{Im}Q$

$T_{g}=-\Delta+\operatorname{Re}Q+{\rm i}g\operatorname{Im}Q,\qquad
g\to+\infty,$ (1.2)

arising in various contexts like enhanced dissipation, _cf._ [23, 38], or
$\mathcal{P}\mathcal{T}$-symmetric phase transitions, _cf._ [13, 5] (see
Section 7 for details).

In particular in example (1.1), our results show that the truncations $T_{n}$
contain asymptotically the diverging eigenvalues

$\lambda_{k,n,l}=(2s_{n})^{\frac{2}{3}}\left(\overline{\nu_{k}}+\mathcal{O}_{k,l}\left(s_{n}^{-\frac{4}{3}}\right)\right)+{\rm
i}s_{n}^{2},\quad n\to\infty,$ (1.3)

where $\\{\nu_{k}\\}$ are eigenvalues of the imaginary Airy operator
$-\partial_{x}^{2}+{\rm i}x$ in $L^{2}({\mathbb{R}}_{+})$ with Dirichlet
boundary condition at $0$ (see Section 6.2, Example 5.2, Figures 1.1 and 6.3).
To be more precise, by writing that spectra of operators $\\{A_{n}\\}$ contain
asymptotically the eigenvalues $\\{\lambda_{k,n}\\}_{k}$ we mean that

$\forall k\in\mathbb{N},\quad\exists n_{k}\in\mathbb{N},\quad\forall
n>n_{k},\quad\lambda_{k,n}\in\sigma(A_{n}).$ (1.4)

Figures 1.1 and 6.3, as well as other examples in Section 6, exhibit a good
correspondence of numerics and obtained asymptotics. Moreover, they suggest
that all diverging eigenvalues are described (in these examples); however,
this remains open.

The improvements in the spectral exactness lie in finding the convergence rate
for the resolvent norm in terms of the decay of $|Q|^{-1}$, establishing
convergence in Schatten norms and estimating the constants in the convergence
rates. Moreover, we can also treat truncations of operators with non-empty
essential spectrum like $-\partial_{x}^{2}+x{\rm e}^{x}$ in
$L^{2}({\mathbb{R}})$ where one truncates the part of the domain where the
potential is unbounded, _e.g._ to $(-\infty,s_{n})$ with
$s_{n}\nearrow+\infty$, see Example 4.4.

An example of our results for the operators (1.2) with a strongly coupled
$\operatorname{Im}Q$ are the eigenvalues of operators $T_{g}$ in
$L^{2}({\mathbb{R}})$

$T_{g}=-\partial_{x}^{2}+x^{2}+{\rm i}g(1+|x|^{\kappa})^{-1},$ (1.5)

with $\kappa,g>0$, which are known to satisfy
$\operatorname{Re}\sigma(T_{g})\geq C_{\kappa}\,g^{\frac{2}{\kappa+2}}$ for
$g>0$, see [38]. Our results show that this bound is exhausted as
$g\to+\infty$ since the spectra of $T_{g}$ contain asymptotically the
eigenvalues

$\lambda_{k,g}=g^{\frac{2}{\kappa+2}}(\overline{\nu}_{k,\kappa}+o_{k}(1))+{\rm
i}g,\quad g\to+\infty,$ (1.6)

where $\\{\nu_{k,\kappa}\\}$ are eigenvalues of $-\partial_{x}^{2}+{\rm
i}|x|^{\kappa}$ in $L^{2}({\mathbb{R}})$ (see Example 7.4 for more details and
remainder estimates).

On a more technical side, in this paper we focus on accretive case
($\operatorname{Re}Q\geq 0$) and Dirichlet boundary conditions, nonetheless,
several extensions are possible and straightforward, see remarks after
Assumption 2.1, Theorem 2.2 and Remark 3.3. In Section 2 we collect relevant
known facts about Schrödinger operators with complex potentials, in
particular, the results on the domains, graph norm, compactness of resolvent
and eigenfunctions decay; justifications for some slight extensions are given
in Appendix. We also include several examples used throughout the paper.

In Section 3, we estimate of the resolvent difference for Schrödinger
operators with perturbed potential as well as underlying domain, see Theorem
3.2. The latter can be seen as a generalization of the estimates in [15, 16]
to an accretive case with a variable underlying domain and represents the key
technical step in our analysis. The essential ingredient is the graph norm
separation

$\|(-\Delta+Q)f\|+\|f\|\gtrsim\|\Delta f\|+\|Qf\|+\|f\|,$ (1.7)

see Theorem 2.2, which is known to be valid (for $Q$ unbounded at infinity) if

$|\nabla Q(x)|=o(|Q(x)|^{\frac{3}{2}}),\quad|x|\to\infty,$ (1.8)

see [30], Assumption 2.1 and remarks below for more details and Remark 3.3 for
possible extensions in less regular cases. Theorem 3.2 as well as the related
estimates on the eigenvalues and eigenfunctions in Theorem 3.4 are
reformulated for a sequence of operators in Corollary 3.6 which constitutes
our main technical tool.

In Section 4 we revisit domain truncations in Theorem 4.2 and improve the
previous results in [11] which are based on collective compactness (and a
slightly stronger assumption than (1.8)). In Section 5 we implement the
localization strategy of [6] and reformulate conditions of Corollary 3.6, see
Theorem 5.4. The somewhat implicit conditions on the potential $Q$ can be
further substantially simplified in one dimensional case with purely imaginary
potentials, see Theorem 5.7; interestingly the main condition in Theorem 5.7
is Assumption 5.6 3 is closely related to (1.8). A range of examples is
covered in Section 6, including a multidimensional one in Example 6.4 for
which the assumptions of Theorem 5.4 are verified directly. Finally, in
Theorem 7.3 we employ Corollary 3.6 to analyze operators (1.2).

All plots are produced using build in commands in Mathematica, namely
NDEigenvalues using FiniteElement PDEDiscretization method and improving its
precision by refinement of the mesh with setting MaxCellMeasure to 0.01.

###  1.1. Notation

We use conventions $\mathbb{N}=\\{1,2,\dots\\}$,
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$,
$\mathbb{N}^{*}=\mathbb{N}\cup\\{\infty\\}$, a subscript in the notation
$\mathcal{O}_{a}$ indicates that the constant depends on the parameter $a$,
and $\langle x\rangle:=(1+|x|^{2})^{\frac{1}{2}}$. We write $a\lesssim b$ to
denote that, given $a,b\geq 0$, there exists a constant $C>0$, independent of
any relevant variable or parameter, such that $a\leq Cb$; $a\gtrsim b$ is
analogous and $a\approx b$ means that $a\lesssim b$ and $a\gtrsim b$.

For an open $\Omega\subset{\mathbb{R}}^{d}$ and a measurable function
$m:\Omega\to{\mathbb{C}}$, we define the corresponding multiplication operator
in $L^{2}(\Omega)$ on the maximal domain

${\operatorname{Dom}}(m):=\\{f\in L^{2}(\Omega)\,:\,mf\in L^{2}(\Omega)\\}.$
(1.9)

The Dirichlet Laplacian $-\Delta_{\rm D}$ is defined via its quadratic form,
_i.e._

${\operatorname{Dom}}(\Delta_{\rm D}):=\\{f\in W^{1,2}_{0}(\Omega)\,:\,\Delta
f\in L^{2}(\Omega)\\}.$ (1.10)

The characteristic function of a set $\Sigma$ is denoted by $\chi_{\Sigma}$
and $\widetilde{\chi}_{\Sigma}:=1-\chi_{\Sigma}.$

##  2\. Preliminaries

We collect several known results on Schrödinger operators with complex
potentials, mostly following [3, 30, 11]; precise references are given at
individual claims. We also work out several examples that are used later to
illustrate the results.

###  2.1. Schrödinger operators with complex potentials

The main basic assumption on the potential $Q$ reads as follows.

######  Assumption 2.1.

Let $\emptyset\neq\Omega\subset{\mathbb{R}}^{d}$ be open and let $Q\in
W^{1,\infty}_{\rm loc}(\overline{\Omega};{\mathbb{C}})$ with
$\operatorname{Re}Q\geq 0$ satisfy

$\exists\varepsilon_{\nabla}\in[0,\varepsilon_{\rm crit}),\quad\exists
M_{\nabla}\geq 0,\quad|\nabla
Q|\leq\varepsilon_{\nabla}|Q|^{\frac{3}{2}}+M_{\nabla}\quad\text{a.e.~{}in~{}}\Omega;$
(2.1)

here $\varepsilon_{\rm crit}=2-\sqrt{2}$. $\blacksquare$

The value of $\varepsilon_{\rm crit}$ in Assumption 2.1 is obtained by simple
estimates in Lemma A.1 in Appendix which imply the graph norm estimate (2.5).
The optimal value of $\varepsilon_{\rm crit}$ for the latter in the self-
adjoint case is $\varepsilon_{\rm crit}=2$, see [21, 22]. In examples of $Q$,
(2.1) holds usually with an arbitrarily small $\varepsilon_{\nabla}>0$, see
_e.g._ Example 2.3, and if $Q$ is unbounded it suffices to check (1.8).

We omit explicit claims, nonetheless, as a consequence (2.5) or more precisely
(A.1), the results summarized in this section generalize straightforwardly
when a relatively bounded perturbation with a sufficiently small bound is
added.

The Dirichlet realization $T$ of $-\Delta+Q$ in $L^{2}(\Omega)$ can be
obtained via the form

$\displaystyle t[f]:=\|\nabla f\|^{2}+\int_{\Omega}Q(x)|f(x)|^{2}\,{{\rm
d}}x,\quad{\operatorname{Dom}}(t):=W_{0}^{1,2}(\Omega)\cap{\operatorname{Dom}}(|Q|^{\frac{1}{2}})$
(2.2)

invoking the generalization of Lax-Migram theorem due to Almog and Helffer
[3]. The associated operator is defined in the usual way

$\displaystyle{\operatorname{Dom}}(T)$
$\displaystyle:=\\{f\in{\operatorname{Dom}}(t)\,:\,\exists\eta\in
L^{2}(\Omega),\ \forall g\in{\operatorname{Dom}}(t),\
t(f,g)=\langle\eta,g\rangle\\},$ (2.3) $\displaystyle Tf$
$\displaystyle:=\eta=-\Delta f+Qf.$

Under Assumption 2.1 the form $t$ is bounded with respect to a natural norm
$\|\cdot\|_{t}^{2}:=\|\cdot\|_{W^{1,2}}^{2}+\||Q|^{\frac{1}{2}}\cdot\|^{2}$,
but not coercive in general. Nevertheless, following [3], the form $t$
exhibits a generalized coercivity

$|t_{\alpha}(f,f)|+|t_{\alpha}(\Phi
f,f)|\gtrsim\|f\|_{t}^{2},\quad|t_{\alpha}(f,f)|+|t_{\alpha}(f,\Phi
f)|\gtrsim\|f\|_{t}^{2},\quad f\in{\operatorname{Dom}}(t),$

where $t_{\alpha}:=t+\alpha$ with some $\alpha\geq 0$ and

$\Phi:=\frac{\operatorname{Im}Q}{\sqrt{1+|Q|^{2}}}.$ (2.4)

Alternatively, one can introduce $T$ by Kato’s theorem [20, Thm. VII.2.5], see
[11].

The following theorem summarizes known properties of $T$.

######  Theorem 2.2.

Let $Q$ satisfy Assumption 2.1 and let the operator $T$ be defined as in
(2.3). Then $T$ is $m$-accretive, moreover,

  1. n)

the graph norm of $T$ separates, _i.e._ there is a constant $a_{\nabla}>0$,
depending only on $\varepsilon_{\nabla}$, $M_{\nabla}$, such that for all
$f\in{\operatorname{Dom}}(T)$

$\|Tf\|^{2}+\|f\|^{2}\geq a_{\nabla}(\|\Delta f\|^{2}+\|Qf\|^{2}+\|f\|^{2}),$
(2.5)

hence the domain of $T$ separates, _i.e._
${\operatorname{Dom}}(T)={\operatorname{Dom}}(-\Delta_{\rm
D})\cap{\operatorname{Dom}}(Q)$;

  2. n)

$T$ is $\mathcal{C}$-self-adjoint, _i.e._ $T^{*}=\mathcal{C}T\mathcal{C}$,
where $\mathcal{C}$ is the complex conjugation operator
$\mathcal{C}f=\overline{f}$, $f\in L^{2}(\Omega)$, thus

$\displaystyle
T^{*}=-\Delta+\overline{Q},\quad{\operatorname{Dom}}(T^{*})={\operatorname{Dom}}(T);$
(2.6)

  3. n)

if $\Omega$ is bounded or if $\Omega$ in unbounded and

$\lim_{R\to\infty}\operatorname*{ess\,inf}_{|x|>R,x\in\Omega}|Q(x)|=+\infty,$
(2.7)

then $T$ has compact resolvent, thus the spectrum of $T$ is discrete (consists
of isolated eigenvalues of finite algebraic multiplicity);

  4. n)

denote by $S$ the self-adjoint Dirichlet realization of $-\Delta+|Q|$ in
$L^{2}(\Omega)$, _i.e._ $S:=-\Delta+|Q|$,
${\operatorname{Dom}}(S):={\operatorname{Dom}}(T)$, then (with $p>0$)

$(T+1)^{-1}\in\mathcal{S}_{p}(L^{2}(\Omega))\quad\Longleftrightarrow\quad(S+1)^{-1}\in\mathcal{S}_{p}(L^{2}(\Omega))$
(2.8)

and with $k_{1},k_{2}>0$ depending only on $\varepsilon_{\nabla},M_{\nabla}$,

$k_{1}\|(S+1)^{-1}\|_{\mathcal{S}_{p}}\leq\|(T+1)^{-1}\|_{\mathcal{S}_{p}}\leq
k_{2}\|(S+1)^{-1}\|_{\mathcal{S}_{p}}.$ (2.9)

Moreover, let $\partial\Omega\in C^{2,\alpha}$ for some $\alpha>0$. If for
$p>0$

$\int_{\Omega\times{\mathbb{R}}^{d}}(|\xi|^{2}+|Q(x)|+1)^{-p}\,{{\rm d}}x{{\rm
d}}\xi<\infty,$ (2.10)

then $(S+1)^{-1}\in\mathcal{S}_{p}(L^{2}(\Omega))$.

The proofs can be found in [3, 30] where more general forms of $T$ are
analyzed (_e.g._ with a real magnetic field, complex rotated coefficients, a
relatively bounded negative real part of the potentials or singular
perturbations controlled by the Laplacian). Estimates on the constant
$a_{\nabla}$ are in Lemma A.1. The equivalences (2.8), (2.9) are a consequence
of the characterization of ${\operatorname{Dom}}(T)$, in particular (2.5).
Assuming a suitable regularity of $\Omega$, one can also include different
boundary conditions (Neumann, Robin), see [11] for some details.

######  Example 2.3.

Let $Q$ with $\operatorname{Re}Q\geq 0$ be such that

$|Q(x)|\approx\langle x\rangle^{\gamma},\qquad x\in{\mathbb{R}}^{d}.$ (2.11)

with some $\gamma>0$ and let (1.8) and hence (2.1) be satisfied; notice that
_e.g._ in the special case of $Q(x)={\rm i}\langle x\rangle^{\gamma}$, we have

$|\nabla Q(x)|=\mathcal{O}(|Q(x)||x|^{-1}),\qquad|x|\to\infty.$ (2.12)

From Theorem 2.2, the corresponding Schrödinger operator $T$ is m-accretive
and has a compact resolvent, moreover, for any $\varepsilon>0$,

$(T+1)^{-1}\in\mathcal{S}_{p_{\gamma,d}+\varepsilon}(L^{2}({\mathbb{R}}^{d})),\qquad
p_{\gamma,d}=\frac{2+\gamma}{2\gamma}d;$ (2.13)

the latter follows from (2.10) by Young’s inequality. $\blacksquare$

###  2.2. Decay of eigenfunctions

Result on the eigenfunctions decay for accretive Schrödinger operators can be
found in [30]. A slight adaptation of [30, Prop. 4.1] yields the following,
(see Appendix for details).

######  Theorem 2.4.

Let Assumption 2.1 be satisfied, $T$ be the Dirichlet realizations of
$-\Delta+Q$ in $L^{2}(\Omega)$, $\Phi$ be as in (2.4) and define

$\widetilde{Q}:=\operatorname{Re}Q+\Phi\operatorname{Im}Q-\frac{1}{2}|\nabla\Phi|^{2}.$
(2.14)

Let $\lambda\in\sigma_{\rm p}(T)$ and
$\psi,\psi_{0}\in{\operatorname{Dom}}(T)$ satisfy
$T\psi=\lambda\psi+\psi_{0}$. Suppose that for this $\lambda$, there exist an
open $\Omega_{1}\subset\Omega$, a constant $\delta>0$ and a weight $W\in
W^{1,\infty}_{\rm loc}(\Omega;{\mathbb{R}})$ such that $e^{W}\psi_{0}\in
L^{2}(\Omega)$, $e^{W}+|\widetilde{Q}|+|\nabla W|\in
L^{\infty}(\Omega\setminus\Omega_{1})$ and

$\widetilde{Q}-3|\nabla
W|^{2}-\operatorname{Re}\lambda-|\operatorname{Im}\lambda|\geq
2\delta>0,\quad\text{a.e. in }\Omega_{1}.$ (2.15)

Then there exists $C=C(\lambda,\delta,\Omega_{1},W,\widetilde{Q})>0$ such that

$\|e^{W}\psi\|\leq C\|\psi\|+\delta^{-2}\|{\rm e}^{W}\psi_{0}\|.$ (2.16)

(For an estimates of $C$ see the proof Theorem 2.4 in Appendix.)

######  Example 2.5.

Let $Q$ and $T$ be as in Example 2.3. It follows from (2.11), (1.8) and a
straightforward estimate of $|\nabla\Phi|$ that there exist $\beta,R>0$ such
that

$\widetilde{Q}(x)\geq\beta|x|^{\gamma},\quad|x|>R.$ (2.17)

Suppose that we can find a weight $W\in W^{1,\infty}_{\rm
loc}(\mathbb{R}^{d};{\mathbb{R}})$ such that

$|\nabla W(x)|^{2}\leq\frac{1-\kappa}{3}\widetilde{Q}(x),\quad|x|>R,$ (2.18)

where $\kappa\in(0,1)$. Then for all $|x|>R$

$\widetilde{Q}(x)-3|\nabla
W(x)|^{2}-\operatorname{Re}\lambda-|\operatorname{Im}\lambda|\geq\kappa\beta|x|^{\gamma}-\operatorname{Re}\lambda-|\operatorname{Im}\lambda|.$
(2.19)

Thus for each $\lambda\in{\mathbb{C}}$, $\operatorname{Re}\lambda\geq 0$, we
can find $r_{\lambda}\geq R$ such that (2.15) is satisfied with
$\Omega_{1}=\mathbb{R}^{d}\setminus\overline{B_{r_{\lambda}}(0)}$ (and some
$\delta>0$).

A simple weight $W$ satisfying (2.18) can be selected as a radial function

$W(x):=\sqrt{\beta\frac{1-\kappa}{3}}\int_{0}^{|x|}s^{\frac{\gamma}{2}}\,{{\rm
d}}s.$ (2.20)

Thus eigenfunctions $\psi$ of $T$ satisfy $e^{W}\psi\in
L^{2}({\mathbb{R}}^{d})$ and the same follows for generalized eigenfunctions
by a repeated application of Theorem 2.4.

