Spectral enclosures for non-self-adjoint extensions of symmetric operators

To recall the notions of ordinary and quasi boundary triples in more detail, let $S$ be a densely defined, closed, symmetric operator in a Hilbert space $({\mathcal{H}},(\cdot,\cdot)_{\mathcal{H}})$ and let $S^{*}$ denote its adjoint; then $\{{\mathcal{G}},\Gamma_{0},\Gamma_{1}\}$ is said to be an ordinary boundary triple for $S^{*}$ if $\Gamma_{0},\Gamma_{1}:\operatorname{dom}S^{*}\rightarrow{\mathcal{G}}$ are linear mappings from the domain of $S^{*}$ into an auxiliary Hilbert space $({\mathcal{G}},(\cdot,\cdot)_{\mathcal{G}})$ that satisfy the abstract Lagrange or Green identity

and a certain maximality condition. The corresponding Weyl function $M$ is an operator-valued function in ${\mathcal{G}}$, which is defined by

where $A_{0}=S^{*}\upharpoonright\ker\Gamma_{0}$ is a self-adjoint operator in ${\mathcal{H}}$. For a singular Sturm–Liouville expression $-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}x^{2}}+V$ in $L^{2}(0,\infty)$ with a real-valued potential $V\in L^{\infty}(0,\infty)$ the operators $S$ and $S^{*}$ can be chosen as the minimal and maximal operators, respectively, together with ${\mathcal{G}}={\mathbb{C}}$ and $\Gamma_{0}f=f(0)$, $\Gamma_{1}f=f^{\prime}(0)$ for $f\in\operatorname{dom}S^{*}$; in this case the corresponding abstract Weyl function coincides with the classical Titchmarsh–Weyl $m$-function.

The notion of quasi boundary triples is a natural generalization of the concept above, inspired by, and developed for, the treatment of elliptic differential operators. The main difference is, that the boundary maps $\Gamma_{0}$ and $\Gamma_{1}$ are only defined on a subspace $\operatorname{dom}T$ of $\operatorname{dom}S^{*}$, where $T$ is an operator in ${\mathcal{H}}$ which satisfies $\overline{T}=S^{*}$. The identities (1.1) and (1.2) are only required to hold for elements in $\operatorname{dom}T$; see Section 2 for precise definitions. For the Schrödinger operator $-\Delta+V$ in $L^{2}(\Omega)$ with a real-valued potential $V\in L^{\infty}(\Omega)$ on a domain $\Omega\subset{\mathbb{R}}^{n}$ with a sufficiently regular boundary $\partial\Omega$, the operators $S$ and $S^{*}$ can again be taken as the minimal and maximal operator, respectively, and a convenient choice for the domain of $T=-\Delta+V$ is $H^{2}(\Omega)$. Then ${\mathcal{G}}=L^{2}(\partial\Omega)$ and $\Gamma_{0}f=\partial_{\nu}f|_{\partial\Omega}$, $\Gamma_{1}f=f|_{\partial\Omega}$ (where the latter denote the normal derivative and trace) form a quasi boundary triple, and the corresponding Weyl function is the energy-dependent Neumann-to-Dirichlet map.

The main focus of this paper is on non-self-adjoint extensions of $S$ that are restrictions of $S^{*}$ parameterized by an ordinary or quasi boundary triple and an (in general non-self-adjoint) boundary parameter, and to describe their spectral properties. For a quasi boundary triple $\{{\mathcal{G}},\Gamma_{0},\Gamma_{1}\}$ and a linear operator $B$ in ${\mathcal{G}}$ we consider the operator

in ${\mathcal{H}}$. The principal results of this paper include (a) a sufficient condition for $A_{[B]}$ to be m-sectorial and (b) enclosures for the numerical range and the spectrum of the operator $A_{[B]}$ in parabola-type regions. The latter make use of decay properties of the Weyl function $M$ along the negative half-axis or inside sectors in the complex plane; in order to make these results easily applicable, we provide (c) an abstract sufficient condition for the Weyl function to decay appropriately. We point out that, to the best of our knowledge, these results are also new in the special case of ordinary boundary triples. While the operator $A_{[B]}$ can be regarded as a perturbation of the self-adjoint operator $A_{0}$ in the resolvent sense, let us mention that the spectra of additive non-self-adjoint perturbations of self-adjoint operators were studied recently in, e.g. [48, 49, 50, 51, 71]. In the second half of the present paper, we provide applications of these results to several classes of operators, namely to elliptic differential operators with local and non-local Robin boundary conditions on domains with possibly non-compact boundaries, to Schrödinger operators with $\delta$-interactions of complex strength supported on hypersurfaces, to infinitely many point $\delta$-interactions on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