Finally, we remark that the result on the eigenfunctions decay will remain
valid also in other underlying domains than $\mathbb{R}^{d}$, _e.g._ for cones
in Example 5.2 below. $\blacksquare$

######  Example 2.6.

Let $T$ be the Schrödinger operator in $L^{2}({\mathbb{R}})$ with the
potential $Q:{\mathbb{R}}\to{\mathbb{C}}$ that satisfies with some $\gamma>0$
and $R_{0}\geq 0$,

$|Q(x)|\approx|x|^{\gamma}{\rm e}^{x},\quad|x|>R_{0}$ (2.21)

and $|Q^{\prime}(x)|=o(|Q(x)|^{3/2})$ as $x\to+\infty$ (hence Assumption 2.1
is satisfied with arbitrarily small $\varepsilon_{\nabla}>0$.) Theorem 2.4
applies with $\Omega_{1}=(r_{\lambda},+\infty)$ where $r_{\lambda}>0$ is
sufficiently large so that (2.15) holds for the weight (with a sufficiently
small $\kappa>0$)

$W(x)=\kappa\,\chi_{{\mathbb{R}}_{+}}(x)\int_{0}^{x}s^{\frac{\gamma}{2}}{\rm
e}^{\frac{s}{2}}\,{{\rm d}}s,\quad x\in{\mathbb{R}}.$ (2.22)

Thus the (generalized) eigenfunctions of $T$ decay fast as $x\to+\infty$.
$\blacksquare$

##  3\. Perturbations in domain and potential

For $j=1,2$, we consider the Dirichlet realizations $T_{j}=-\Delta+Q_{j}$ in
$L^{2}(\Omega_{j})$ with open $\Omega_{j}\subset{\mathbb{R}}^{d}$ as in
Section 2.1 and study the distance of resolvents and spectra.

######  Assumption 3.1.

For $j=1,2$, let $\Omega_{j}\subset{\mathbb{R}}^{d}$ be open and let $Q_{j}$
satisfy Assumption 2.1 with constants $\varepsilon_{\nabla,j}$,
$M_{\nabla,j}$. Let $T_{j}=-\Delta+Q_{j}$ be the Dirichlet realizations in
$L^{2}(\Omega_{j})$ defined via the forms $t_{j}$ given in Theorem 2.2. Let
$\Omega_{0}:=\Omega_{1}\cup\Omega_{2}$ and suppose that $\Omega_{1}$ and
$\Omega_{2}$ are such that there exists a cut-off $\xi:\Omega_{0}\to[0,1]$
satisfying that $\chi_{\Omega_{1}\cap\Omega_{2}}\xi=\xi$ on $\Omega_{0}$,
$|\nabla\xi|+\Delta\xi\in L^{\infty}(\Omega_{0})$ and

$\displaystyle\forall f\in{\operatorname{Dom}}(T_{1}),\quad\xi
f\in{\operatorname{Dom}}(t_{2}),$ (3.1) $\displaystyle\forall
g\in{\operatorname{Dom}}(T_{2}),\quad\xi g\in{\operatorname{Dom}}(t_{1});$

here we understand $\xi f$ as

$\begin{cases}\xi(x)f(x),&x\in\Omega_{2}\cap\Omega_{1},\\\
0,&x\in\Omega_{2}\setminus\Omega_{1},\end{cases}$ (3.2)

and analogously for $\xi g$. $\blacksquare$

An illustration of a choice of suitable cut-off $\xi$ is in Figure 3.1.

Figure 3.1. The domains $\Omega_{1}$ (blue) and $\Omega_{2}$ (yellow) are
taken as a part of sector and parabola, respectively. One can construct
$\xi\in C^{\infty}(\Omega_{0})$ with $\Omega_{0}:=\Omega_{1}\cup\Omega_{2}$
such that $\xi=1$ on $\Omega_{4}\subset\Omega_{1}\cap\Omega_{2}$ (orange) and
$\xi=0$ on the complement of $\Omega_{3}$ (green) in $\Omega_{0}$. Since
$\operatorname{supp}\xi$ is bounded, the conditions (3.1) are satisfied for
any admissible $Q_{1}$, $Q_{2}$ (which is not the case in general for
unbounded $\Omega_{1},\Omega_{2}$ and unbounded $\operatorname{supp}\xi$).

###  3.1. Resolvent difference estimate

For $j=1,2$ and $z\in\rho(T_{j})$, we write $R_{j}(z):=(T_{j}-z)^{-1}$,
$j=1,2$. We introduce

$\widetilde{\xi}:=1-\xi,\qquad\zeta:=\chi_{\operatorname{supp}\widetilde{\xi}}$
(3.3)

where $\xi$ is as in Assumption 3.1; see also Figure 3.1. Notice that

$\zeta\widetilde{\xi}=\widetilde{\xi},\quad\zeta\nabla\xi=\nabla\xi,\quad\zeta\Delta\xi=\Delta\xi.$
(3.4)

In $L^{2}(\Omega_{0})$, let $P_{j}$, $P$ and $\widetilde{P}$ be the following
orthogonal projections

$P_{j}f=\chi_{\Omega_{j}}f,\quad
Pf=\chi_{\Omega_{1}\cap\Omega_{2}}f,\quad\widetilde{P}:=I-P,\qquad f\in
L^{2}(\Omega_{0}).$ (3.5)

######  Theorem 3.2.

For $j=1,2$, let $\Omega_{j}$, $T_{j}$ and $\xi$ be as in Assumption 3.1, let
$\zeta$ be as in (3.3) and let $P_{j}$ be as in (3.5). Then there exists a
constant $K\geq 0$, depending only on $\|\nabla\xi\|_{L^{\infty}}$ and
$\varepsilon_{\nabla,j}$, $M_{\nabla,j}$, such that

$\displaystyle\|R_{1}(-1)P_{1}-R_{2}(-1)P_{2}\|_{\mathcal{B}(L^{2}(\Omega_{0}))}$
(3.6) $\displaystyle\qquad\leq
K\left(\left\|\frac{\xi(Q_{1}-Q_{2})}{(Q_{1}+1)(Q_{2}+1)}\right\|_{L^{\infty}(\Omega_{1}\cap\Omega_{2})}+\sum_{j=1}^{2}\left\|\frac{\zeta}{Q_{j}+1}\right\|_{L^{\infty}(\Omega_{j})}\right).$

If in addition $R_{j}(-1)\in\mathcal{S}_{p}(L^{2}(\Omega_{j}))$ with some
$p>0$, then for every $q\in(p,\infty)$ there exists a constant $K_{q}>0$ such
that

$\|R_{1}(-1)P_{1}-R_{2}(-1)P_{2}\|_{\mathcal{S}_{q}}\leq
K_{q}\|R_{1}(-1)P_{1}-R_{2}(-1)P_{2}\|_{\mathcal{B}(L^{2}(\Omega_{0}))}^{1-\frac{p}{q}}.$
(3.7)

######  Remark 3.3.

The proof of (3.6) is based on the “maximal estimate” (2.5) for the graph
norms which results in terms with $|Q_{j}+1|^{-1}$. An analogue of (3.6) holds
if weaker lower estimates of the graph norms are available, _e.g._ in the
sectorial case with $Q_{j}\in L^{1}_{\rm loc}(\Omega_{j})$, the denominators
in (3.6) would contain $|Q_{j}+1|^{-1/2}$ instead.

The claim (3.6) can be extended to all $z\in\rho(T_{1})\cap\rho(T_{2})$

$\displaystyle\|R_{1}(z)P_{1}-R_{2}(z)P_{2}\|$
$\displaystyle\leq\|(I-(z+1)R_{1}(z)P_{1})^{-1}\|\|I+(z+1)R_{2}(z)P_{2}\|\times$
(3.8) $\displaystyle\qquad\|R_{1}(-1)P_{1}-R_{2}(-1)P_{2}\|;$

the proof employs resolvent identities in a straightforward way.
$\blacksquare$

###### Proof of Theorem 3.2.

We establish a resolvent-type identity (for all $f,g\in L^{2}(\Omega_{0})$)

$\displaystyle\langle(R_{1}(z)P_{1}-R_{2}(z)P_{2})f,g\rangle=$ (3.9)
$\displaystyle\quad\langle R_{2}(z)\xi(Q_{2}-Q_{1})R_{1}(z)Pf,Pg\rangle$
$\displaystyle\quad+\langle(\nabla\xi)R_{1}(z)Pf,\nabla
R_{2}(z)^{*}Pg\rangle-\langle\nabla
R_{1}(z)Pf,(\nabla\xi)R_{2}(z)^{*}Pg\rangle$
$\displaystyle\quad+\langle\widetilde{\xi}R_{1}(z)Pf,g\rangle-\langle
f,\widetilde{\xi}R_{2}(z)^{*}Pg\rangle+\langle
R_{1}(z)P_{1}\widetilde{P}f,g\rangle-\langle
f,R_{2}(z)^{*}P_{2}\widetilde{P}g\rangle.$

To this end, we analyze the first term after the following splitting

$\displaystyle\langle(R_{1}(z)P_{1}-R_{2}(z)P_{2})f,g\rangle$
$\displaystyle=\langle(R_{1}(z)-R_{2}(z))Pf,Pg\rangle$ (3.10)
$\displaystyle\quad+\langle(R_{1}(z)P_{1}-R_{2}(z)P_{2})\widetilde{P}f,g\rangle$
$\displaystyle\quad+\langle(R_{1}(z)-R_{2}(z))Pf,\widetilde{P}g\rangle.$

Let $F:=R_{1}(z)Pf$ and $G:=R_{2}(z)^{*}Pg$, then, inserting
$1=\xi+\widetilde{\xi}$, we obtain

$\displaystyle\langle(R_{1}(z)-R_{2}(z))Pf,Pg\rangle$
$\displaystyle=\langle\xi F,T_{2}^{*}G\rangle-\langle T_{1}F,\xi G\rangle$
(3.11) $\displaystyle\quad+\langle\widetilde{\xi}R_{1}(z)Pf,Pg\rangle-\langle
Pf,\widetilde{\xi}R_{2}(z)^{*}Pg\rangle.$

Using the assumption (3.1), we get

$\displaystyle\langle\xi F,T_{2}^{*}G\rangle-\langle T_{1}F,\xi
G\rangle=t_{2}(\xi F,G)-t_{1}(F,\xi G)$ (3.12) $\displaystyle\quad=\langle
R_{2}(z)\xi(Q_{2}-Q_{1})R_{1}(z)Pf,Pg\rangle+\langle(\nabla\xi)R_{1}(z)Pf,\nabla
R_{2}(z)^{*}Pg\rangle$ $\displaystyle\quad\quad-\langle\nabla
R_{1}(z)Pf,(\nabla\xi)R_{2}(z)^{*}Pg\rangle$

Putting together (3.10), (3.11), (3.12) and using
$\widetilde{\xi}\widetilde{P}=\widetilde{P}$ we arrive at (3.9).

Next, we employ (3.9) with $z=-1$ and

$\|h\|=\sup_{g\neq 0}\frac{|\langle h,g\rangle|}{\|g\|}.$ (3.13)

Denoting $R_{1}:=R_{1}(-1)$, $R_{2}:=R_{2}(-1)$ and considering $\zeta$ in
(3.3), we get

$\displaystyle\|R_{1}P_{1}-R_{2}P_{2}\|$
$\displaystyle\leq\|R_{2}\xi(Q_{2}-Q_{1})R_{1}P\|+\|\nabla\xi\|_{L^{\infty}}(\|\zeta
R_{1}\|\|\nabla R_{2}^{*}\|+\|\zeta R_{2}^{*}\|\|\nabla R_{1}\|)$
$\displaystyle\quad+\|\zeta R_{1}\|+\|\zeta R_{2}\|+\|\zeta
R_{1}^{*}\|+\|\zeta R_{2}^{*}\|.$

In the sequel $j=1,2$. We show that $(Q_{j}+1)R_{j}$, $(Q_{j}+1)R_{j}^{*}$ and
$\nabla R_{j}$, $\nabla R_{j}^{*}$ are bounded operators. To this end, observe
that it follows from (2.5) and (2.6) that there are constants $C_{T_{j}}>0$
such that for all
$f_{j}\in{\operatorname{Dom}}(T_{j})={\operatorname{Dom}}(T_{j}^{*})$

$\displaystyle\|(T_{j}+1)f_{j}\|+\|f_{j}\|$ $\displaystyle\geq
C_{T_{j}}(\|\Delta f_{j}\|+\|(Q_{j}+1)f_{j}\|+\|f_{j}\|),$ (3.14)
$\displaystyle\|(T_{j}^{*}+1)f_{j}\|+\|f_{j}\|$ $\displaystyle\geq
C_{T_{j}}(\|\Delta f_{j}\|+\|(Q_{j}+1)f_{j}\|+\|f_{j}\|),$

and $C_{T_{j}}$ depend only on $\varepsilon_{\nabla,j}$ and $M_{\nabla,j}$
from (2.1).

Using a numerical range argument, we have $\|R_{1}\|\leq 1$, thus

$\|(Q_{1}+1)R_{1}f\|\leq\frac{1+\|R_{1}\|}{C_{T_{1}}}\|f\|\leq\frac{2}{C_{T_{1}}}\|f\|,\quad
f\in L^{2}(\Omega_{1});$ (3.15)

the estimate of other similar terms is analogous. Furthermore, as for all
$g\in{\operatorname{Dom}}(T_{1})$ we have $\|\nabla g\|^{2}\leq\|\Delta
g\|\|g\|$, we get from (3.14) that

$\|\nabla R_{1}f\|^{2}\leq\|\Delta
R_{1}f\|\|R_{1}f\|\leq\frac{2}{C_{T_{1}}}\|f\|^{2},\quad f\in
L^{2}(\Omega_{1});$ (3.16)

the estimate of other similar terms is analogous.

Finally, by inserting $1=(Q_{1}+1)^{-1}(Q_{1}+1)$, we get

$\displaystyle\|\zeta R_{1}\|$
$\displaystyle\leq\|\zeta(Q_{1}+1)^{-1}\|\|(Q_{1}+1)R_{1}\|\leq\frac{2}{C_{T_{1}}}\|\zeta(Q_{1}+1)^{-1}\|$
(3.17)

and analogously for the remaining terms. The claim (3.6) follows by putting
the estimates above together. The estimate with Schatten norms (3.7) follows
by Hölder’s inequality in $\mathcal{S}_{p}$, see  _e.g._ [19, Lem. XI.9]. ∎

###  3.2. Eigenvalues and eigenfunctions convergence

Let $T_{j}$, $j=,1,2$, be as in Assumption 3.1 and let $\mu\in\sigma(T_{1})$
be an isolated eigenvalue of finite algebraic multiplicity $m\in\mathbb{N}$.
If the gap distance of $T_{1}$ and $T_{2}$ is sufficiently small (or
equivalently the norm of the difference of resolvents estimated in Theorem
3.2), then $\sigma(T_{2})$ contains exactly $m$ eigenvalues
$\\{\mu_{k}\\}_{k=1}^{m}$ in a neighborhood of $\mu$ (counting with
multiplicities). This follows by estimating the norm of difference of spectral
projections (with a suitable contour $\gamma_{\mu}$)

$E_{1}:=\frac{1}{2\pi{\rm i}}\int_{\gamma_{\mu}}(z-T_{1})^{-1}P_{1}\,{{\rm
d}}z,\qquad E_{2}:=\frac{1}{2\pi{\rm
i}}\int_{\gamma_{\mu}}(z-T_{2})^{-1}P_{2}\,{{\rm d}}z;$ (3.18)

for details see _e.g._ [11, Thm. 5.1], [36], [29, Chap. IV].

Our goal is to estimate the distance of $\mu$ and the average of $\mu_{k}$

$\hat{\mu}:=\frac{1}{m}\sum_{k=1}^{m}\mu_{k}$ (3.19)

and the distance of eigenfunctions. Notice that the estimate (3.6) relates the
resolvent difference with a decay of $|Q|^{-1}$. Nevertheless, the convergence
rate of eigenvalues and eigenfunctions is typically much faster as these are
related to the decay of eigenfunctions, for an illustration see Corollary 3.6
and Theorem 4.2 below.

######  Theorem 3.4.

For $j=1,2$, let $\Omega_{j}$, $T_{j}$ and $\xi$ be as in Assumption 3.1, let
$\zeta$ be as in (3.3), let $P_{j}$, $P$ and $\widetilde{P}$ be as in (3.5).
Let $\mu\in\sigma(T_{1})$ be an isolated eigenvalue of finite algebraic
multiplicity $m\in\mathbb{N}$. Suppose further that $\Omega_{j}$ and $Q_{j}$,
$j=1,2$, are such that the spectral projections $E_{j}$, $j=1,2$, in (3.18)
satisfy $\|E_{1}-E_{2}\|<1$. Then the following hold.

  1. n)

Let $\hat{\mu}$ be as in (3.19), then

$|\mu-\hat{\mu}|\leq
C_{1,\mu}\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\left\|\frac{\xi(Q_{1}-Q_{2})}{(Q_{1}+1)(Q_{2}+1)}\phi\right\|+C_{2,\mu}\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\|\zeta\phi\|,$ (3.20)

  2. n)

For all $\psi\in{\operatorname{Ran}}(E_{1})$, we have

$\displaystyle\|\psi-E_{2}\psi\|$ $\displaystyle\leq
D_{1,\mu}\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\left\|\frac{\xi(Q_{1}-Q_{2})}{(Q_{1}+1)(Q_{2}+1)}\phi\right\|+D_{2,\mu}\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\|\zeta\phi\|$ (3.21)
$\displaystyle\quad+\|E_{2}\|\max_{\begin{subarray}{c}\phi_{2}\in{\operatorname{Ran}}(E_{2})\\\
\|\phi_{2}\|=1\end{subarray}}\|\widetilde{P}\phi_{2}\|,$

(For estimates of the constants $C_{j,\mu}$ and $D_{j,\mu}$, $j=1,2$, see
(3.28) below.)

###### Proof.

We follow standard regular perturbation theory arguments, see  _e.g._ [36,
Thm. 2] or [37, Chap. XII.2] for details.

  1. n)

Since $E_{1}$ and $E_{2}$ are sufficiently close,
$E_{2}\restriction{\operatorname{Ran}}(E_{1}):{\operatorname{Ran}}(E_{1})\to{\operatorname{Ran}}(E_{2})$
is bijective and for
$F_{2}:=(E_{2}\restriction{\operatorname{Ran}}(E_{1}))^{-1}:\,{\operatorname{Ran}}(E_{2})\to{\operatorname{Ran}}(E_{1})$
we get from $\|E_{1}f\|-\|E_{2}f\|\leq\|(E_{1}-E_{2})f\|$ that
$\|F_{2}\|\leq(1-\|E_{1}-E_{2}\|)^{-1}$. Moreover,
$F_{2}E_{2}\restriction{\operatorname{Ran}}(E_{1})=I\restriction{\operatorname{Ran}}(E_{1})$
and
$E_{2}F_{2}\restriction{\operatorname{Ran}}(E_{2})=I\restriction{\operatorname{Ran}}(E_{2})$.
We define $\hat{T}_{1}:=T_{1}E_{1}$ and
$\hat{T}_{2}:=F_{2}T_{2}E_{2}\restriction{\operatorname{Ran}}(E_{1})$ to
obtain

$\mu-\hat{\mu}=\frac{1}{m}\operatorname{Tr}(\hat{T}_{1}-\hat{T}_{2})=\frac{1}{m}\sum_{k=1}^{m}\langle(\hat{T}_{1}-\hat{T}_{2})f_{k},f_{k}\rangle,$
(3.22)

where $\\{f_{k}\\}_{k=1}^{m}$ is an orthonormal basis of
${\operatorname{Ran}}(E_{1})$. Using the assumption (3.1), one can check that
$\xi f_{k}\in{\operatorname{Dom}}(T_{j})$, $j=1,2$, $k=1,\dots,m$, thus (with
$\widetilde{\xi}:=1-\xi$)

$\displaystyle\hat{T}_{1}f_{k}-\hat{T}_{2}f_{k}$
$\displaystyle=F_{2}E_{2}T_{1}\xi f_{k}-F_{2}E_{2}T_{2}\xi
f_{k}+F_{2}E_{2}T_{1}\widetilde{\xi}f_{k}-F_{2}T_{2}E_{2}\widetilde{\xi}f_{k}$
(3.23) $\displaystyle=F_{2}E_{2}(Q_{1}-Q_{2})\xi
f_{k}-F_{2}T_{2}E_{2}\widetilde{\xi}f_{k}$
$\displaystyle\qquad+F_{2}E_{2}(\widetilde{\xi}T_{1}f_{k}-2\nabla\xi.\nabla
f_{k}-(\Delta\xi)f_{k}).$

Since $f_{k}\in{\operatorname{Ran}}(E_{1})$, we can estimate

$\|\widetilde{\xi}T_{1}f_{k}\|\leq\frac{\|\widetilde{\xi}T_{1}f_{k}\|}{\|T_{1}f_{k}\|}\|T_{1}E_{1}f_{k}\|\leq\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\|\zeta\phi\|\|T_{1}E_{1}\|.$ (3.24)

The remaining terms are estimated in a straightforward way; we use that
$\nabla\xi.\nabla f_{k}=\nabla\xi.\nabla(\zeta f_{k})$ and
$E_{2}\nabla\xi.\nabla$ has a bounded extension.