Let us explain in more detail the structure, methodology, and results of this paper. After the preliminary Section 2, our first main result is Theorem 3.1, where it is shown that, under certain assumptions on the Weyl function and the boundary parameter $B$, the operator $A_{[B]}$ in (1.3) is sectorial, and a sector containing the numerical range of $A_{[B]}$ is specified. However, in applications it is essential to ensure that a sectorial operator is m-sectorial; hence the next main objective is to prove that the resolvent set of the operator $A_{[B]}$ in (1.3) is non-empty, which is a non-trivial question particularly for quasi boundary triples. This problem is treated in Section 4. The principal result here is Theorem 4.1, in which we provide sufficient conditions for $\lambda_{0}\in\rho(A_{[B]})$ in terms of the operator $M(\lambda_{0})$ and the parameter $B$. In this context also a Krein-type resolvent formula is obtained, and the adjoint of $A_{[B]}$ is related to a dual parameter $B^{\prime}$; cf. [27, 29] for the special case of symmetric $B$. We list various corollaries of Theorem 4.1 for more specialized situations. We point out that an alternative description of sectorial and m-sectorial extensions of a symmetric operator can be found in [14, 114]; see also the review article [15] and [16, 17, 18]. Section 4 is complemented by two propositions on Schatten–von Neumann properties for the resolvent difference of $A_{[B]}$ and $A_{0}$; cf. [27, 55] for related abstract results and, e.g. [21, 26, 33, 85, 112, 115] for applications to differential operators. Such estimates can be used, for instance, to get bounds on the discrete spectrum of $A_{[B]}$; cf. [51]. In Section 5 we consider the situation when the Weyl function $M$ converges to $0$ in norm along the negative half-axis or in some sector in the complex plane. The most important result in this section is Theorem 5.6 where, under the assumption that $\|M(\lambda)\|$ decays like a power of $\frac{1}{|\lambda|}$, the numerical range and the spectrum of $A_{[B]}$ are contained in a parabola-type region. Spectral enclosures of this type with more restrictive assumptions on $B$ were obtained for elliptic partial differential operators in [19, 20, 73]; similar enclosures for Schrödinger operators with complex-valued regular potentials can be found in [2, 71, 108]. They also appear in the abstract settings of so-called $p$-subordinate perturbations [131]. Finally, as the last topic within the abstract part of this paper, we prove in Theorem 6.1 that the Weyl function decays along the negative real line or in suitable complex sectors with a certain rate if the map $\Gamma_{1}|A_{0}-\mu|^{-\alpha}$ is bounded for some $\alpha\in(0,\tfrac{1}{2}]$ and some $\mu\in\rho(A_{0})$, where the rate of the decay depends on $\alpha$. Example 6.4 shows the sharpness of this result.

Our abstract results are applied in Section 7 to elliptic partial differential operators with (in general non-local) Robin boundary conditions on domains with possibly non-compact boundaries; the class of admissible unbounded domains includes, for instance, domains of waveguide-type as considered in [34, 67]. In Section 8 we apply our abstract results to Schrödinger operators in ${\mathbb{R}}^{n}$ with $\delta$-potentials of complex strength supported on (not necessarily bounded) hypersurfaces. We indicate also how our abstract methods can be combined with very recent norm estimates from [75] in order to obtain further spectral enclosures and to establish absence of non-real spectrum for ‘weak’ complex $\delta$-interactions in space dimensions $n\geq 3$ for compact hypersurfaces. Finally, we apply our machinery to Schrödinger operators on the real line with non-Hermitian $\delta$-interactions supported on infinitely many points in Section 9, and to Laplacians on finite (not necessarily compact) graphs with non-self-adjoint vertex couplings in Section 10. Each of these sections has the same structure: after the problem under consideration is explained, first a quasi (or ordinary) boundary triple and its Weyl function are provided; next a lemma on the decay of the Weyl function is proved, and then a main result on spectral properties and enclosures is formulated, which can be derived easily from that decay together with the abstract results in the first part of this paper in each particular situation. To illustrate the different types of boundary conditions and interactions, more specialized cases and explicit examples are included in Sections 7–10.