  2. n)

Consider $\psi\in{\operatorname{Ran}}(E_{1})$. Using (3.18) and the resolvent
identity (3.9), we get

$\langle\psi-E_{2}\psi,g\rangle=-\frac{1}{2\pi{\rm
i}}\int_{\gamma_{\mu}}\langle(R_{1}(z)P_{1}-R_{2}(z)P_{2})\psi,g\rangle\,{{\rm
d}}z$ (3.25)

for all $g\in L^{2}(\Omega)$. The claim (3.21) follows by (3.13), (3.4) and
the following manipulations. The term with $Q_{1}-Q_{2}$ is estimated in a
straightforward way. In terms with $\nabla\xi$, we integrate by parts and get

$\displaystyle\langle(\nabla\xi)R_{1}(z)Pf,\nabla
R_{2}(z)^{*}Pg\rangle-\langle\nabla
R_{1}(z)Pf,(\nabla\xi)R_{2}(z)^{*}Pg\rangle$ (3.26)
$\displaystyle\quad=2\langle(\nabla\xi)R_{1}(z)Pf,\nabla
R_{2}(z)^{*}Pg\rangle+\langle(\Delta\xi)R_{1}(z)Pf,R_{2}(z)^{*}Pg\rangle.$

We further rewrite $P\psi=\psi-\widetilde{P}\psi$, use that
$R_{1}(z)\psi\in{\operatorname{Ran}}(E_{1})$ and obtain

$\|\zeta
R_{1}(z)\psi\|\leq\|R_{1}(z)\psi\|\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}(E_{1})\\\
\|\phi\|=1\end{subarray}}\|\zeta\phi\|.$ (3.27)

In the other terms, the integration leads to formulas with spectral
projections. ∎

From the proof we can deduce the following estimates on the appearing
constants

$\displaystyle C_{1,\mu}$
$\displaystyle=\|(E_{2}\restriction{\operatorname{Ran}}(E_{1}))^{-1}\|\|E_{2}(Q_{2}+1)\|\|(Q_{1}+1)E_{1}\|,$
(3.28) $\displaystyle C_{2,\mu}$
$\displaystyle=\|(E_{2}\restriction{\operatorname{Ran}}(E_{1}))^{-1}\|(\|E_{2}\|\|T_{1}E_{1}\|+\|T_{2}E_{2}\|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+2\|\nabla\xi\|_{L^{\infty}}\|\nabla
E_{2}^{*}\|+\|\Delta\xi\|_{L^{\infty}}\|E_{2}^{*}\|),$ $\displaystyle
D_{1,\mu}$
$\displaystyle=\frac{|\gamma_{\mu}|}{2\pi}\max_{z\in\gamma_{\mu}}\left(\|(Q_{1}+1)R_{1}(z)\|\|(Q_{2}+1)R_{2}(z)^{*}\|\right),$
$\displaystyle D_{2,\mu}$
$\displaystyle=\frac{|\gamma_{\mu}|}{\pi}\max_{z\in\gamma_{\mu}}(2\|\nabla\xi\|_{L^{\infty}}\|R_{1}(z)\|\|\nabla
R_{2}(z)^{*}\|+\|\Delta\xi\|_{L^{\infty}}\|R_{1}(z)\|\|R_{2}(z)\|)$
$\displaystyle\quad+1+\|E_{1}\|+\|E_{2}\|.$

###  3.3. Sequence of operators

In next sections, we use Theorems 3.2 and 3.4 for a sequence of operators
$\\{T_{n}\\}$ converging to $T_{\infty}$, in the setting summarized as
follows.

######  Assumption 3.5.

Suppose that

  1. n)

domains $\\{\Omega_{n}\\}_{n\in\mathbb{N}^{*}}\subset\mathbb{R}^{d}$ are open
(non-empty) and $\Omega_{n}\subset\Omega_{\infty}$, $n\in\mathbb{N}$;

  2. n)

potentials $Q_{n}\in W^{1,\infty}_{\rm loc}(\overline{\Omega_{n}})$ with
$\operatorname{Re}Q_{n}\geq 0$, $n\in\mathbb{N}^{*}$, satisfy (2.1) uniformly,

$\exists\varepsilon_{\nabla}\in[0,\varepsilon_{\rm crit}),\ \exists
M_{\nabla}\geq 0,\ \forall n\in\mathbb{N}^{*},\ |\nabla
Q_{n}|\leq\varepsilon_{\nabla}|Q_{n}|^{\frac{3}{2}}+M_{\nabla}\quad\text{a.e.~{}in~{}}\Omega_{n};$
(3.29)

  3. n)

operators $T_{n}=-\Delta+Q_{n}$ in $L^{2}(\Omega_{n})$ (introduced via forms
$t_{n}$, $n\in\mathbb{N}^{*}$, as in Section 2.1) and cut-offs
$\\{\xi_{n}\\}_{n\in\mathbb{N}}$ are such that

$\sup_{n\in\mathbb{N}}\left(\||\nabla\xi_{n}|\|_{L^{\infty}}+\|\Delta\xi_{n}\|_{L^{\infty}}\right)<\infty$
(3.30)

and that the conditions of Assumption 3.1 are satisfied for $\Omega_{1}$,
$\Omega_{2}$, $\xi$, $T_{1}$, $T_{2}$, $t_{1}$, $t_{2}$ replaced by
$\Omega_{n}$, $\Omega_{\infty}$, $\xi_{n}$, $T_{n}$, $T_{\infty}$, $t_{n}$,
$t_{\infty}$, respectively, for every $n\in\mathbb{N}$;

  4. n)

potentials $\\{Q_{n}\\}$ converge in the following sense

$\displaystyle\tau_{n}$
$\displaystyle:=\left\|\frac{\xi_{n}(Q_{n}-Q_{\infty})}{(Q_{n}+1)(Q_{\infty}+1)}\right\|_{L^{\infty}(\Omega_{n})}\\!\\!\\!\\!+\left\|\frac{\zeta_{n}}{Q_{n}+1}\right\|_{L^{\infty}(\Omega_{n})}\\!\\!\\!\\!+\left\|\frac{\zeta_{n}}{Q_{\infty}+1}\right\|_{L^{\infty}(\Omega_{\infty})}$
(3.31) $\displaystyle=o(1),\quad n\to\infty,$

where $\widetilde{\xi}_{n}:=1-\xi_{n}$,
$\zeta_{n}:=\chi_{\operatorname{supp}\widetilde{\xi}_{n}}$, $n\in\mathbb{N}$.
$\blacksquare$

A typical situation we analyze is with $\Omega_{\infty}$ being
${\mathbb{R}}^{d}$, an exterior domain in ${\mathbb{R}}^{d}$ or a cone in
$\mathbb{R}^{d}$ and $\Omega_{n}$ are expanding subsets eventually exhausting
$\Omega_{\infty}$, _e.g._ expanding balls. Moreover, note that in the setting
of Assumption 3.5, the domain $\Omega_{0}$ in Assumption 3.1 corresponds to
$\Omega_{\infty}$; the projections $P_{1}$, $P_{2}$ in (3.5) correspond to

$P_{n}:=\chi_{\Omega_{n}}\cdot,\quad P_{\infty}=I_{L^{2}(\Omega_{\infty})},$
(3.32)

respectively and the projection $P$ in (3.5) corresponds to $P_{n}$.

To formulate the result we introduce notation $\\{\nu_{k}\\}$ for the isolated
eigenvalues of $T_{\infty}$ with finite algebraic multiplicities
$\\{m_{a}(\nu_{k})\\}$,

$\sigma_{\rm disc}(T_{\infty})=\\{\nu_{k}\\},\quad m_{a}(\nu_{k})<\infty,$
(3.33)

and for spectral projections of $T_{\infty}$ and $\\{T_{n}\\}$

$E_{k}:=\frac{1}{2\pi{\rm i}}\int_{\gamma_{k}}(z-T_{\infty})^{-1}\,{{\rm
d}}z,\qquad E_{k,n}:=\frac{1}{2\pi{\rm
i}}\int_{\gamma_{k}}(z-T_{n})^{-1}P_{n}\,{{\rm d}}z,$ (3.34)

where $\\{\gamma_{k}\\}$ are suitable contours around $\\{\nu_{k}\\}$.
Moreover, we use notation

$\kappa_{n}:=\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}E_{k}\\\
\|\phi\|=1\end{subarray}}\left(\left\|\frac{\xi_{n}(Q_{n}-Q_{\infty})}{(Q_{n}+1)(Q_{\infty}+1)}\phi\right\|+\|\zeta_{n}\phi\|\right).$
(3.35)

The statements of Corollary 3.6 follow in a straightforward way from Theorems
3.2 and 3.4. The details on the claim on no pollution can be found in [29,
Chap. IV.6] or [11, Rem. 4.2]. Notice that the last term in (3.21) disappears
since $\widetilde{P}$ corresponds to $I-P_{n}$. Finally, if all $T_{n}$,
$n\in\mathbb{N}^{*}$, have compact resolvents, Corollary 3.6 yields spectral
exactness.

######  Corollary 3.6.

Let Assumption 3.5 be satisfied and let $P_{n}$, $\nu_{k}$, $E_{k}$, $E_{k,n}$
and $\kappa_{n}$ be as in (3.32), (3.33), (3.34) and (3.35). Then the
following hold as $n\to\infty$.

  1. n)

$\\{T_{n}\\}$ converge to $T_{\infty}$ in the norm resolvent sense (hence
there is no spectral pollution): for every $z\in\rho(T_{\infty})$, there is
$n_{z}>0$ such that $z\in\rho(T_{n})$, $n>n_{z}$, and

$\|(T_{n}-z)^{-1}P_{n}-(T_{\infty}-z)^{-1}\|_{\mathcal{B}(L^{2}(\Omega_{\infty}))}=\mathcal{O}_{z}(\tau_{n});$
(3.36)

  2. n)

spectral projections converge in norm:

$\|E_{k,n}-E_{k}\|_{\mathcal{B}(L^{2}(\Omega_{\infty}))}=\mathcal{O}_{k}(\tau_{n});$
(3.37)

  3. n)

there is spectral inclusion for isolated eigenvalues with finite algebraic
multiplicities: for every $\nu_{k}\in\sigma_{\rm disc}(T_{\infty})$, as
$n\to\infty$, there are exactly $m_{a}(\nu_{k})$ eigenvalues
$\mu_{k,n}^{(j)}$, $j=1,\dots,m_{a}(\nu_{k})$, of $T_{n}$ in a neighborhood of
$\nu_{k}$ (repeated according to their algebraic multiplicities) and

$|\nu_{k}-\hat{\mu}_{k,n}|=\mathcal{O}_{k}\left(\kappa_{n}\right),\quad\hat{\mu}_{k,n}:=\frac{1}{m_{a}(\nu_{k})}\sum_{j=1}^{m_{a}(\nu_{k})}\mu_{k,n}^{(j)};$
(3.38)

  4. n)

(generalized) eigenfunctions converge in norm: for very
$\psi\in{\operatorname{Ran}}(E_{k})$

$\|\psi-E_{k,n}\psi\|=\mathcal{O}_{k}\left(\kappa_{n}\right),\quad
n\to\infty.$ (3.39)

######  Remark 3.7.

Based on (3.8) and an additional Neumann series argument, one can give an
estimate on $n_{z}$, the constant in (3.36) and hence also in the other
estimates. Namely, using notation
$\varrho_{n}(z):=\|(T_{n}-z)^{-1}P_{n}-(T_{\infty}-z)^{-1}\|_{\mathcal{B}(L^{2}(\Omega_{\infty}))}$,
we obtain

$\varrho_{n}(z)\leq\frac{\|(T_{\infty}-z)(T_{\infty}+1)^{-1}\|\|(T_{\infty}+1)(T_{\infty}-z)^{-1}\|}{1-|z+1|\|(T_{\infty}+1)(T_{\infty}-z)^{-1}\|\varrho_{n}(-1)}\varrho_{n}(-1).$
(3.40)

if the denominator is positive. $\blacksquare$

##  4\. Domain truncations

We apply first Corollary 3.6 to domain truncations of a given Schrödinger
operator with the underlying initial domain $\mathbb{R}^{d}$ to bounded
expanding domains $\\{\Omega_{n}\\}$. It is easy to verify that the results
can be reformulated for other initial domains like exterior domains in
$\mathbb{R}^{d}$, cones in $\mathbb{R}^{d}$ etc.

######  Assumption 4.1.

Let $\Omega_{\infty}:=\mathbb{R}^{d}$, let
$\\{\Omega_{n}\\}\subset\mathbb{R}^{d}$ be bounded and open sets and let there
exist a sequence $\\{r_{n}\\}\subset(0,\infty)$ such that
$r_{n}\nearrow+\infty$ and

$B_{r_{n}+2}(0)\subset\Omega_{n},\qquad n\in\mathbb{N}.$ (4.1)

Let $Q_{\infty}$ satisfy Assumption 2.1 (with $Q:=Q_{\infty}$) and suppose in
addition that

$\lim_{R\to\infty}\operatorname*{ess\,inf}_{|x|>R,x\in\Omega}|Q_{\infty}(x)|=+\infty.$
(4.2)

$\blacksquare$

Note that (4.2) implies that all operators in Theorem 2.2 have compact
resolvent, thus we obtain spectral exactness. Under a stronger version of
Assumption 2.1, the spectral exactness and estimate on the convergence rate of
eigenvalues and eigenfunctions were proved in [11]; the norm resolvent
convergence was based on a collective compactness which did not provide the
convergence rate.

######  Theorem 4.2.

Let Assumption 4.1 be satisfied and let $T_{n}$ be the Dirichlet realizations
of $-\Delta+Q$ in $L^{2}(\Omega_{n})$, $n\in\mathbb{N}^{*}$, respectively.
Then the statements of Corollary 3.6 hold with
$\\{\nu_{k}\\}:=\sigma(T_{\infty})$ and

$\tau_{n}=\left\|\widetilde{\chi}_{B_{r_{n}}(0)}|Q|^{-1}\right\|_{L^{\infty}},\quad\kappa_{n}=\max_{\begin{subarray}{c}\phi\in{\operatorname{Ran}}E_{k}\\\
\|\phi\|=1\end{subarray}}\|\widetilde{\chi}_{B_{r_{n}}(0)}\phi\|_{L^{2}},\quad
n\in\mathbb{N}.$ (4.3)

Suppose in addition that
$(T_{\infty}+1)^{-1}\in\mathcal{S}_{p}(L^{2}(\mathbb{R}^{d}))$ for some $p>0$.
Then, for any fixed $q>p$ and for any $m\in\mathbb{N}$ with $m>p$

$\displaystyle\|(T_{n}+1)^{-1}P_{n}-(T_{\infty}+1)^{-1}\|_{\mathcal{S}_{q}}$
$\displaystyle=\mathcal{O}\left(\tau_{n}^{1-\frac{p}{q}}\right),$ (4.4)
$\displaystyle\left|\operatorname{Tr}\Big{(}(T_{n}+1)^{-m}P_{n}-(T_{\infty}+1)^{-m}\Big{)}\right|$
$\displaystyle=\mathcal{O}\left(\tau_{n}^{m-p}\right),\quad n\to\infty.$

###### Proof.

The conditions 1 and 2 in Assumption 3.5 are clearly satisfied, in particular
since $Q_{n}=Q_{\infty}\restriction\Omega_{n}$. The cut-offs $\xi_{n}$ can be
constructed by a suitable mollification of $\chi_{B_{r_{n}}(0)}$ so that
$\operatorname{supp}(\xi_{n})\subset B_{r_{n}+1}(0)$ and the condition (3.30)
hold. Notice that then we have also $\chi_{\Omega_{n}}\xi_{n}=\xi_{n}$ in
$\mathbb{R}^{d}$. Moreover, since $\Omega_{n}$ are bounded, we have
${\operatorname{Dom}}(t_{n})=W^{1,2}_{0}(\Omega_{n})$, $n\in\mathbb{N}$, thus
every $f\in{\operatorname{Dom}}(T_{n})\subset W^{1,2}_{0}(\Omega_{n})$
satisfies
$\xi_{n}f\in{\operatorname{Dom}}(t_{\infty})=W^{1,2}(\mathbb{R}^{d})\cap{\operatorname{Dom}}(|Q_{\infty}|^{\frac{1}{2}})$.
On the other hand, every
$g\in{\operatorname{Dom}}(T_{\infty})=W^{2,2}({\mathbb{R}}^{d})\cap{\operatorname{Dom}}(Q_{\infty})$
satisfies $\xi_{n}g\in{\operatorname{Dom}}(t_{n})$ using the properties of
$\xi_{n}$. Thus the conditions of Assumption 3.5.3 are satisfied as well. The
functions $\\{\zeta_{n}\\}$ from (3.31) satisfy
$\zeta_{n}\leq\chi_{B_{r_{n}}(0)}$, $n\in\mathbb{N}$. Obviously
$\xi_{n}(Q_{n}-Q_{\infty})=0$ in our case, thus we get from (4.2) that the
condition Assumption 3.5.4 is satisfied with $\tau_{n}$ and $\kappa_{n}$ as in
(4.3). In summary the assumptions of Corollary 3.6 are satisfied.

By the equivalence (2.9) and a min-max argument, see _e.g._ [37, Prop. 4 in
Sec. XIII.15], applied to the self-adjoint Dirichlet realizations
$S_{n}:=-\Delta+|Q|$ in $L^{2}(\Omega_{n})$ and $S_{\infty}:=-\Delta+|Q|$ in
$L^{2}(\mathbb{R}^{d})$, we obtain that
$\|(T_{n}+1)^{-1}\|_{\mathcal{S}_{p}}\lesssim\|(T_{\infty}+1)^{-1}\|_{\mathcal{S}_{p}}$
and so the convergence in $\mathcal{S}_{q}$ follows from (3.7) and the already
established norm resolvent convergence.

Next we use Hölder inequality for Schatten norms and the telescope sum formula
with $r\in\mathbb{N}$ (valid for any two bounded operators $A$, $B$, see
_e.g._ [7])

$A^{r}-B^{r}=\sum_{j=0}^{r-1}A^{r-1-j}(A-B)B^{j}.$ (4.5)

Denoting
$c_{p}:=\sup_{n\in\mathbb{N}_{*}}\|(T_{n}+1)^{-1}\|_{\mathcal{S}_{p}}$, we
arrive at

$\|(T_{n}+1)^{-r}P_{n}-(T_{\infty}+1)^{-r}\|_{\mathcal{S}_{1}}\leq
rc_{p}^{r-1}\|(T_{n}+1)^{-1}P_{n}-(T_{\infty}+1)^{-1}\|_{\mathcal{S}_{q}}$
(4.6)

where $(r-1)/p+1/q=1$. Finally, the second claim in (4.4) follows by
$|\operatorname{Tr}A|=\|A\|_{\mathcal{S}_{1}}$ for $A\in\mathcal{S}_{1}$, see
_e.g._ [40, Chap. 3]. ∎

######  Example 4.3.

For $Q$ from Example 2.3, consider $T_{\infty}=-\Delta+Q$ in
$L^{2}({\mathbb{R}}^{d})$ and its truncations $T_{n}$, $n\in\mathbb{N}$, for
which (4.1) holds with $r_{n}=n$. Then Theorem 4.2 and Example 2.5 yield that
(with some $c_{k}>0$)

$\tau_{n}=\mathcal{O}(n^{-\gamma}),\quad\kappa_{n}=\mathcal{O}(\exp(-c_{k}n^{\frac{\gamma}{2}+1})).$
(4.7)

Moreover, for any $r\in\mathbb{N}$ such that $r>p$ for $p>p_{\gamma,d}$, see
(2.13),

$\left|\sum_{j=1}^{\infty}\frac{1}{(\lambda_{j,n}+1)^{r}}-\sum_{j=1}^{\infty}\frac{1}{(\lambda_{j,\infty}+1)^{r}}\right|=\mathcal{O}\left(n^{-\gamma(r-p)}\right),\quad
n\to\infty,$ (4.8)

where $\sigma(T_{n})=\\{\lambda_{j,n}\\}_{j}$, $n\in\mathbb{N}_{*}$. For real
$Q$, Hoffman-Wielandt inequality yields

$\sum_{j=1}^{\infty}\left|\frac{1}{\widetilde{\lambda}_{j,n}+1}-\frac{1}{\lambda_{j,\infty}+1}\right|^{q}=\mathcal{O}\left(n^{-\gamma(q-p)}\right),\quad
n\to\infty,$ (4.9)

for any $q>p>p_{\gamma,d}$, and suitable enumerations
$\\{\widetilde{\lambda}_{k,n}\\}_{k}$ of $\\{\sigma(T_{n})\\}$, see [28].
$\blacksquare$

Our results are applicable for truncations of operators $T$ without compact
resolvent to suitable unbounded domains. Roughly speaking, one can truncate
the parts of domain where the potential $Q$ is unbounded as $x\to\infty$. We
illustrate this on simple self-adjoint one dimensional example;
generalizations to complex potentials and more dimensions are straightforward.

######  Example 4.4.

Consider

$T_{\infty}=-\frac{{{\rm d}}^{2}}{{{\rm d}}x^{2}}+x{\rm
e}^{x},\quad{\operatorname{Dom}}(T_{\infty}):=W^{2,2}({\mathbb{R}})\cap{\operatorname{Dom}}(x{\rm
e}^{x}),$ (4.10)

and (Dirichlet) truncations $T_{n}:=-\partial_{x}^{2}+x{\rm e}^{x}$ to
$\Omega_{n}=(-\infty,n+2)$, $n\in\mathbb{N}$. Notice that standard arguments
show that the essential spectrum of $T_{n}$ is $[0,\infty)$ for all
$n\in\mathbb{N}^{*}$ and that $T_{\infty}$ has at least one negative
eigenvalue.

Regarding Assumption 3.5, we construct cut-offs $\xi_{n}$ as mollifications of
$\chi_{(-\infty,n]}$ such that $\xi_{n}\restriction(-\infty,n]=1$,
$\operatorname{supp}\xi_{n}\subset(-\infty,n+1)$ and
$\sup_{n}(\|\nabla\xi_{n}\|_{L^{\infty}}+\|\Delta\xi_{n}\|_{L^{\infty}})<\infty$.
The conditions in Assumption 3.5 are then easy to verify. Hence Corollary 3.6
yields the norm resolvent convergence of $T_{n}$ to $T_{\infty}$ with
$\tau_{n}=n^{-1}{\rm e}^{-n}$. Thereby we obtain also no spectral pollution
and spectral inclusion of discrete eigenvalues of $T_{\infty}$ with the
eigenvalue convergence rate determined by $\kappa_{n}=\exp(-c_{\lambda}n{\rm
e}^{\frac{n}{2}})$ with some $c_{\lambda}>0$, see Example 2.6. $\blacksquare$

At the first sight, the spectral exactness of the domain truncations
established in Theorem 4.2, does not seem to be troublesome for complex
potentials. Nevertheless, it is crucial to observe that while the obtained
rates depend on the decay of $|Q|^{-1}$ and of (generalized) eigenfunctions at
infinity, the estimate of $z$-dependent constants, indicated in the
$\mathcal{O}$-notation in Theorem 4.2, is very different for real and complex
potentials. The occurring quantities are related to the norm of resolvent of
$T$ and $T_{n}$ in $z$ as well as to the norms of the spectral projections
corresponding to a given eigenvalue $\lambda$, see Theorem 3.3, Remarks 3.3,
3.7, Theorem 3.4 and (3.28). The behavior of these can be very far from the
self-adjoint case already for the simplest one dimensional cases with
$Q(x)={\rm i}x^{n}$, $n\in\mathbb{N}$, where we have exponential growth when
$z\to\infty$ along relevant rays in ${\mathbb{C}}$, see _e.g._ [17, 18, 26,
27, 31].

The second observation and the main motivation of this paper is the presence
of diverging eigenvalues of $T_{n}$ as $n\to\infty$, see Figure 1.1 and
Section 6 for examples.

##  5\. Diverging eigenvalues in domain truncations

In this section we analyze diverging eigenvalues in truncations $T_{n}$ of
$T=-\Delta+Q$ in $L^{2}(\Omega)$ to a certain type of open domains
$\\{\Omega_{n}\\}\subset{\mathbb{R}}^{d}$ with corners. More precisely, we
assume that after suitable shifts and rescalings the domains
$\\{\Omega_{n}\\}$ lie in a cone $\Gamma$ in ${\mathbb{R}}^{d}$ symmetric
around positive $x_{1}$-axis and they exhaust $\Gamma$ eventually. If domains
$\\{\Omega_{n}\\}$ and $Q$ satisfy Assumption 4.1 in addition, then the
approximation is spectrally exact, however, we do not make such assumptions
here.

To identify the diverging eigenvalues, a combination of suitable unitary
transforms (translation and scaling) is performed, following the ideas in [6,
Thm. 3.1]. This procedure explained in the model case in Example 5.1 reveals a
suitable limiting operator and hence asymptotic formulas for diverging
eigenvalues.

######  Example 5.1 (Imaginary Airy operator).

Consider $\Omega_{n}:=(-s_{n},s_{n})$ with some
$\\{s_{n}\\}\subset{\mathbb{R}}_{+}$ with $s_{n}\nearrow+\infty$ and

$T_{n}=-\partial_{x}^{2}+{\rm
i}x,\quad{\operatorname{Dom}}(S_{n})=W^{2,2}(\Omega_{n})\cap
W_{0}^{1,2}(\Omega_{n}).$ (5.1)

The translation $x\mapsto x-s_{n}$ leads to unitarily equivalent operators

$\displaystyle-\partial_{x}^{2}+{\rm i}x-{\rm i}s_{n}=:S_{n}-{\rm
i}s_{n},\quad{\operatorname{Dom}}(S_{n})=W^{2,2}(\Sigma_{n})\cap
W_{0}^{1,2}(\Sigma_{n}),$ (5.2)

where $\Sigma_{n}=(0,2s_{n})$. Theorem 4.2 implies that $S_{n}$ converges to
$S_{\rm A}=-\partial_{x}^{2}+{\rm i}x$ in $L^{2}({\mathbb{R}}_{+})$ in the
norm resolvent convergence sense, hence the approximation is spectrally exact
and so the spectra of $S_{n}$ contain asymptotically the eigenvalues
$\\{\nu_{k}+\rho_{k,n}\\}_{k}$ where $\sigma(S_{\rm
A})=\\{\nu_{k}\\}\neq\emptyset$, see Example 5.2 for more details, and with
some $c_{k}>0$ we have $\rho_{k,n}=\mathcal{O}_{k}(\exp(-c_{k}s_{n}^{3/2}))$
as $n\to\infty$. Thus, by spectral mapping and (5.2), we obtain that spectra
of $T_{n}$ contain asymptotically the eigenvalues

$\lambda_{k,n}=(\nu_{k}+\rho_{k,n})-{\rm i}s_{n},\quad n\to\infty.$ (5.3)

A second set of diverging eigenvalues of $T_{n}$ can be obtained by two
transformations $x\mapsto x+s_{n}$ and $x\mapsto-x$. Isospectral partner of
(5.1) in $L^{2}(\Sigma_{n})$ then takes the form $-\partial_{x}^{2}-{\rm
i}x+{\rm i}s_{n}$, which is the adjoint operator of (5.2). Hence its spectrum
contains asymptotically the conjugate eigenvalues
$\\{\overline{\lambda_{k,n}}\\}$.

In summary, the spectrum of $T_{n}$ contains asymptotically the complex
conjugated eigenvalues $\\{\lambda_{k,n},\overline{\lambda_{k,n}}\\}$, _cf._
Figure 5.1.

Figure 5.1. Real (left) and imaginary (right) parts of eigenvalues of domain
truncations of imaginary Airy operator $-\partial_{x}^{2}+{\rm i}x$ in
$L^{2}({\mathbb{R}})$ to $L^{2}((-s_{n},s_{n}))$, $s_{n}=0.1n$,
$n=5,6,\dots,100$; subject to Dirichlet boundary conditions. Six asymptotic
curves (blue) to which the eigenvalues converge; see Example 5.1 and Section
6.1.

$\blacksquare$

###  5.1. General convergence result

As Example 5.1 suggests, an essential role is played by complex Airy operators
on cones, the eigenvalues of which enter the main asymptotic term of the
diverging eigenvalues.

For $\theta\in[0,\pi/2)$, we introduce notation for cones symmetric with
respect to the positive $x_{1}$-axis,

$\displaystyle\Gamma_{\theta}(x_{0})$
$\displaystyle:=\\{(x_{1},\widetilde{x})\in{\mathbb{R}}^{1+(d-1)}\,:\,x_{1}>x_{0}\text{
and }|\widetilde{x}|<\tan(\theta)|x_{1}-x_{0}|\\},$ (5.4)
$\displaystyle\Gamma_{\theta}$
$\displaystyle:=\Gamma_{\theta}(0),\qquad\Gamma_{\theta}(x_{0})=(x_{0},\infty)\text{
if }d=1.$

######  Example 5.2 (Complex Airy operator in a cone).

Let $\Gamma:=\Gamma_{\theta_{\infty}}\subset{\mathbb{R}}^{d}$ with some
$\theta_{\infty}\in(0,\pi/2)$ if $d>1$. Consider the complex Airy operator in
$L^{2}(\Gamma)$, _i.e._

$\displaystyle S_{\rm A}:=-\Delta+{\rm e}^{{\rm i}\omega}\,\vartheta\cdot
x,\quad{\operatorname{Dom}}(S_{\rm A}):={\operatorname{Dom}}(\Delta_{\rm
D})\cap{\operatorname{Dom}}(|x|),$ (5.5)

with $\operatorname{Re}e^{{\rm i}\omega}\geq 0$ and $|\vartheta|=1$ such that

$\forall\theta\in\overline{\Gamma},\ |\theta|=1,\ \theta\cdot\vartheta>0.$
(5.6)

Notice that (5.6) implies that there is a constant $c>0$ such that

$|\vartheta\cdot x|>c|x|,\quad x\in\Gamma.$ (5.7)

The property (5.7) guarantee that the potential in $S_{\rm A}$ satisfies (2.7)
in $\Gamma$, hence the resolvent of $S_{\rm A}$ is compact. Moreover, the
eigenvalues of $S_{\rm A}$ can be related to the eigenvalues of the
corresponding self-adjoint Airy operator in $L^{2}(\Gamma)$ by complex
scaling, namely

$\sigma(S_{\rm A})={\rm e}^{\frac{2}{3}{\rm i}\omega}\sigma(-\Delta_{\rm
D}+\vartheta\cdot x)=:\\{\nu_{k}\\}_{k\in\mathbb{N}}.$ (5.8)

In particular for $d=1$, the eigenvalues of the (Dirichlet) Airy operator
$-\partial_{x}^{2}+{\rm e}^{{\rm i}\omega}x$ in $L^{2}({\mathbb{R}}_{+})$ are
explicit in terms of the zeros $\\{\mu_{k}\\}$ of Airy function
$\operatorname{Ai}$

$\nu_{k}={\rm e}^{{\rm i}(\frac{2\omega}{3}-\pi)}\mu_{k},\quad
k\in\mathbb{N};$ (5.9)

$\\{\mu_{k}\\}$ are ordered decreasingly and they diverge to $-\infty$, see
_e.g._ [1] for more details.

The eigenfunctions $\\{\psi_{k}\\}$ of $S_{\rm A}$ related to $\\{\nu_{k}\\}$
exhibit a super-exponential decay

$\exp(c_{k}|\cdot|^{\frac{3}{2}})\psi_{k}\in L^{2}(\Gamma)$ (5.10)

with some $\\{c_{k}\\}\subset(0,+\infty)$, see Theorem 2.4, Example 2.5 and
(5.7). $\blacksquare$

Under the following assumptions, we follow the steps in Example 5.1 with an
additional scaling to reveal a suitable limiting Airy operator in a cone.

######  Assumption 5.3.

Let $\Gamma=\Gamma_{\theta_{\infty}}$ be a cone as in (5.4) with
$0<\theta_{\infty}<\pi/2$ if $d>1$. Let
$\\{\Omega_{n}\\}\subset{\mathbb{R}}^{d}$ be open, let
$\Omega:=\cup_{n\in\mathbb{N}}\Omega_{n}$ and let $Q\in
C^{1}(\overline{\Omega})$ satisfy Assumption 2.1 on every $\Omega_{n}$,
$n\in\mathbb{N}$. Suppose further that

  1. n)

there exists $\\{s_{n}\\}\subset{\mathbb{R}}_{+}$ with $s_{n}\nearrow+\infty$
such that for $p_{n}:=(s_{n},0)\in{\mathbb{R}}^{1+(d-1)}$

$|\nabla Q(-p_{n})|\neq 0,\quad n\in\mathbb{N};$ (5.11)

  2. n)

shifted and scaled domains exhaust the cone $\Gamma$ eventually:

there is $\\{r_{n}\\}\subset{\mathbb{R}}_{+}$ with $r_{n}\nearrow+\infty$ such
that for

$\Sigma_{n}:=\sigma_{n}^{-1}\left(p_{n}+\Omega_{n}\right),\quad\sigma_{n}:=|\nabla
Q(-p_{n})|^{-\frac{1}{3}},\quad n\in\mathbb{N},$ (5.12)

we have

$\Sigma_{n}\cap B_{r_{n}+2}(0)=\Gamma\cap B_{r_{n}+2}(0),\quad
n\in\mathbb{N};$ (5.13)

  3. n)

the potentials

$Q_{n}(x):=\sigma_{n}^{2}\left(Q(\sigma_{n}x-p_{n})-Q(-p_{n})\right),\quad
x\in\Sigma_{n},$ (5.14)

satisfy Assumption 3.5.2;

  4. n)

linear approximation of $\\{Q_{n}\\}$ converge: there is
$\omega\in[-\pi/2,\pi/2]$ and $\vartheta$ is such that for all
$\theta\in\overline{\Gamma}$ with $|\theta|=1$, we have
$\theta\cdot\vartheta>0$ and such that

$\lim_{n\to\infty}\frac{\nabla Q(-p_{n})}{|\nabla Q(-p_{n})|}={\rm e}^{{\rm
i}\omega}\vartheta$ (5.15)

and

$\iota_{n}:=\left\|\frac{Q_{n}(x)-{\rm e}^{{\rm i}\omega}\vartheta\cdot
x}{(Q_{n}(x)+1)({\rm e}^{{\rm i}\omega}\vartheta\cdot
x+1)}\right\|_{L^{\infty}(\Sigma_{n})}=o(1),\quad n\to\infty.$ (5.16)

$\blacksquare$

To check the condition (3.29), it is convenient to use $\sigma_{n}x=:y\in
p_{n}+\Omega_{n}$ as

$|\nabla Q_{n}(x)|=\frac{|\nabla Q(y-p_{n})|}{|\nabla
Q(-p_{n})|},\qquad\frac{|\nabla
Q_{n}(x))|}{|Q_{n}(x)|^{\frac{3}{2}}}=\frac{|\nabla
Q(y-p_{n})|}{|Q(y-p_{n})-Q(-p_{n})|^{\frac{3}{2}}}.$ (5.17)

Moreover, assuming that (5.15) is satisfied, it suffices to estimate

$\frac{|Q(y-p_{n})-Q(-p_{n})-\nabla Q(-p_{n})\cdot
y|}{|Q(y-p_{n})-Q(-p_{n})|(|\nabla Q(-p_{n})|^{\frac{1}{3}}|y|+1)},\quad y\in
p_{n}+\Omega_{n}$ (5.18)

to verify (5.16).

######  Theorem 5.4.

Let Assumption 5.3 be satisfied, let $T_{n}$ be the Dirichlet realizations of
$-\Delta+Q$ in $L^{2}(\Omega_{n})$, $n\in\mathbb{N}$, and let $\\{\nu_{k}\\}$
be the eigenvalues of the complex Airy operator $S_{\rm A}$ with $\omega$ and
$\vartheta$ as in (5.15). Then the spectra of $T_{n}$ contain asymptotically
the eigenvalues (with $k\in\mathbb{N}$ and $j\in\\{1,\dots,m_{a}(\nu_{k})\\}$)

$\lambda_{k,n}^{(j)}=|\nabla
Q(-p_{n})|^{\frac{2}{3}}\left(\nu_{k}+\rho_{k,n}^{(j)}\right)+Q(-p_{n}),\quad
n\to\infty,$ (5.19)

where $\rho_{k,n}^{(j)}=o_{j,k}(1)$ as $n\to\infty$ and (with $c_{k}>0$ as in
(5.10))

$\frac{1}{m_{a}(\nu_{k})}\left|\sum_{j=1}^{m_{a}(\nu_{k})}\rho_{k,n}^{(j)}\right|=\mathcal{O}_{k}(\iota_{n}+\exp(-c_{k}r_{n}^{\frac{3}{2}})),\quad
n\to\infty.$ (5.20)

###### Proof.

We consider $T_{n}$ and perform transformations analogous to those in Example
5.1 to obtain suitable $S_{n}$. Then we use Assumption 5.3 to apply Corollary
3.6 and finally obtain (5.19) by spectral mapping.

We shift the point $-p_{n}$ to the origin, _i.e._ we translate $x\mapsto
x-p_{n}$, and transform $T_{n}$ to the unitarily equivalent operator
$-\Delta+Q(x-p_{n})$ in $L^{2}(p_{n}+\Omega_{n})$. Next we subtract the
“absolute term” $Q(-p_{n})$ and employ the scaling $x\mapsto\sigma_{n}x$
leading to the operator in $L^{2}(\Sigma_{n})$

$\sigma_{n}^{-2}S_{n}:=\sigma_{n}^{-2}\left[-\Delta+\sigma_{n}^{2}\Big{(}Q(\sigma_{n}x-p_{n})-Q(-p_{n}))\Big{)}\right].$
(5.21)

The Taylor expansion of the potential leads further to

$S_{n}=-\Delta+Q_{n}(x)=-\Delta+\frac{\nabla Q(-p_{n})\cdot x}{|\nabla
Q(-p_{n})|}+\sigma_{n}R(x),\quad n\in\mathbb{N}.$ (5.22)

In the next step, we apply Corollary 3.6 where we replace $T_{n}$,
$n\in\mathbb{N}^{*}$, by $S_{n}$, $n\in\mathbb{N}$, $S_{\infty}:=S_{\rm A}$
and $\Omega_{n}$ by $\Sigma_{n}$, $n\in\mathbb{N}^{*}$. In detail, the
condition 1 of Assumption 3.5 is satisfied with $\Sigma_{\infty}:=\Gamma$ as
in Assumption 5.3. The condition 2 holds by Assumption 5.3.3.

Regarding the condition 3 of Assumption 3.5, let $P_{n}$ being the orthogonal
projection in $L^{2}(\Gamma)$ to $L^{2}(\Sigma_{n}\cap B_{r_{n}+2})$. Since
(5.13) holds, we can construct cut-offs $\xi_{n}\in C^{2}(\overline{\Gamma})$
with $0\leq\xi_{n}\leq 1$ be such that $\xi_{n}\equiv 1$ on
$\overline{B_{r_{n}}(0)\cap\Gamma}$, $\xi_{n}\equiv 0$ on
$\overline{\Gamma}\setminus\overline{B_{r_{n}+1}(0)}$, $n\in\mathbb{N}$, and
such that (3.30) is satisfied. Moreover, using the properties of
$\\{\xi_{n}\\}$, it is straightforward to verify that every
$f\in{\operatorname{Dom}}(S_{n})\subset
W^{1,2}_{0}(\Sigma_{n})\cap{\operatorname{Dom}}(|Q_{n}|^{\frac{1}{2}})$
satisfies
$\xi_{n}f\in{\operatorname{Dom}}(s_{\infty})=W_{0}^{1,2}(\Gamma)\cap{\operatorname{Dom}}(|x|^{\frac{1}{2}})$
and also every $g\in{\operatorname{Dom}}(S_{\infty})$ satisfies
$\xi_{n}g\in{\operatorname{Dom}}(s_{n})$.

Using (5.16), we can estimate

$\left\|\frac{\zeta_{n}(x)}{Q_{n}(x)+1}\right\|_{L^{\infty}(\Sigma_{n})}\leq\iota_{n}+\left\|\frac{\zeta_{n}(x)}{{\rm
e}^{{\rm i}\omega}\vartheta\cdot x+1}\right\|_{L^{\infty}(\Sigma_{n})},$
(5.23)

thus the condition 4 of Assumption 3.5 is satisfied with
$\tau_{n}:=\iota_{n}+r_{n}^{-1}$.

Corollary 3.6 yields the convergence of $S_{n}$ to $S_{\rm A}$ and the
eigenvalue convergence with the rate given by
$\kappa_{n}=\iota_{n}+\exp(-c_{k}r_{n}^{3/2})$ where the second term
originates in the decay of eigenfunctions of $S_{\rm A}$, see Example 5.2.

Finally, (5.19) follows by spectral mapping since

$T_{n}=\sigma_{n}^{-2}U_{n}S_{n}U_{n}^{-1}+Q(-p_{n})$ (5.24)

where $U_{n}$ are unitary transformations (the shift and scaling) described
above. ∎

Theorem 5.4, based on Corollary 3.6.iii), provides a result for diverging
eigenvalues, obtained by the spectral mapping and convergence of operators
$S_{n}$ to an Airy operator, _cf._ (5.22) and (5.24). By Corollary 3.6.iv),
the corresponding eigenfunctions (after appropriate transformations) converge
to the eigenfunctions of the Airy operator, thus they localize to the corner
$-p_{n}$.

######  Remark 5.5.

The claim of Theorem 5.4 remains valid (with an additional term
$\iota_{n}^{\prime}$ as below in (5.20)) for perturbations of $T_{n}$. In
detail, assume that the transformed operator $S_{n}$, see the proof of Theorem
5.4, is perturbed by $W_{n}$ satisfying

$|W_{n}(x)|\leq a|Q_{n}(x)|+b,\quad x\in\Sigma_{n},\quad n\to\infty,$ (5.25)

with a sufficiently small $a>0$ (depending on $\varepsilon_{\nabla}$) and some
$b>0$, and

$\iota_{n}^{\prime}:=\left\|\frac{W_{n}(x)}{(Q_{n}(x)+1)({\rm e}^{{\rm
i}\omega}\vartheta\cdot x+1)}\right\|_{L^{\infty}(\Sigma_{n})}=o(1),\quad
n\to\infty.$ (5.26)

The condition (5.25) guarantees that the graph norms of $S_{n}$ and
$S_{n}+W_{n}$ are equivalent (uniformly in $n$) and we obtain (for a
sufficiently large $z_{0}>0$) that

$\|(S_{n}+W_{n}+z_{0})^{-1}P_{n}-(S_{\rm
A}+z_{0})^{-1}\|=\mathcal{O}\left(\iota_{n}+\iota_{n}^{\prime}+r_{n}^{-1}\right)=o(1)$
(5.27)

as $n\to+\infty$, see (3.6). $\blacksquare$

###  5.2. One dimensional imaginary potentials

The conditions on potential $Q$ in Assumption 5.3 are expressed implicitly in
terms of $Q_{n}$. In one dimensional case and when $Q$ is imaginary, we give
explicit conditions on $Q$ which guarantee that the Assumption 5.3 is
satisfied. To avoid working with conditions at $-\infty$ we express $Q$ in a
specific way in terms of a new function $U$, namely as

$Q(x)=-{\rm i}U(-x),\quad x\in{\mathbb{R}}.$ (5.28)

######  Assumption 5.6.

Let $U\in C^{1}({\mathbb{R}};{\mathbb{R}})\cap C^{2}((x_{0},\infty))$ with a
sufficiently large $x_{0}>0$ as below satisfy the condition (2.1) on
${\mathbb{R}}$ with an abitrarily small $\varepsilon_{\nabla}>0$ (where we
replace $Q$ by $U$). Let $\Omega_{n}=(-s_{n},t_{n})$ with
$s_{n}\nearrow+\infty$ and

$\lim_{n\to\infty}(s_{n}+t_{n})\sigma_{n}^{-1}=+\infty,$ (5.29)

where $\sigma_{n}:=|U(s_{n})|^{-\frac{1}{3}}$. Suppose further that

  1. n)

$U$ is eventually increasing and unbounded at $+\infty$:

$U^{\prime}(x)>0,\quad x>x_{0},\qquad\lim_{x\to+\infty}U(x)=+\infty;$ (5.30)

  2. n)

$U$ has controlled derivatives: there is $\nu\geq-1$ such that

$U^{\prime}(x)\lesssim U(x)x^{\nu},\qquad|U^{\prime\prime}(x)|\lesssim
U^{\prime}(x)x^{\nu},\qquad x>x_{0},$ (5.31)

  3. n)

$U$ grows sufficiently fast at $+\infty$:

$\Upsilon(x):=\frac{x^{\nu}}{U^{\prime}(x)^{\frac{1}{3}}}\to 0,\quad
x\to+\infty$ (5.32)

  4. n)

$U$ is relatively smaller on $(-\infty,x_{0})$: there exists
$\delta_{0}\in(0,1)$ such that

$\sup_{y\in(s_{n}-x_{0},s_{n}+t_{n})}U(s_{n}-y)\leq(1-\delta_{0})U(s_{n}),\quad
n\to\infty.$ (5.33)

$\blacksquare$

By Gronwalls inequality, (5.31) implies that for all sufficiently large $x>0$

$U(x)\lesssim\begin{cases}x^{\gamma},&\nu=-1,\\\ \exp(\gamma
x^{\nu+1}),&\nu>-1.\end{cases}$ (5.34)

with some $\gamma>0$. Moreover, there exist constants $c_{1},c_{2}>0$ such
that for all sufficiently large $x>0$ and for all
$|\delta|\leq\tfrac{1}{4}|x|^{-\nu}$, we have

$c_{1}U^{(j)}(x)\leq U^{(j)}(x+\delta)\leq c_{2}U^{(j)}(x),\qquad j=0,1;$
(5.35)

for details see [32, Sec. 3.1] and [34, Sec. 2]. We also remark that
Assumption 5.6.3 is related with the condition (1.8) since, by (5.31),

$\frac{U^{\prime}(x)}{U(x)^{\frac{3}{2}}}\lesssim\frac{U^{\prime}(x)}{(U^{\prime}(x)x^{-\nu})^{\frac{3}{2}}}=\Upsilon^{\frac{3}{2}}(x)\to
0,\quad x\to+\infty.$ (5.36)

######  Theorem 5.7.

Let Assumption 5.6 be satisfied, let $Q$, $U$ be as in (5.28) and let

$\sigma_{n}=U^{\prime}(s_{n})^{-\frac{1}{3}},\qquad\Sigma_{n}:=(0,(s_{n}+t_{n})\sigma_{n}^{-1}).$
(5.37)

Then the potentials (with $x\in\Sigma_{n}$)

$Q_{n}(x)=\sigma_{n}^{2}\big{(}Q(\sigma_{n}x-s_{n})-Q(-s_{n})\big{)}={\rm
i}\sigma_{n}^{2}\big{(}U(s_{n})-U(s_{n}-\sigma_{n}x)\big{)},$ (5.38)

satisfy Assumption 5.3 with $\omega=\pi/2$, $\vartheta=1$,
$\Gamma={\mathbb{R}}_{+}$ and $\iota_{n}\lesssim\Upsilon(s_{n})$.

Hence the spectra of Dirichlet realizations $T_{n}=-\partial_{x}^{2}+Q$ in
$L^{2}(\Omega_{n})$, $n\in\mathbb{N}$, contain asymptotically as $n\to\infty$
the eigenvalues

$\lambda_{k,n}=U^{\prime}(s_{n})^{\frac{2}{3}}\left(\nu_{k}+\rho_{k,n}\right)-{\rm
i}U(s_{n}),\quad\rho_{k,n}=\mathcal{O}_{k}\big{(}\Upsilon(s_{n})+\exp(-c_{k}r_{n}^{\frac{3}{2}})\big{)},$
(5.39)

where $c_{k}>0$ are as in (5.10) and $r_{n}=(s_{n}+t_{n})\sigma_{n}^{-1}-2$.

###### Proof.

Due to (5.29), Assumption 5.3.1 and 2 are satisfied and we have $\omega=\pi/2$
and $\vartheta=1$, see (5.15). Next we verify Assumption 5.3.3, in particular
the condition (3.29). We use the variable variable
$y=\sigma_{n}x\in(0,s_{n}+t_{n})$ and formula (5.17).

$|Q_{n}^{\prime}(x)|\lesssim\frac{|U^{\prime}(s_{n}-y)|}{U^{\prime}(s_{n})}\lesssim
1,\qquad n\to\infty.$ (5.40)

Next let $y\in[\tfrac{1}{4}s_{n}^{-\nu},s_{n}-x_{1})\cap(0,s_{n}+t_{n})$ with
$x_{1}\geq x_{0}$ sufficiently large so that in particular (5.35) holds for
all $x>x_{1}$; a further restriction on the choice $x_{1}$ is below. First
note that $s_{n}-y+\tfrac{1}{4}(s_{n}-y)^{-\nu}\leq s_{n}$, hence using (5.30)
and (5.35), we obtain

$U(s_{n})-U(s_{n}-y)\geq\int_{s_{n}-y}^{s_{n}-y+\tfrac{1}{4}(s_{n}-y)^{-\nu}}U^{\prime}(t)\,{{\rm
d}}t\gtrsim U^{\prime}(s_{n}-y)(s_{n}-y)^{-\nu}.$ (5.41)

Thus

$\frac{|U^{\prime}(s_{n}-y)|}{|U(s_{n})-U(s_{n}-y)|^{\frac{3}{2}}}\lesssim\Upsilon^{\frac{3}{2}}(s_{n}-y)$
(5.42)

and so by (5.32), $x_{1}$ can be selected so large that, for all $y$ in the
considered range, the right hand side of (5.42) is arbitrarily small.

For $y\in[s_{n}-x_{1},s_{n}-x_{0})\cap(0,s_{n}+t_{n})$, it suffices to use
that $U$ and $U^{\prime}$ are locally bounded and $U$ is unbounded at
$+\infty$, see (5.30).

In the last case, $y\in[s_{n}-x_{0},s_{n}+t_{n})$, we use that $Q$ and thus
$U$ satisfy (2.1) and that $U(s_{n})-U(s_{n}-y)\geq\delta_{0}U(s_{n})$ from
Assumption 5.6.4,

$\displaystyle\frac{|U^{\prime}(s_{n}-y)|}{|U(s_{n})-U(s_{n}-y)|^{\frac{3}{2}}}$
$\displaystyle\leq\varepsilon_{\nabla}\left|\frac{U(s_{n}-y)}{U(s_{n})-U(s_{n}-y)}\right|^{\frac{3}{2}}+\frac{M_{\nabla}}{|U(s_{n})-U(s_{n}-y)|^{\frac{3}{2}}}$
$\displaystyle\leq\varepsilon_{\nabla}\left(1+\frac{1}{\delta_{0}}\right)^{\frac{3}{2}}+\frac{M_{\nabla}}{\delta_{0}U(s_{n})};$

note that $\varepsilon_{\nabla}$ can be taken arbitrarily small by assumption
and the second term decays. Hence, putting estimates above together we obtain
that the condition (3.29) is indeed satisfied.

Finally, we show that Assumption 5.3.4 is satisfied using (5.18). In the first
case, $y\in(0,\tfrac{1}{4}s_{n}^{-\nu})\cap(0,s_{n}+t_{n})$, Taylor’s theorem,
(5.31) and (5.35) yield

$\left|\frac{U(s_{n})-U(s_{n}-y)-U^{\prime}(s_{n})y}{U^{\prime}(s_{n})^{\frac{1}{3}}(U(s_{n})-U(s_{n}-y))y}\right|\lesssim\frac{U^{\prime}(s_{n})s_{n}^{\nu}y^{2}}{U^{\prime}(s_{n})^{\frac{4}{3}}y^{2}}\lesssim\Upsilon(s_{n}).$
(5.43)

For the remaining steps, we use the estimate

$\left|\frac{U(s_{n})-U(s_{n}-y)-U^{\prime}(s_{n})y}{U^{\prime}(s_{n})^{\frac{1}{3}}(U(s_{n})-U(s_{n}-y))y}\right|\leq\frac{1}{U^{\prime}(s_{n})^{\frac{1}{3}}y}+\frac{U^{\prime}(s_{n})^{\frac{2}{3}}}{U(s_{n})-U(s_{n}-y)}.$
(5.44)

In the case $y\in[\tfrac{1}{4}s_{n}^{-\nu},s_{n}-x_{0})\cap(0,s_{n}+t_{n})$,
using $U^{\prime}(x)>0$ for $x>x_{0}$, mean value theorem and (5.35), we get

$\frac{1}{U^{\prime}(s_{n})^{\frac{1}{3}}y}+\frac{U^{\prime}(s_{n})^{\frac{2}{3}}}{U(s_{n})-U(s_{n}-y)}\lesssim\Upsilon(s_{n})+\frac{U^{\prime}(s_{n})^{\frac{2}{3}}}{U^{\prime}(s_{n})s_{n}^{-\nu}}\lesssim\Upsilon(s_{n}).$
(5.45)

Finally, if $y\in[s_{n}-x_{0},s_{n}+t_{n})$, we obtain from (5.33) and (5.31)
that

$\frac{1}{U^{\prime}(s_{n})^{\frac{1}{3}}y}+\frac{U^{\prime}(s_{n})^{\frac{2}{3}}}{U(s_{n})-U(s_{n}-y)}\lesssim\left(\frac{s_{n}^{3\nu}}{U^{\prime}(s_{n})}\right)^{\frac{1}{3}}\frac{1}{s_{n}^{1+\nu}}+\frac{U^{\prime}(s_{n})^{\frac{2}{3}}}{\delta_{0}U(s_{n})}\lesssim\Upsilon(s_{n}).$

Thus in summary, we obtain the estimate on $\iota_{n}$ in the claim. ∎

In the next step, following Remark 5.5, we determine a class of admissible
perturbations of $U$ as in Assumption 5.6.

######  Proposition 5.8.

Let Assumption 5.6 be satisfied. Suppose that $U_{1}\in L_{\rm
loc}^{\infty}({\mathbb{R}};{\mathbb{C}})$, $U_{1}^{\prime}\in L_{\rm
loc}^{\infty}((x_{1},\infty);{\mathbb{C}})$ for some $x_{1}>0$ and (using
notation of Assumption 5.6),

$U_{1}^{\prime}(x)=o(U^{\prime}(x)),\quad
x\to+\infty,\quad\|U_{1}\|_{L^{\infty}((-t_{n},s_{n}))}=o(U(s_{n})),$ (5.46)

Then, with $\sigma_{n}$ as in (5.38),

$W_{n}(x):=\sigma_{n}^{2}(U_{1}(s_{n})-U_{1}(s_{n}-\sigma_{n}x)),\qquad
x\in(0,(s_{n}+t_{n})\sigma_{n}^{-1})$ (5.47)

satisfies the conditions (5.25) and (5.26) with respect to $Q_{n}$ as in
(5.38). Hence the claim of Theorem 5.7 remains valid with (see Remark 5.5)

$\displaystyle\lambda_{k,n}$
$\displaystyle=U^{\prime}(s_{n})^{\frac{2}{3}}\left(\nu_{k}+\rho_{k,n}^{\prime}\right)-{\rm
i}U(s_{n})-U_{1}(s_{n}),$ (5.48) $\displaystyle\rho_{k,n}^{\prime}$
$\displaystyle=\mathcal{O}_{k}\big{(}\Upsilon(s_{n})+\iota_{n}^{\prime}+\exp(-c_{k}r_{n}^{\frac{3}{2}})\big{)}.$

In particular if the support of $U_{1}$ is bounded, then

$\iota_{n}^{\prime}=\mathcal{O}(U(s_{n})^{-1}\Upsilon(s_{n})s_{n}^{\nu-1}),\quad
n\to\infty.$ (5.49)

###### Proof.

Using the first assumption in (5.46), for any $\varepsilon_{0}>0$, there
exists a sufficiently large $y_{0}>0$ such that (with $\sigma_{n}x=y$)

$\frac{|W_{n}(x)|}{|Q_{n}(x)|}\leq\frac{\int_{s_{n}-y}^{s_{n}}\frac{|U_{1}^{\prime}(t)|}{U^{\prime}(t)}U^{\prime}(t)\,{{\rm
d}}t}{U(s_{n})-U(s_{n}-y)}\leq\varepsilon_{0},\quad y\in[0,s_{n}-y_{0}].$
(5.50)

From (5.33) we have that (with some $\delta(y_{0})>0$)

$U(s_{n})-U(s_{n}-y)\geq\delta(y_{0})U(s_{n}),\quad
y\in[s_{n}-y_{0},s_{n}+t_{n}],$ (5.51)

thus

$\frac{|W_{n}(x)|}{|Q_{n}(x)|}\leq\frac{2\|U_{1}\|_{L^{\infty}((-t_{n},s_{n}))}}{\delta(y_{0})U(s_{n})},\quad
y\in[s_{n}-y_{0},s_{n}+t_{n}].$ (5.52)

Using the second assumption in (5.46), the right hand side of (5.52) decays as
$n\to\infty$. In summary, the condition (5.25) is satisfied.

Similarly, for a given $\varepsilon_{0}>0$, we split the estimate to
$[0,s_{n}-y_{0}]$ and $[s_{n}-y_{0},s_{n}+t_{n}]$ with a suitable $y_{0}$ so
that

$\left|\frac{U_{1}(s_{n})-U_{1}(s_{n}-y)}{(U(s_{n})-U(s_{n}-y))(|U^{\prime}(s_{n})|^{\frac{1}{3}}y+1)}\right|\leq\varepsilon_{0}+\frac{2\|U_{1}\|_{L^{\infty}((-t_{n},s_{n}))}}{\delta(y_{0})U(s_{n})(|U^{\prime}(s_{n})|^{\frac{1}{3}}y+1)}.$
(5.53)

The validity of (5.26) follows from the second assumption in (5.46) and since
$\varepsilon_{0}$ can be taken arbitrarily small. In the case of bounded
support of $U_{1}$, we can take fixed $y_{0}$ and obtain (5.49). ∎

######  Remark 5.9 (First eigenvalue correction).

In Theorem 5.7, the eigenvalues $\\{\nu_{k}\\}$ of $S_{\rm A}$ are simple. We
introduce the truncated imaginary Airy operator in $L^{2}(\Sigma_{n})$

$\widetilde{S}_{n}:=-\partial_{x}^{2}+{\rm
i}x,\qquad{\operatorname{Dom}}(\widetilde{S}_{n})=W^{2,2}(\Sigma_{n})\cap
W_{0}^{1,2}(\Sigma_{n})\cap{\operatorname{Dom}}(x)$ (5.54)

and write

$\lambda_{k,n}-\nu_{k}=\lambda_{k,n}-\widetilde{\lambda}_{k,n}+\widetilde{\lambda}_{k,n}-\nu_{k},$
(5.55)

where $\widetilde{\lambda}_{k,n}$ are eigenvalues of $\widetilde{S}_{n}$ which
converge to $\nu_{k}$ as $n\to\infty$. Recall that the rate of
$\widetilde{\lambda}_{k,n}-\nu_{k}$ is super-exponential since it corresponds
to the eigenvalue convergence of domain truncations for the imaginary Airy
operator, see Theorem 4.2. Thus the main term in $\lambda_{k,n}-\nu_{k}$
usually arises from $\lambda_{k,n}-\widetilde{\lambda}_{k,n}$, _i.e._ from the
difference of eigenvalues of the truncated imaginary Airy operator
$\widetilde{S}_{n}$ and the “perturbed” truncated imaginary Airy operator
$S_{n}$.

The standard perturbation theory can be used to express
$\lambda_{k,n}-\widetilde{\lambda}_{k,n}$ as

$\lambda_{k,n}-\widetilde{\lambda}_{k,n}=\frac{\langle(Q_{n}(x)-{\rm
i}x)\psi_{k,n},\psi_{k,n}^{*}\rangle}{\langle\psi_{k,n},\psi_{k,n}^{*}\rangle}+\widetilde{\rho}_{k,n},$
(5.56)

where $\psi_{k,n}$, $\psi_{k,n}^{*}$ are eigenfunctions of $\widetilde{S}_{n}$
and $\widetilde{S}_{n}^{*}$ associated with $\widetilde{\lambda}_{k,n}$ and
$\widetilde{\lambda}_{k,n}^{*}$, respectively. Finally, in (5.56) one can
further replace $\psi_{k,n}$ by $\psi_{k}$, the eigenfunctions of $S_{\rm A}$
associated with $\nu_{k}$, producing typically only an exponentially small
error due to the fast convergence of $\psi_{k,n}$ to $\psi_{k}$, see Theorem
4.2 and decay estimates on eigenfunctions, see Theorem 2.4 and Examples 2.5,
5.2. We implement these observations in Example 6.1 below. $\blacksquare$

##  6\. Examples

We illustrate the results on several examples. We start with the one
dimensional ones, where Theorem 5.7 and Proposition 5.8 are employed. Next we
analyze a radially symmetric cases truncated to annuli, still using the one
dimensional results. Finally, Theorem 5.4 is used in a two dimensional example
with truncations to domains with corners.

###  6.1. One dimensional examples

######  Example 6.1 (Odd imaginary potentials).

Let $U:{\mathbb{R}}\to{\mathbb{R}}$ be odd and satisfy Assumption 5.6; note
that (5.33) holds automatically if the previous conditions are satisfied. We
consider Dirichlet realizations $T_{n}=-\partial_{x}^{2}+{\rm i}U$ in
$L^{2}((-s_{n},s_{n}))$ with $s_{n}\nearrow+\infty$. Since $U$ is odd, (5.28)
corresponds to the relation $Q={\rm i}U$, thus by Theorem 5.7, the spectra of
$T_{n}$ contain asymptotically the eigenvalues $\\{\lambda_{k,n}\\}_{k}$ in
(5.39). Due to the antilinear symmetry of $T_{n}$ ($x\mapsto-x$ together with
complex conjugation, the so-called $\mathcal{P}\mathcal{T}$-symmetry), the
spectra of $T_{n}$ contain also $\\{\overline{\lambda}_{k,n}\\}_{k}$.

In particular, $U(x)=\operatorname{sgn}(x)|x|^{\alpha}$ with $\alpha>0$,
satisfies Assumption 5.6 with $\nu=-1$ and a possible lack of
differentiability of $U$ at $0$ can be treated by splitting $U=\eta
U+(1-\eta)U$ with $\eta\in C_{0}^{\infty}((-2,2))$ and $\eta=1$ on $(-1,1)$.
Notice that $U_{1}=\eta U$ satisfies assumptions of Proposition 5.8. Hence we
obtain

$\lambda_{k,n}=\alpha^{\frac{2}{3}}s_{n}^{\frac{2(\alpha-1)}{3}}\left(\nu_{k}+\mathcal{O}_{k}\left(s_{n}^{-\frac{2+\alpha}{3}}\right)\right)-{\rm
i}s_{n}^{\alpha},$ (6.1)

and their complex conjugates; see Figures 5.1 and 6.1 for illustration in two
well-known special cases (the imaginary Airy operator and imaginary cubic
oscillator).

Figure 6.1. $Q(x)={\rm i}x^{3}$: Real (left) and imaginary (right) part of the
eigenvalues (red) of truncated operators $T_{n}$, defined on
$L^{2}((-s_{n},s_{n}))$ with $s_{n}=0.1n$, $n=5,6,\dots,60$. Asymptotic curves
(blue) for $\lambda_{k,n}$, $\overline{\lambda_{k,n}}$ with first corrections
for $k=1,2,\dots,5$.

In Figure 6.1 we plot the asymptotic curves taking into account the first
correction with $\psi_{k}(y)=\operatorname{Ai}({\rm e}^{\frac{{\rm
i}\pi}{6}}y+\mu_{k})$, see Example 5.2 with $d=1$ and Remark 5.9.
$\blacksquare$

######  Example 6.2 (Even imaginary potentials).

Let $V:{\mathbb{R}}\to{\mathbb{R}}$ be an even and with $V^{\prime}(x)>0$ for
$x>0$ and consider Dirichlet realizations $T_{n}=-\partial_{x}^{2}+{\rm i}V$
in $L^{2}((-s_{n},s_{n}))$ with $s_{n}\nearrow+\infty$. Theorem 5.7 is not
directly applicable because of the condition (5.33). Nonetheless, due to the
symmetry of $V$, eigenfunctions of $T_{n}$ satisfy either Dirichlet or Neumann
boundary conditions at $0$. Therefore we can split the spectral problem and
analyze separately the spectra of

$\displaystyle T_{n}^{\rm DD}$ $\displaystyle=-\partial_{x}^{2}+{\rm i}V(x),$
$\displaystyle{\operatorname{Dom}}(T_{n}^{\rm DD})=W^{2,2}((-s_{n},0))\cap
W_{0}^{1,2}((-s_{n},0)),$ $\displaystyle T_{n}^{\rm DN}$
$\displaystyle=-\partial_{x}^{2}+{\rm i}V(x),$
$\displaystyle{\operatorname{Dom}}(T_{n}^{\rm DN})=\\{f\in
W^{2,2}((-s_{n},0))\,:f^{\prime}(0)=f(-s_{n})=0\\}.$

Introducing $U:=V\chi_{{\mathbb{R}}_{+}}$, we obtain that $(T_{n}^{\rm
DD})^{*}=-\partial_{x}^{2}+Q$ in $L^{2}((-s_{n},0))$ with $Q(x)=-{\rm i}U(-x)$
as in (5.28).

We assume that this $U$ satisfies Assumption 5.6, possibly with perturbations
as in Example 6.1, and notice that (5.33) is satisfied automatically. Then
Theorem 5.7 yields that the spectra of $T_{n}^{\rm DD}$ contain asymptotically
the eigenvalues

$\lambda_{k,n}^{\rm
DD}=V^{\prime}(s_{n})^{\frac{2}{3}}\Big{(}\overline{\nu_{k}}+\rho_{k,n}^{\rm
DD}\Big{)}+{\rm i}V(s_{n}),\quad n\to\infty.$ (6.2)

It is not difficult to see that the claim of Theorem 5.7 holds also for
Neumann boundary conditions at the endpoints as well as for the combinations
of Dirichlet and Neumann boundary conditions. Depending on the boundary
condition at $0$, the limiting operator is Dirichlet or Neuman imaginary Airy
operator in $L^{2}({\mathbb{R}}_{+})$, in the Neumann case with eigenvalues
$\\{\nu_{k}^{\prime}\\}=\\{{\rm e}^{{\rm
i}(\frac{2\omega}{3}-\pi)}\mu_{k}^{\prime}\\}$ where $\\{\mu_{k}^{\prime}\\}$
are zeros of $\operatorname{Ai}^{\prime}$. Thus we obtain that the spectra of
$T_{n}^{\rm ND}$ contain asymptotically the eigenvalues

$\lambda_{k,n}^{\rm
DN}=V^{\prime}(s_{n})^{\frac{2}{3}}\Big{(}\overline{\nu_{k}}+\rho_{k,n}^{\rm
ND}\Big{)}+{\rm i}V(s_{n}),\quad n\to\infty.$ (6.3)

These two sets of eigenvalues have the same main asymptotic terms, however,
the corresponding eigenfunctions of $T_{n}$ are very different (odd and even).
$\blacksquare$

######  Example 6.3 (Imaginary exponential potential with non-empty essential
spectrum).

Consider the operator $T=-\partial_{x}^{2}+{\rm i}{\rm e}^{x}$ and its
truncations $T_{n}$ to $(-\infty,s_{n})$ with $s_{n}\nearrow+\infty$. Defining
$U(x):={\rm e}^{x}$ and $Q(x):=-{\rm i}U(-x)$ as in (5.28), we obtain that
$T_{n}^{*}$ is unitarily equivalent via the reflection $x\mapsto-x$ to
$-\partial_{x}^{2}+Q$ in $L^{2}((-s_{n},\infty))$. This $U$ satisfies
Assumption 5.6 with $t_{n}=+\infty$ and $\nu=0$, thus by Theorem 5.7, the
spectra of $T_{n}$ contain asymptotically the eigenvalues

$\lambda_{k,n}={\rm
e}^{\frac{2}{3}s_{n}}\left(\overline{\nu_{k}}+\mathcal{O}_{k}\left({\rm
e}^{-\frac{1}{3}s_{n}}\right)\right)+{\rm i}{\rm e}^{s_{n}},\quad n\to\infty.$
(6.4)

In fact, since Assumption 5.6 is satisfied also with $t_{n}=s_{n}$, the
eigenvalues (6.4), with possibly different remainders, are asymptotically
contained in the spectra of operators $T_{n}=-\partial_{x}^{2}+{\rm i}{\rm
e}^{x}$ subject to Dirichlet boundary conditions in $L^{2}((-s_{n},s_{n}))$;
spectra of these are illustrated in Figure 6.2.

Figure 6.2. $U(x)={\rm i}{\rm e}^{x}$: Real (left) and imaginary (right) part
of the eigenvalues (red) of truncated operators $T_{n}$, defined on
$L^{2}((-s_{n},s_{n}))$ with $s_{n}=0.1n$, $n=5,6,\dots,60$. Asymptotic curves
(blue) for $\lambda_{k,n}$ with first correction for $k=1,2,3$.

$\blacksquare$

###  6.2. Radially symmetric potentials on annuli

Consider the exterior domain
$\Omega={\mathbb{R}}^{d}\setminus\overline{B_{1}(0)}$, a radial potential
$V:\Omega\to{\mathbb{C}}$ satisfying Assumption 2.1 (with $Q$ replaced by $V$)
and the Dirichlet realization of $T=-\Delta+V$ in $L^{2}(\Omega)$. Consider
also the truncated operators $T_{n}=-\Delta+V$ in $L^{2}(\Omega_{n})$ with
$\Omega_{n}=\Omega\cap B_{s_{n}}(0)$ and $s_{n}\nearrow+\infty$, subject to
Dirichlet boundary conditions both on $\partial B_{1}(0)$ and $\partial
B_{s_{n}}(0)$. Truncations of a specific problem of this type were originally
considered in [12, Sec. 3.1] and it was shown in [11, Sec. 6] that such domain
truncation is spectrally exact, see also Theorem 4.2. Our aim here is to
investigate the diverging eigenvalues.

We transform $T_{n}$ in spherical coordinates with $r\in(1,s_{n})$,
$\Theta\in\mathcal{S}^{d-1}$, employ the usual unitary transform in the radial
part (see _e.g._ [41, Chap. 18])

$L^{2}({\mathbb{R}}_{+};r^{d-1}{{\rm d}}r)\to L^{2}({\mathbb{R}}_{+},{{\rm
d}}r)\,:\,h(r)\mapsto r^{(d-1)/2}h(r),$ (6.5)

and use the spherical harmonics $\\{Y_{l,j}\\}_{j=1}^{N(l,d)}$,
$l\in\mathbb{N}_{0}$, $N(l,d)=\frac{(2l+d-2)(l+d-3)!}{l!(d-2)!}$ in $d-1$
dimensions, which satisfy
$-\Delta_{\mathcal{S}^{d-1}}Y_{l,j}(\Theta)=l(l+d-2)Y_{l,j}(\Theta).$ Thereby
we obtain a decomposition of $T_{n}$ to one dimensional operators

$T_{n,l}:=-\partial_{r}^{2}+{\rm
i}U(r)+U_{1}(r),\quad{\operatorname{Dom}}(T_{n,l}):=W^{2,2}((1,s_{n}))\cap
W^{1,2}_{0}((1,s_{n})),$ (6.6)

where $U(r)=V(x)$ for $|x|=r$ and

$U_{1}(r)=\frac{(d-1)(d-3)+4l(l+d-2)}{4r^{2}}.$ (6.7)

Similarly as in Examples 6.2, 6.3, $T_{n,l}^{*}$ is unitarily equivalent via
the reflection $r\mapsto-r$ to $-\partial_{r}^{2}+Q$ in $L^{2}((-s_{n},-1))$
with $Q(r)=-{\rm i}U(-r)-U_{1}(-r)$.

We suppose that $U\chi_{[1,+\infty]}$ satisfies Assumption 5.6 (with
perturbations as in Example 6.1) and note that $U_{1}$ satisfies conditions of
Proposition 5.8. Then Theorem 5.7, Proposition 5.8 and Theorem 5.4 yield that
the spectra of $T_{n,l}$ in (6.6) contain asymptotically the eigenvalues

$\lambda_{k,n,l}=U^{\prime}(s_{n})^{\frac{2}{3}}\left(\overline{\nu_{k}}+\rho_{k,n,l}\right)+{\rm
i}U(s_{n})-U_{1}(s_{n}),\quad n\to\infty.$ (6.8)

In particular for $V(x)={\rm i}|x|^{2}$ with
$x\in{\mathbb{R}}^{d}\setminus\overline{B_{1}(0)}$ we obtain from (6.8) that
the spectral of the one dimensional operators $T_{n,l}$, see (6.6), contain
asymptotically the eigenvalues

$\lambda_{k,n,l}=(2s_{n})^{\frac{2}{3}}\left(\overline{\nu_{k}}+\mathcal{O}_{k,l}\left(s_{n}^{-\frac{4}{3}}\right)\right)+{\rm
i}s_{n}^{2},\quad n\to\infty;$ (6.9)

Figures 6.3 and 1.1 illustrate this result.

Figure 6.3. $V(x)={\rm i}|x|^{2}$: Real (left) and imaginary (right) part of
the eigenvalues of truncated operators $T_{n,l}$ with $d=3$ and $l=1,\dots,5$
(red, pink, green, purple, brown), defined on $L^{2}((1,s_{n}))$ with
$s_{n}=0.1n$, $n=15,16,\dots,115$. Asymptotic curves (blue) for
$\lambda_{k,n}$ with the first correction for $k=1,2,\dots,6$.

###  6.3. Two dimensional rotated squares and polynomial potential

Finally we show that Theorem 5.4 can be applied directly in more dimensional
problems. The verification of the assumptions is analogous to the steps in
proof of Theorem 5.7 in the one dimensional case.

######  Example 6.4.

We consider the potential

$Q(x_{1},x_{2})={\rm i}(x_{1}^{3}+x_{2}^{4})+x_{1}^{2}x_{2}^{2},\quad
x:=(x_{1},x_{2})\in{\mathbb{R}}^{2}$ (6.10)

and a sequence of domains $\\{\Omega_{n}\\}\subset{\mathbb{R}}^{2}$, which are
expanding squares rotated by $\pi/4$ with the left-most corner at
$(-s_{n},0)$, _i.e._ with some $\\{s_{n}\\}$ with $s_{n}\nearrow+\infty$,

$\displaystyle\Omega_{n}=\\{x\in{\mathbb{R}}^{2}\,:$
$\displaystyle(x_{1}\in(-s_{n},0]\text{ and }|x_{2}|<x_{1}+s_{n})\text{ or }$
(6.11) $\displaystyle(x_{1}\in(0,s_{n})\text{ and }|x_{2}|<(s_{n}-x_{1}))\\}.$

Using Theorems 4.2 and 5.4 we explain below that the Dirichlet truncations of
$T=-\Delta+Q$ in $L^{2}({\mathbb{R}}^{2})$ to $T_{n}=-\Delta+Q$ in
$L^{2}(\Omega_{n})$, $n\in\mathbb{N}$, are spectrally exact and the spectra of
$T_{n}$ contain asymptotically the eigenvalues

$\lambda_{k,n}^{(j)}=(3s_{n}^{2})^{\frac{2}{3}}\Big{(}\nu_{k}+\rho_{k,n}^{(j)}\Big{)}-{\rm
i}s_{n}^{3},\quad n\to\infty,$ (6.12)

where $\\{\nu_{k}\\}$ are eigenvalues of the complex Airy operator $S_{\rm A}$
with $\Gamma=\Gamma_{\pi/4}$, $\omega=\pi/2$ and $\vartheta=(1,0)$, see
Example 5.2, and

$\frac{1}{m_{a}(\nu_{k})}\left|\sum_{j=1}^{m_{a}(\nu_{k})}\rho_{k,n}^{(j)}\right|=\mathcal{O}(s_{n}^{-\frac{2}{3}}),\quad
n\to\infty.$ (6.13)

Clearly $Q\in C^{1}({\mathbb{R}}^{d})$ satisfies Assumption 2.1 on each
$\Omega_{n}$, $n\in\mathbb{N}$, and

$\nabla Q(x_{1},x_{2})=(2x_{1}x_{2}^{2}+3{\rm
i}x_{1}^{2},2x_{1}^{2}x_{2}+4{\rm i}x_{2}^{3}),\ \ |\nabla
Q(-p_{n})|=3s_{n}^{2},\ p_{n}=(s_{n},0).$ (6.14)

We first check that $Q$ satisfies Assumption 2.1 and (2.7) on
${\mathbb{R}}^{2}$, so the truncations on $\\{\Omega_{n}\\}$ are spectrally
exact by Theorem 4.2. Indeed, (1.8) holds since

$\displaystyle|x_{2}|\leq\tfrac{1}{2}|x_{1}|^{\frac{4}{3}},|x_{1}|\gtrsim 1:$
$\displaystyle\quad|Q(x)|\gtrsim|x_{1}|^{3},$ $\displaystyle|\nabla Q(x)|$
$\displaystyle\lesssim x_{1}^{4},$ (6.15)
$\displaystyle|x_{2}|\geq\tfrac{3}{2}|x_{1}|^{\frac{4}{3}},|x_{2}|\gtrsim 1:$
$\displaystyle\quad|Q(x)|\gtrsim x_{2}^{4},$ $\displaystyle|\nabla Q(x)|$
$\displaystyle\lesssim|x_{2}|^{3},$
$\displaystyle|x_{2}|\in(\tfrac{1}{2}|x_{1}|^{\frac{4}{3}},\tfrac{3}{2}|x_{1}|^{\frac{4}{3}}),|x_{1}|\gtrsim
1:$ $\displaystyle\quad|Q(x)|\gtrsim|x_{1}|^{\frac{14}{3}},$
$\displaystyle|\nabla Q(x)|$ $\displaystyle\lesssim|x_{1}|^{4}.$

To apply Theorem 5.4 we check conditions in Assumption 5.3. The conditions 1
and 2 are satisfied with $\Gamma=\Gamma_{\pi/4}$ and $r_{n}\approx
s_{n}^{5/3}$. Moreover, we have $\omega=\pi/2$ and $\vartheta=(0,1)$, see
(5.15). To verify the condition 3, we employ formulas (5.17), (5.18) and
proceed similarly as in the one dimensional case (see the proof of Theorem
5.7).

Using the variable $\sigma_{n}x=y$, if $(y_{1},y_{2})\in p_{n}+\Omega_{n}$
with $0<y_{1}<1$, then (since $|y_{2}|\leq y_{1}$)

$|(\nabla Q_{n})(x)|=\frac{|\nabla Q(y-p_{n})|}{|\nabla
Q(-p_{n})|}\lesssim\frac{s_{n}^{2}}{s_{n}^{2}}=1.$ (6.16)

Next, for $(y_{1},y_{2})\in p_{n}+\Omega_{n}$ with $y_{1}\in[1,s_{n}]$, by the
elementary identity for $s_{n}^{3}-(s_{n}-x)^{3}$, we obtain

$|Q(y_{1}-s_{n},y_{2})-Q(-s_{n},0)|\geq
y_{1}(s_{n}^{2}+s_{n}(s_{n}-y_{1})+(s_{n}-y_{1})^{2})\geq y_{1}s_{n}^{2}.$
(6.17)

Thus, (recall $|y_{2}|\leq y_{1}$)

$\frac{|(\nabla Q_{n})(x)|}{|Q_{n}(x)|^{\frac{3}{2}}}=\frac{|(\nabla
Q)(y_{1}-s_{n},y_{2})|}{|Q(y_{1}-s_{n},y_{2})-Q(-s_{n},0)|^{\frac{3}{2}}}\lesssim\frac{s_{n}y_{1}^{2}+s_{n}^{2}+s_{n}^{2}y_{1}+y_{1}^{3}}{y_{1}^{\frac{3}{2}}s_{n}^{3}}\lesssim\frac{1}{s_{n}}.$

Finally, for $(y_{1},y_{2})\in p_{n}+\Omega_{n}$ with
$y_{1}\in(s_{n},2s_{n})$, we have

$|Q(y_{1}-s_{n},y_{2})-Q(-s_{n},0)|\gtrsim s_{n}^{3}$ (6.18)

and so (again since $|y_{2}|\leq y_{1}$)

$\frac{|(\nabla
Q_{n})(x)|}{|Q_{n}(x)|^{\frac{3}{2}}}\lesssim\frac{s_{n}^{3}}{s_{n}^{\frac{9}{2}}}=\frac{1}{s_{n}^{\frac{3}{2}}}.$
(6.19)

Finally, using (5.18)

$\iota_{n}\lesssim\frac{|(s_{n}-y_{1})^{2}y_{2}^{2}+{\rm
i}(y_{2}^{4}-3s_{n}y_{1}^{2}+y_{1}^{3})|}{s_{n}^{\frac{2}{3}}(s_{n}^{3}-(s_{n}-y_{1})^{3})y_{1}},\quad(y_{1},y_{2})\in
p_{n}+\Omega_{n}.$ (6.20)

For $0<y_{1}<s_{n}$, by the identity for $s_{n}^{3}-(s_{n}-y_{1})^{3}$ and
$|y_{2}|/y_{1}\leq 1$, we obtain

$\frac{|(s_{n}-y_{1})^{2}y_{2}^{2}+{\rm
i}(y_{2}^{4}-3s_{n}y_{1}^{2}+y_{1}^{3})|}{s_{n}^{\frac{2}{3}}(s_{n}^{3}-(s_{n}-y_{1})^{3})y_{1}}\lesssim\frac{s_{n}^{2}}{s_{n}^{\frac{2}{3}+2}}=\frac{1}{s_{n}^{\frac{2}{3}}}$
(6.21)

and for $s_{n}\leq y_{1}<2s_{n}$, we arrive at

$\frac{|(s_{n}-y_{1})^{2}y_{2}^{2}+{\rm
i}(y_{2}^{4}-3s_{n}y_{1}^{2}+y_{1}^{3})|}{s_{n}^{\frac{2}{3}}(s_{n}^{3}-(s_{n}-y_{1})^{3})y_{1}}\lesssim\frac{s_{n}^{4}}{s_{n}^{\frac{2}{3}+4}}=\frac{1}{s_{n}^{\frac{2}{3}}}.$
(6.22)

Thus $\iota_{n}=\mathcal{O}(s_{n}^{-\frac{2}{3}})$, $n\to\infty$.
$\blacksquare$

##  7\. Remarks on strong coupling

We consider a family of Dirichlet realizations of

$T_{g}=-\Delta+Q_{1}+{\rm i}gQ_{2},\qquad g>0,$ (7.1)

in $L^{2}(\Omega)$ where $\Omega$ is open, functions $Q_{i}$, $i=1,2$, are
real valued and $g\to+\infty$.

Operators with this structure arise in several contexts, in particular, in
enhanced dissipation, see Example 7.4, in $\mathcal{PT}$-symmetric phase
transitions, see Examples 7.5 and 7.6, or when $Q_{1}=0$ as semi-classical
problems with purely imaginary potentials, see _e.g._ [25, 4, 2], in
particular in the context of Bloch-Torrey equation.

We focus here on the case when $\Omega$ is (typically) unbounded and $|Q_{2}|$
has a global minimum inside of $\Omega$, see Assumption 7.1 for details. As an
application of Corollary 3.6, we describe some of the diverging eigenvalues as
$g\to+\infty$. In Section 7.1 we show how Theorem 7.3 can be implemented and
indicate its possible further extensions.

######  Assumption 7.1.

Let $\Omega\subset\mathbb{R}^{d}$ be open with $0\in\Omega$, let
$\overline{B_{R}(0)}\subset\Omega$ for some $R>0$ and let $Q_{1}\in
C^{1}(\overline{\Omega};{\mathbb{R}})$ with $Q_{1}\geq 0$, $Q_{2}\in
C^{1}(\overline{\Omega}\setminus\\{0\\};{\mathbb{R}})$. Suppose further that

  1. n)

for some $\varepsilon>0$, the condition (2.1) is satisfied with $\Omega$
replaced by $\Omega\setminus\overline{B_{\varepsilon}(0)}$ and $Q$ replaced
separately by $Q_{1}$ and by $Q_{2}$;

  2. n)

$Q_{2}(0)=0$ and $|Q_{2}|$ attains the global minimum at $0$, _i.e._ for every
$\delta>0$

$\inf_{x\in\Omega\setminus\overline{B_{\delta}(0)}}|Q_{2}(x)|>0;$ (7.2)

  3. n)

there exists $Q_{\infty}\in C(\mathbb{R}^{d})\cap
C^{1}(\mathbb{R}^{d}\setminus\\{0\\};{\mathbb{R}})$ with
$\min_{|x|=1}|Q_{\infty}(x)|>0$ such that for some $\kappa>0$

$\displaystyle Q_{\infty}(tx)$ $\displaystyle=t^{\kappa}Q_{\infty}(x),$
$\displaystyle x\in\mathbb{R}^{d},\ t>0,$ (7.3) $\displaystyle
Q_{2}(x)-Q_{\infty}(x)$ $\displaystyle=|x|^{\kappa}h_{0}(x),$ $\displaystyle
h_{0}(x)=o(1),\quad|x|\to 0,$ $\displaystyle|\nabla Q_{2}(x)-\nabla
Q_{\infty}(x)|$ $\displaystyle=|x|^{\kappa-1}h_{1}(x),$ $\displaystyle
h_{1}(x)=o(1),\quad|x|\to 0,$

and the discrete spectrum of $S_{\infty}:=-\Delta+{\rm i}Q_{\infty}$ in
$L^{2}(\mathbb{R}^{d})$ is non-empty. $\blacksquare$

######  Example 7.2.

Typical examples of $S_{\infty}$ in Assumption 7.1 in one dimension are

$-\partial_{x}^{2}+{\rm i}x^{n},\quad
n\in\mathbb{N}\setminus\\{1\\},\qquad-\partial_{x}^{2}+{\rm
i}|x|^{\kappa},\quad\kappa>0.$ (7.4)

The spectra of the former for $n={2k+1}$, $k\in\mathbb{N}$, are real, see
[39], and the spectra of the remaining operators with even potential can be
obtained by complex scaling (after possibly reducing the problem to
Dirichlet/Neumann operators in $L^{2}({\mathbb{R}}_{+})$). A typical case in
more dimensions is an imaginary oscillator with potential ${\rm i}\langle
Ax,x\rangle_{\mathbb{R}^{d}}$ and a positive definite matrix $A$.
$\blacksquare$

######  Theorem 7.3.

Let Assumption 7.1 be satisfied and let $T_{g}$, $g>0$, be as in (7.1). Then
the spectra of $T_{g}$ contain asymptotically the eigenvalues (with
$k\in\mathbb{N}$ and $j\in\\{1,\dots,m_{a}(\nu_{k})\\}$)

$\lambda_{k,g}^{(j)}=g^{\frac{2}{2+\kappa}}\left(\nu_{k}+\rho_{k,g}^{(j)}\right),\quad
g\to+\infty,$ (7.5)

where $\\{\nu_{k}\\}=\sigma_{\rm disc}(S_{\infty})$ and, as $g\to+\infty$,
$\rho_{k,g}^{(j)}=o_{j,k}(1)$ and for any $\beta\in(0,1)$,

$\frac{1}{m_{a}(\nu_{k})}\left|\sum_{j=1}^{m_{a}(\nu_{k})}\rho_{k,g}^{(j)}\right|=\mathcal{O}_{k}\left(g^{-\frac{\min\\{2,\kappa(1-\beta)\\}}{2+\kappa}}+\sup_{|y|\leq
g^{-\frac{\beta}{2+\kappa}}}|h_{0}(y)|\right).$ (7.6)

###### Proof.

We select $\sigma=\sigma(g)$ to satisfy

$g\sigma^{2+\kappa}=1,\quad g>1$ (7.7)

so $\sigma\to 0$ as $g\to+\infty$. By scaling $x\mapsto\sigma x$, we obtain
operators in $L^{2}(\sigma^{-1}\Omega)$

$\frac{1}{\sigma^{2}}\left(-\Delta+\sigma^{2}\Big{(}Q_{1}(\sigma x)+{\rm
i}gQ_{2}(\sigma x)\Big{)}\right)=:\frac{1}{\sigma^{2}}S_{g},$ (7.8)

which are unitarily equivalent to $T_{g}$. In the following we apply Corollary
3.6 to the operators $S_{g}$ and $S_{\infty}=-\Delta+{\rm i}Q_{\infty}$ in
$L^{2}(\mathbb{R}^{d})$.

We define

$Q_{\sigma}(x):=\sigma^{2}\Big{(}Q_{1}(\sigma x)+{\rm i}gQ_{2}(\sigma
x)\Big{)},\quad x\in\sigma^{-1}\Omega$ (7.9)

and verify conditions in Assumption 3.5.

1 From the scaling we have $B_{\sigma^{-1}R}(0)\subset\sigma^{-1}\Omega$ and
so $\sigma^{-1}\Omega$ exhaust $\Omega_{\infty}:=\mathbb{R}^{d}$.

2 We first split $Q_{\sigma}$ as $Q_{\sigma}=\eta
Q_{\sigma}+(1-\eta)Q_{\sigma}$ where $\eta\in C_{0}^{\infty}(B_{2\alpha}(0))$
with $\eta=1$ on $B_{\alpha}(0)$ and where sufficiently large $\alpha>0$,
independent of $g$, will be fixed later. We show that $\eta Q_{\sigma}$ is
uniformly bounded (and so can be treated as a perturbation, see remarks after
Assumption 2.1) and $(1-\eta)Q_{\sigma}$ satisfies (3.29).

For $|x|\leq 2\alpha$, using (7.3) and (7.7) we have

$|Q_{\sigma}(x)|\lesssim\sigma^{2}+\sigma^{2+\kappa}g\left(|Q_{\infty}(x)|+|x|^{\kappa}h_{0}(\sigma
x)\right)\lesssim 1.$ (7.10)

Since $\nabla(1-\eta)Q_{\sigma}=-(\nabla\eta)Q_{\sigma}+(1-\eta)\nabla
Q_{\sigma}$ and $\operatorname{supp}\eta\subset B_{2\alpha}(0)$, it suffices
to further analyze $|\nabla Q_{\sigma}(x)|$ for $|x|>\alpha$. Namely, we
estimate

$\frac{|\nabla Q_{\sigma}(x)|}{|Q_{\sigma}(x)|^{\frac{3}{2}}}=\frac{|(\nabla
Q_{1})(\sigma x)+{\rm i}g(\nabla Q_{2})(\sigma x)|}{|Q_{1}(\sigma x)+{\rm
i}gQ_{2}(\sigma x)|^{\frac{3}{2}}}.$ (7.11)

At first we focus on the region $\alpha<|x|\leq\delta\sigma^{-1}$ with a
sufficiently small $\delta>0$ which will be selected later. Using (7.3) and
homogeneity of $Q_{\infty}$, we obtain

$|(\nabla Q_{2})(\sigma x)|\leq|(\nabla Q_{\infty})(\sigma
x)|\left(1+\frac{|h_{1}(\sigma x)|}{\displaystyle\min_{|z|=1}|\nabla
Q_{\infty}(z)|}\right)\lesssim|(\nabla Q_{\infty})(\sigma x)|;$ (7.12)

in the last step we use Euler’s homogeneous function theorem, the homegeneity
of $Q_{\infty}$ and that the assumption $\min_{|z|=1}|Q_{\infty}(z)|>0$
implies that $\min_{|z|=1}|\nabla Q_{\infty}(z)|>0$ as well. Similarly,

$|Q_{2}(\sigma x)|\geq|Q_{\infty}(\sigma x)|\left(1-\frac{|h_{0}(\sigma
x)|}{\displaystyle\min_{|z|=1}|Q_{\infty}(z)|}\right),$ (7.13)

thus, if $\delta>0$ is sufficiently small

$|Q_{2}(\sigma x)|\gtrsim|Q_{\infty}(\sigma
x)|,\quad\alpha<|x|<\delta\sigma^{-1}.$ (7.14)

Hence, writing $x=tz$ with $|z|=1$, $t>\alpha$ and using (7.7), we arrive at

$\frac{|\nabla
Q_{\sigma}(x)|}{|Q_{\sigma}(x)|^{\frac{3}{2}}}\lesssim\frac{1+g(\sigma
t)^{\kappa-1}|\nabla
Q_{\infty}(z)|}{(g\sigma^{\kappa}t^{\kappa})^{\frac{3}{2}}|Q_{\infty}(z)|^{\frac{3}{2}}}\lesssim\frac{\sigma^{3}}{\alpha^{\frac{3\kappa}{2}}}+\frac{1}{\alpha^{\frac{\kappa}{2}+1}},\qquad\alpha<|x|<\delta\sigma^{-1}$
(7.15)

and so we can select $\alpha>0$ so that the right hand side is sufficiently
small for all $x$ in the considered region.

Next let $\delta\sigma^{-1}\leq|x|\leq\varepsilon\sigma^{-1}$ with
$\varepsilon>0$ from the assumption. Then, from (7.2),

$\frac{|\nabla
Q_{\sigma}(x)|}{|Q_{\sigma}(x)|^{\frac{3}{2}}}\lesssim\frac{1+g}{g^{\frac{3}{2}}|Q_{2}(\sigma
x)|^{\frac{3}{2}}}\lesssim\frac{1}{g^{\frac{1}{2}}}.$ (7.16)

Finally, let $\varepsilon\sigma^{-1}\leq|x|$. Here we use (7.2) and the
$Q_{1}$ and $Q_{2}$ satisfy separately (2.1) outside $B_{\varepsilon}(0)$.
Thus writing $y=\sigma x$, we get

$\frac{|(\nabla
Q_{\sigma})(x)|}{|Q_{\sigma}(x)|^{\frac{3}{2}}}\leq\frac{|\nabla
Q_{1}(y)|_{+}g|\nabla Q_{2}(y)|}{|Q_{1}(y)+{\rm
i}gQ_{2}(y)|^{\frac{3}{2}}}\leq\varepsilon_{\nabla}+\frac{M_{\nabla}}{g^{\frac{3}{2}}|Q_{2}(y)|^{\frac{3}{2}}}+\frac{\varepsilon_{\nabla}}{g^{\frac{1}{2}}}+\frac{M_{\nabla}}{g^{\frac{1}{2}}|Q_{2}(y)|^{\frac{3}{2}}}.$

3 We consider projection $P_{\sigma}:=\chi_{B_{\sigma^{-1}R}(0)}$ and cut-offs
$\xi_{\sigma}\in C_{0}^{\infty}(B_{\sigma^{-1}R}(0))$ so that
$\xi_{\sigma}(x)=1$ for $|x|\leq\sigma^{-1}R-1$ and
$\||\nabla\xi_{\sigma}|\|_{L^{\infty}}$, $\|\Delta\xi_{\sigma}\|_{L^{\infty}}$
are uniformly bounded. The conditions on the operator and form domains in 3
can be verified easily since the support of $\xi_{\sigma}$ is bounded (see
_e.g._ the proof of Theorem 4.2).

4 We split the estimate of

$\left\|\frac{\xi_{\sigma}(Q_{\sigma}-{\rm
i}Q_{\infty})}{(Q_{\infty}+1)(Q_{\sigma}+1)}\right\|_{L^{\infty}(\sigma^{-1}\Omega)}$
(7.17)

to three regions. First let $\sigma|x|\leq\sigma^{\beta}$ with
$\beta\in(0,1)$. Then, using homogeneity of $Q_{\infty}$,
$\min_{|z|=1}|Q_{\infty}(z)|>0$, (7.7) and (7.3),

$\left|\frac{Q_{\sigma}(x)-{\rm
i}Q_{\infty}(x)}{(Q_{\infty}(x)+1)(Q_{\sigma}(x)+1)}\right|\lesssim\sigma^{2}+\frac{|\sigma^{2+\kappa}gQ_{2}(\sigma
x)-Q_{\infty}(\sigma x)|}{|Q_{\infty}(\sigma
x)|}\lesssim\sigma^{2}+|h_{0}(\sigma x)|.$

Next, when $\sigma^{\beta}\leq|\sigma x|\leq\delta$ with $\delta>0$ fixed, but
sufficiently small, we use inequality (7.13) and the properties of
$Q_{\infty}$ similarly as above to arrive at

$\left|\frac{Q_{\sigma}(x)-{\rm
i}Q_{\infty}(x)}{(Q_{\infty}(x)+1)(Q_{\sigma}(x)+1)}\right|\lesssim\frac{1}{|Q_{\infty}(x)|}+\frac{1}{|Q_{\sigma}(x)|}\lesssim\sigma^{\kappa(1-\beta)}.$
(7.18)

Finally, for $\delta\leq|\sigma x|\leq R$, we use in addition (7.2) and obtain

$\left|\frac{Q_{\sigma}(x)-{\rm
i}Q_{\infty}(x)}{(Q_{\infty}(x)+1)(Q_{\sigma}(x)+1)}\right|\lesssim\frac{1}{|Q_{\infty}(x)|}+\frac{1}{|Q_{\sigma}(x)|}\lesssim\sigma^{\kappa}+\frac{\sigma^{\kappa}}{|Q_{2}(\sigma
x)|}\lesssim\sigma^{\kappa}.$ (7.19)

The estimate of the remaing terms in (3.31) is similar. Namely, denoting
$\zeta_{\sigma}$ the characteristic function of
$\operatorname{supp}(1-\xi_{\sigma})$, we obtain

$\left\|\frac{\zeta_{\sigma}}{Q_{\sigma}}\right\|_{L^{\infty}(\sigma^{-1}\Omega)}+\left\|\frac{\zeta_{\sigma}}{Q_{\infty}}\right\|_{L^{\infty}(\mathbb{R}^{d})}\lesssim\sigma^{\kappa}.$
(7.20)

Since all assumptions of Corollary 3.6 are satisfied, we obtain that
$S_{\sigma}\to S_{\infty}$ as $g\to+\infty$ and the conclusion for eigenvalues
(7.5) follows by spectral mapping. Since the eigenfunctions of $S_{\infty}$
decay exponentially, see Example 2.5, the second term in (3.35), entering the
estimate of $\rho_{k,g}^{(j)}$, can be omitted. ∎

###  7.1. Examples

######  Example 7.4 (Enhanced dissipation).

For operators $T_{g}$, sufficient conditions for the divergence of the real
parts of all eigenvalues of $T_{g}$ as $g\to+\infty$ were found _cf._ [14, 23,
38]. In [38], the specific operator

$T_{g}=-\partial_{x}^{2}+x^{2}+{\rm i}g(1+|x|^{\kappa})^{-1},$ (7.21)

in $L^{2}({\mathbb{R}})$ and with $\kappa>0$ was analyzed and an estimate on
the divergence rate of the real part of eigenvalues
$\operatorname{Re}\sigma(T_{g})\gtrsim g^{\frac{2}{\kappa+2}}$ was proved,
_cf._ [38, Thm. 1.2]. Similar problem and result was also established in [23,
Thm. 1.9].

Figure 7.1. $Q_{1}(x)=x^{2}$, $Q_{2}(x)=(1+|x|^{\kappa})^{-1}$: Real (left)
and imaginary (right) part of the eigenvalues (red) of operators $T_{g}$ with
$\kappa=3.15$ and $g=5,10,\dots,200$. Asymptotic curves (blue) for
$\lambda_{k,g}$ for $k=1,2,\dots,5$.

Note that the conjugated and shifted operator $T_{g}^{*}+{\rm i}g$ satisfies
Assumption 7.1 with $Q_{2}(x)=|x|^{\kappa}/(1+|x|^{\kappa})$,
$Q_{\infty}(x)=|x|^{\kappa}$ and $h_{0}(x)=-Q_{2}(x)$. Therefore by Theorem
7.3, spectra of $T_{g}$ contain asymptotically the eigenvalues

$\lambda_{k,g}=g^{\frac{2}{\kappa+2}}(\overline{\nu_{k}+\rho_{k,g}})+{\rm
i}g,\quad g\to+\infty,$ (7.22)

where $\\{\nu_{k}\\}$ are the eigenvalues of operator in (7.4) with the
potential ${\rm i}|x|^{\kappa}$. The remainder decays as
$\rho_{k,g}=\mathcal{O}(g^{\frac{-\kappa}{2(2+\kappa)}})$ for $\kappa\in(0,4)$
and $\rho_{k,g}=\mathcal{O}(g^{\frac{-2}{2+\kappa}})$ for $\kappa\geq 4$. This
result shows that the estimate in [38, Thm. 1.2] is optimal (see Figure 7.1).
$\blacksquare$

######  Example 7.5 ($\mathcal{P}\mathcal{T}$-symmetric phase transitions I).

Let $\Omega={\mathbb{R}}$, $Q_{1}$ be even, $Q_{2}$ odd and such that
Assumption 7.1 is satisfied. As in Example 6.1, the operators $T_{g}$ in (7.1)
with such $Q_{1}$, $Q_{2}$ have the antilinear
$\mathcal{P}\mathcal{T}$-symmetry and so the spectra of $T_{g}$ consists of
complex conjugate pairs. The spectrum of $T_{0}$ is real due to the self-
adjointness, however, as $g\to\infty$, a graduate appearance of complex
conjugated (non-real) spectral points pairs, called
$\mathcal{P}\mathcal{T}$-symmetric phase transitions, was observed in many
examples, see  _e.g._ [42] for one of the first works.

For $Q_{1}(x)=x^{2}$, upper estimates on the number of non-real eigenvalues
are given in [33] and precise spectral analysis of the double $\delta$
potential (with a fixed $b>0$)

$-\partial_{x}^{2}+x^{2}+{\rm i}g(\delta(x-b)-\delta(x+b))$ (7.23)

is performed in [35, 5]. In particular it is showed in [5] that the number of
non-real eigenvalues of (7.23) diverges as $g\to+\infty$.

We consider here

$T_{g}=-\partial_{x}^{2}+x^{2}+{\rm i}gx^{3}{\rm e}^{-x^{2}},$ (7.24)

in $L^{2}({\mathbb{R}})$ which can be viewed as a ”smooth version” of (7.23).
In this case, we can apply Theorem 7.3 in three stationary points of
$Q_{2}(x)=x^{3}{\rm e}^{-x^{2}}$, namely, $x_{0}=0$, $x_{1}=-\sqrt{3/2}$ and
$x_{2}=-x_{1}$.

The operator $T_{g}$ satisfies Assumption 7.1 with $Q_{1}(x)=x^{2}$,
$Q_{2}(x)=x^{3}{\rm e}^{-x^{2}}$, $Q_{\infty}(x)=x^{3}$, $\kappa=3$,
$h_{0}(x)={\rm e}^{-x^{2}}-1$. Therefore the eigenvalues

$\lambda_{k,g}^{(x_{0})}=g^{\frac{2}{5}}(\nu_{k}+\mathcal{O}_{k}(g^{-\frac{6}{25}})),\quad
g\to+\infty,$ (7.25)

where $\nu_{k}$ are (real) eigenvalues of the imaginary cubic oscillator (the
potential ${\rm i}x^{3}$), _cf._ Example 7.2, lie asymptotically in the
spectra of $T_{g}$.

Further sets of eigenvalues can be obtained by applying the Theorem 7.3 to the
operator $\widetilde{T}_{g}-{\rm i}g(3/(2{\rm e}))^{\frac{3}{2}}$, where
$\widetilde{T}_{g}$ is the operator obtained from $T_{g}$ by the translation
$x\mapsto x+x_{1}$. It satisfies the Assumption 7.1 with $\kappa=2$ and

$\displaystyle Q_{1}(x)$ $\displaystyle=(x+x_{1})^{2},$ $\displaystyle
Q_{2}(x)$ $\displaystyle=(x+x_{1})^{3}{\rm
e}^{-(x+x_{1})^{2}}+(\tfrac{3}{2{\rm e}})^{\frac{3}{2}},$ (7.26)
$\displaystyle Q_{\infty}(x)$ $\displaystyle=(\tfrac{27}{2{\rm
e}^{3}})^{\frac{1}{2}}x^{2},$ $\displaystyle h_{0}(x)$
$\displaystyle=\frac{Q_{2}(x)}{x^{2}}-(\tfrac{27}{2{\rm
e}^{3}})^{\frac{1}{2}}.$

Therefore the eigenvalues

$\lambda_{k,g}^{(x_{1})}=g^{\frac{1}{2}}(\nu_{k}+\mathcal{O}_{k}(g^{-\frac{1}{8}}))-{\rm
i}g(2{\rm e})^{-\frac{1}{2}}+\tfrac{3}{2},\quad g\to+\infty,$ (7.27)

where $\nu_{k}=(\frac{27}{2{\rm e}^{3}})^{\frac{1}{4}}{\rm e}^{{\rm
i}\frac{\pi}{4}}(2k+1)$, $k\in\mathbb{N}_{0}$, lie asymptotically in the
spectra of $T_{g}$. Analogous steps can be implemented on the conjugate
operator $T_{g}^{*}$ and we obtain the second set of eigenvalues
$\lambda_{k,g}^{(x_{2})}=\overline{\lambda_{k,g}^{(x_{1})}}$, _cf._ Figure
7.2.

Figure 7.2. $Q_{1}(x)=x^{2}$, $Q_{2}(x)=x^{3}{\rm e}^{-x^{2}}$: Real (left)
and imaginary (right) part of the eigenvalues (red) of operators $T_{g}$ with
$g=5,10,\dots,500$. Asymptotic curves (blue)
$\lambda_{k,g}^{(x_{1})},\lambda_{k,g}^{(x_{2})}$ and (green)
$\lambda_{k,g}^{(x_{0})}$ for $k=1,2,\dots,5$.

$\blacksquare$

######  Example 7.6 ($\mathcal{P}\mathcal{T}$-symmetric phase transitions II).

$\mathcal{P}\mathcal{T}$-symmetric phase transitions were studied in [13] for
operators in $L^{2}({\mathbb{R}})$ with polynomial potentials

$-\partial_{x}^{2}+\frac{x^{2M}}{2M}+{\rm i}g\frac{x^{M-1}}{M-1},\quad M\in
2\mathbb{N},$ (7.28)

and the eventual transition of each eigenvalue was established, see [13, Thm.
1.1] for precise claims.

For $M\geq 4$, Theorem 7.3 used for the stationary point of $Q_{2}$ at
$x_{0}=0$ yields that spectra of operators (7.28) contain asymptotically the
eigenvalues

$\lambda_{k,g,M}^{(x_{0})}=g^{-\frac{2}{M+1}}(\nu_{k,M}+\mathcal{O}(g^{-\frac{2}{M+1}})),\quad
g\to+\infty,$ (7.29)

where $\nu_{k,M}=(\tfrac{1}{M-1})^{\frac{2}{M+1}}\mu_{k,M}$, and
$\\{\mu_{k,M}\\}_{k}$ are (positive) eigenvalues of $-\partial_{x}^{2}+{\rm
i}x^{M-1}$, see Example 7.2. Notice that the leading term of the asymptotic
expansion of these eigenvalues is real and also that no such sequence is
obtained for $M=2$ when $Q_{2}(x)=x$ since the spectrum of imaginary Airy
operator is empty. Nonetheless, the (diverging) non-real eigenvalues found in
[13] are clearly visible in Figure 7.3 for $M=2$ and in similar plots for
higher $M$. To obtain asymptotics of these we use other stationary points of
the potential outside real axis.

Figure 7.3. $Q_{1}(x)=x^{4}/4$, $Q_{2}(x)=x$: Real (left) and imaginary
(right) part of the eigenvalues (red) of operators $T_{g}$ in (7.32) with
$g=5,10,\dots,500$. Asymptotic curves (blue) for $\lambda_{k,g}^{(x_{2})}$,
$\lambda_{k,g}^{(x_{3})}$ with $k=1,2,\dots,5$.

Consider first a simpler shifted oscillator $-\partial_{x}^{2}+x^{2}+2{\rm
i}gx$ where Theorem 7.3 is not applicable for the stationary point $x_{0}=0$
directly either. Nevertheless, writing $x^{2}+2{\rm i}gx=(x+{\rm
i}g)^{2}+g^{2}$ and the complex shift $x\mapsto x-{\rm i}g$, _i.e._ to the
complex stationary point $x_{1}=-{\rm i}g$, reveals the well-known diverging
eigenvalues $\\{2k+1+g^{2}\\}_{k\in\mathbb{N}_{0}}$. Notice that the complex
shift leaves the spectrum invariant by an argument similar to complex scaling.
Namely, the shift $x\mapsto x+\theta$ generates a holomorphic family (in
$\theta$) of operators of type A since the operator domains are constant,
moreover, for $\theta\in{\mathbb{R}}$, the spectra stay clearly invariant
(such shifts induce a unitary transform).

For operators (7.28), we first rescale $x\mapsto g^{2M/(M+1)}x$ to obtain

$\frac{1}{g^{\frac{2}{M+1}}}\left[-\partial_{x}^{2}+g^{2}\left(\frac{x^{2M}}{2M}+{\rm
i}\frac{x^{M-1}}{M-1}\right)\right]$ (7.30)

The stationary points of the potential read

$x_{0}=0,\qquad x_{k}={\rm e}^{{\rm i}\frac{4k-1}{2(M+1)}\pi},\quad
k=1,\dots,M+1.$ (7.31)

In particular for $M=2$, besides $x_{0}=0$, which was already covered above,
we have $x_{1}={\rm i}$, $x_{2}={\rm e}^{{\rm i}\frac{7}{6}\pi}$ and
$x_{3}={\rm e}^{{\rm i}\frac{11}{6}\pi}$. The shift to $x_{3}$ leads to the
operator

$T_{g}=\frac{1}{g^{\frac{2}{3}}}\left(-\partial_{x}^{2}+\frac{g^{2}}{4}\left[x^{2}(x+\sqrt{3})^{2}-{\rm
i}x^{2}(2x+3\sqrt{3})\right]+\frac{3}{4}g^{2}{\rm e}^{{\rm
i}\frac{\pi}{3}}\right)$ (7.32)

which is not directly covered by Theorem 7.3 as $g^{2}$ multiplies the whole
potential. Nonetheless, Theorem 7.3 can be generalized in a straightforward
way if the real part of the potential is non-negative and it yields that
eigenvalues

$\lambda_{k,g}^{(x_{3})}=\sqrt{\frac{3}{2}}g^{\frac{1}{3}}(\nu_{k}+\mathcal{O}_{k}(g^{-\frac{1}{6}}))+\frac{3}{4}{\rm
e}^{{\rm
i}\frac{5\pi}{3}}g^{\frac{4}{3}},\quad\lambda_{k,g}^{(x_{2})}=\overline{\lambda_{k,g}^{(x_{3})}},\quad
g\to+\infty,$ (7.33)

where $\nu_{k}={\rm e}^{{\rm i}\frac{\pi}{6}}(2k+1)$, $k\in\mathbb{N}_{0}$,
lie asymptotically in the spectra of $T_{g}$, see Figure 7.3 for illustration.
The shift to $x_{1}$ gives the potential with the quadratic term $-3x^{2}/2$
which does not correspond to a suitable limit operator.

The situation is more complicated for $M>2$, there are more stationary points
and in general the real part of the potential after the shift is not non-
negative (although bounded from below). Moreover, numerics suggests that only
two stationary points lead to diverging eigenvalues. Namely the points for
$k=\frac{M}{2}+1$ and $k=M+1$, _i.e._ ${\rm e}^{{\rm i}\frac{2M+3}{2M+2}}$ and
${\rm e}^{{\rm i}\frac{4M+3}{2M+2}}$ (the points where the shifted potential
has a global extreme of imaginary part). $\blacksquare$

##  Appendix A appendix

######  Lemma A.1.

Let Assumption 2.1 be satisfied and let $T=-\Delta+Q$ be the Schrödinger
operator defined as in Section 2.1. Then for each $\varepsilon_{1}>0$ there
exists $C\geq 0$, depending only on $\varepsilon_{1}$, $\varepsilon_{\nabla}$
and $M_{\nabla}$ such that for all $f\in{\operatorname{Dom}}(T)$

$\|Tf\|^{2}\geq\left(\frac{2-\varepsilon_{\nabla}(2+\sqrt{2})-\varepsilon_{1}}{2-\varepsilon_{\nabla}}\right)(\|\Delta
f\|^{2}+\|Qf\|^{2})-C_{\varepsilon_{1},\varepsilon_{\nabla},M_{\nabla}}\|f\|^{2},$
(A.1)

###### Proof.

By a standard approximation argument, see _e.g._ [30], it suffices to
establish (A.1) for $f\in{\operatorname{Dom}}(-\Delta_{\rm D})$ with a bounded
support. Integrating by parts we get

$\|(-\Delta+Q)f\|^{2}\geq\|\Delta f\|^{2}+\|Qf\|^{2}+2\langle\nabla
f,(\operatorname{Re}Q)\nabla f\rangle-2\langle|\nabla f|,|\nabla
Q||f|\rangle.$ (A.2)

Employing (2.1), Cauchy-Schwartz and Young inequalities we get (with
$\delta_{1}>0$)

$\displaystyle|\langle|\nabla f|,|\nabla Q||f|\rangle|$
$\displaystyle\leq\varepsilon_{\nabla}\langle|Q|^{\frac{1}{2}}|\nabla
f|,|f|\rangle+M_{\nabla}\langle|\nabla f|,|f|\rangle$ (A.3)
$\displaystyle\leq\varepsilon_{\nabla}\delta_{1}\||Q|^{\frac{1}{2}}\nabla
f\|^{2}+\frac{\varepsilon_{\nabla}}{4\delta_{1}}\|Qf\|^{2}+M_{\nabla}\|\nabla
f\|\|f\|.$

Next, integrating by parts and using that $|\nabla|Q||\leq|\nabla Q|$,

$\||Q|^{\frac{1}{2}}\nabla f\|^{2}=\langle|Q|\nabla f,\nabla
f\rangle\leq|\langle-\Delta f,|Q|f\rangle|+\langle|\nabla f|,|\nabla
Q||f|\rangle.$ (A.4)

Thus combining (A.3) and (A.4), using Young inequality (with $\delta_{2}>0$)
and

$\|\nabla f\|^{2}=\langle-\Delta f,f\rangle\leq\delta_{3}\|\Delta
f\|^{2}+\frac{1}{4\delta_{3}}\|f\|^{2},$ (A.5)

we obtain

$\displaystyle|\langle|\nabla f|,|\nabla Q||f|\rangle|$
$\displaystyle\leq\frac{1}{1-\delta_{1}\varepsilon_{\nabla}}\left(\delta_{1}\varepsilon_{\nabla}\|\Delta
f\|\|Qf\|+\frac{\varepsilon_{\nabla}}{4\delta_{1}}\|Qf\|^{2}+M_{\nabla}\|\nabla
f\|\|f\|\right)$
$\displaystyle\leq\frac{\varepsilon_{\nabla}\delta_{1}\delta_{2}+M_{\nabla}\delta_{3}}{1-\delta_{1}\varepsilon_{\nabla}}\|\Delta
f\|^{2}+\frac{\varepsilon_{\nabla}}{1-\delta_{1}\varepsilon_{\nabla}}\left(\frac{\delta_{1}}{4\delta_{2}}+\frac{1}{4\delta_{1}}\right)\|Qf\|^{2}$
$\displaystyle\quad+\frac{M_{\nabla}}{1-\delta_{1}\varepsilon_{\nabla}}\left(\frac{1}{4\delta_{3}}+\frac{1}{4}\right)\|f\|^{2}.$

Inserting the last inequality in (A.2) and since $\operatorname{Re}Q\geq 0$ by
assumption, we get

$\displaystyle\|Tf\|^{2}$
$\displaystyle\geq\left(1-\frac{2\varepsilon_{\nabla}\delta_{1}\delta_{2}}{1-\delta_{1}\varepsilon_{\nabla}}-\frac{2M_{\nabla}\delta_{3}}{1-\delta_{1}\varepsilon_{\nabla}}\right)\|\Delta
f\|^{2}$ (A.6)
$\displaystyle\quad+\left(1-\frac{2\varepsilon_{\nabla}}{1-\delta_{1}\varepsilon_{\nabla}}\left(\frac{\delta_{1}}{4\delta_{2}}+\frac{1}{4\delta_{1}}\right)\right)\|Qf\|^{2}-C\|f\|^{2}.$

Notice that $\delta_{3}>0$ can be chosen arbitrarily small and it is hidden in
$\varepsilon_{1}$ in (A.1). Simple manipulations show that (A.1) holds if both

$\varepsilon_{\nabla}<\frac{1}{\delta_{1}(1+2\delta_{2})},\qquad\varepsilon_{\nabla}<\frac{2\delta_{1}\delta_{2}}{\delta_{1}^{2}(1+2\delta_{2})+\delta_{2}}$
(A.7)

are satisfied (the first inequality implies that
$1-\delta_{1}\varepsilon_{\nabla}>0$). Equating the right sides of these
inequalities, we have $\delta_{1}^{2}(4\delta_{2}^{2}-1)=\delta_{2}$ and our
goal is to maximize $\delta_{1}(1+2\delta_{2})$. By a simple calculus we
obtain the values $\delta_{1}=1/2$ and $2\delta_{2}=1+\sqrt{2}$, which yields
the constants in (A.1) ∎

###### Proof of Theorem 2.4.

The crucial step in the proof is the following generalized weighted coercivity
of the form $t$ proved in [30, Thm. 3.3]. Let $w\in
W^{1,\infty}(\Omega;{\mathbb{R}})$, then for every $\alpha\in(0,1]$ and for
every $f\in{\operatorname{Dom}}(t)$

$\displaystyle\operatorname{Re}t(f,e^{2w}f)+\operatorname{Im}t(f,\Phi
e^{2w}f)\geq$ (A.8) $\displaystyle\qquad(1-\alpha)\|\nabla
e^{w}f\|^{2}+\int_{\Omega}\left(\widetilde{Q}_{\alpha}(x)-\left(1+\frac{2}{\alpha}\right)|\nabla
w(x)|^{2}\right)|e^{w(x)}f(x)|^{2}\,{{\rm d}}x,$

where
$\widetilde{Q}_{\alpha}=\Phi^{2}+\operatorname{Re}Q-\frac{1}{2\alpha}|\nabla\Phi|^{2}$
and $\Phi$ is as in (2.4). Hence taking $\alpha=1$,we get

$\operatorname{Re}t(f,e^{2w}f)+\operatorname{Im}t(f,\Phi
e^{2w}f)\geq\int_{\Omega}\left(\widetilde{Q}(x)-3|\nabla
w(x)|^{2}\right)|e^{w(x)}f(x)|^{2}\,{{\rm d}}x.$ (A.9)

Inserting $f=\psi$ in (A.9) leads to

$(\operatorname{Re}\lambda+|\operatorname{Im}\lambda|)\|e^{w}\psi\|^{2}+2\|e^{w}\psi\|\|e^{w}\psi_{0}\|\geq\int_{\Omega}(\widetilde{Q}(x)-3|\nabla
w(x)|^{2})|e^{w(x)}\psi(x)|^{2}\,{{\rm d}}x.$

In the next step, we approximate $W$ in a suitable way. To this end, we define

$\eta_{n}(s)=\begin{cases}s,&0\leq s\leq n,\\\ 2n-s,&n\leq s\leq 2n,\\\
0,&2n\leq s;\end{cases}$ (A.10)

notice that $\|\eta_{n}^{\prime}\|_{L^{\infty}({\mathbb{R}}_{+})}=1$ and
$w_{n}(x):=\eta_{n}(W(x))$, $x\in\Omega$, satisfy

$w_{n}\leq W,\quad|\nabla w_{n}|\leq|\nabla W|,\quad w_{n}\in
W^{1,\infty}(\Omega).$ (A.11)

Then using (2.15) and Young’s inequality with $\delta>0$ applied to
$2\|e^{w}\psi\|\|e^{w}\psi_{0}\|$, we arrive at (with
$\Omega_{2}:=\Omega\setminus\Omega_{1}$)

$\displaystyle\int_{\Omega_{1}}|e^{w_{n}(x)}\psi(x)|^{2}\,{{\rm d}}x$
$\displaystyle\
\leq\frac{|\operatorname{Re}\lambda|+|\operatorname{Im}\lambda|+\||\widetilde{Q}|+3|\nabla
W|^{2}\|_{L^{\infty}(\Omega_{2})}+\delta}{\delta}\|e^{W}\|_{L^{\infty}(\Omega_{2})}^{2}\|\psi\|^{2}+\frac{1}{\delta^{2}}\|e^{W}\psi_{0}\|^{2}.$

The claim (2.16) then follows by Fatou’s lemma and simple manipulations. ∎

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