Finally, let us fix some notation. By $\sqrt{\cdot}$ we denote the branch of the complex square root such that $\operatorname{Im}\sqrt{\lambda}>0$ for all $\lambda\in{\mathbb{C}}\setminus[0,\infty)$. Let us set ${\mathbb{R}}_{+}\mathrel{\mathop{:}}=[0,\infty)$ and ${\mathbb{C}}^{\pm}\mathrel{\mathop{:}}=\{\lambda\in{\mathbb{C}}:\pm\operatorname{Im}\lambda>0\}$. Moreover, for any bounded, complex-valued function $\alpha$ we use the abbreviation $\|\alpha\|_{\infty}\mathrel{\mathop{:}}=\sup|\alpha|$. The space of bounded, everywhere defined operators from a Hilbert space ${\mathcal{H}}_{1}$ to another Hilbert space ${\mathcal{H}}_{2}$ is denoted by ${\mathcal{B}}({\mathcal{H}}_{1},{\mathcal{H}}_{2})$, and we set ${\mathcal{B}}({\mathcal{H}}_{1})\mathrel{\mathop{:}}={\mathcal{B}}({\mathcal{H}}_{1},{\mathcal{H}}_{1})$. The Schatten–von Neumann ideal that consists of all compact operators from ${\mathcal{H}}_{1}$ to ${\mathcal{H}}_{2}$ whose singular values are $p$-summable is denoted by ${\mathfrak{S}}_{p}({\mathcal{H}}_{1},{\mathcal{H}}_{2})$, and we set ${\mathfrak{S}}_{p}({\mathcal{H}}_{1})\mathrel{\mathop{:}}={\mathfrak{S}}_{p}({\mathcal{H}}_{1},{\mathcal{H}}_{1})$; see, e.g. [81] for a detailed study of the ${\mathfrak{S}}_{p}$-classes. Furthermore, for each densely defined operator $A$ in a Hilbert space we write $\operatorname{Re}A\mathrel{\mathop{:}}=\frac{1}{2}(A+A^{*})$ and $\operatorname{Im}A\mathrel{\mathop{:}}=\frac{1}{2i}(A-A^{*})$ for its real and imaginary part, respectively, and, if $A$ is closed, we denote by $\rho(A)$ and $\sigma(A)$ its resolvent set and spectrum, respectively.

In the following let $S$ be a densely defined, closed, symmetric operator in a Hilbert space ${\mathcal{H}}$.

The notion of quasi boundary triples was introduced in [22, Definition 2.1]. The concept of generalized boundary triples appeared first in [56, Definition 6.1]. It follows from [56, Lemma 6.1] that each generalized boundary triple is also a quasi boundary triple. We remark that the converse is in general not true. A quasi or generalized boundary triple reduces to an ordinary boundary triple if the map $\Gamma$ in condition (ii) is onto (see [22, Corollary 3.2]). In this case $T$ is closed and coincides with $S^{*}$, and $A_{0}$ in condition (iii) is automatically self-adjoint. For the convenience of the reader we recall the usual definition of ordinary boundary triples.

We refer the reader to [22, 24] for a detailed study of quasi boundary triples, to [52, 56] for generalized boundary triples and to [43, 44, 55, 82, 96] for ordinary boundary triples. For later purposes we recall the following result, which is useful to determine the adjoint and a (quasi) boundary triple for a given symmetric operator; see [22, Theorem 2.3].

In the following let $\{{\mathcal{G}},\Gamma_{0},\Gamma_{1}\}$ be a quasi boundary triple for $T\subset S^{*}$. Since $A_{0}=T\upharpoonright\ker\Gamma_{0}$ is self-adjoint, we have ${\mathbb{C}}\setminus{\mathbb{R}}\subset\rho(A_{0})$, and for each $\lambda\in\rho(A_{0})$ the direct sum decomposition

holds. In particular, the restriction of the map $\Gamma_{0}$ to $\ker(T-\lambda)$ is injective. This allows the following definition.

The values $\gamma(\lambda)$ of the $\gamma$-field are operators defined on the dense subspace $\operatorname{ran}\Gamma_{0}\subset{\mathcal{G}}$ which map onto $\ker(T-\lambda)\subset{\mathcal{H}}$. The values $M(\lambda)$ of the Weyl function are densely defined operators in ${\mathcal{G}}$ mapping $\operatorname{ran}\Gamma_{0}$ into $\operatorname{ran}\Gamma_{1}$. In particular, if $\{\mathcal{G},\Gamma_{0},\Gamma_{1}\}$ is a generalized or ordinary boundary triple, then $\gamma(\lambda)$ and $M(\lambda)$ are defined on ${\mathcal{G}}=\operatorname{ran}\Gamma_{0}$, and it can be shown that $\gamma(\lambda)\in{\mathcal{B}}({\mathcal{G}},{\mathcal{H}})$ and $M(\lambda)\in{\mathcal{B}}({\mathcal{G}})$ in this case.

Next we list some important properties of the $\gamma$-field and the Weyl function corresponding to a quasi boundary triple $\{\mathcal{G},\Gamma_{0},\Gamma_{1}\}$, which can be found in [22, Proposition 2.6] or [24, Propositions 6.13 and 6.14]. These properties are well known for the $\gamma$-field and Weyl function corresponding to a generalized or ordinary boundary triple. Let $\lambda,\mu\in\rho(A_{0})$. Then the adjoint operator $\gamma(\lambda)^{*}$ is bounded and satisfies

hence also $\gamma(\lambda)$ is bounded and $\overline{\gamma(\lambda)}=\gamma(\lambda)^{**}\in{\mathcal{B}}({\mathcal{G}},{\mathcal{H}})$. One has the useful identity

for $\lambda,\mu\in\rho(A_{0})$, which implies

With the help of the functional calculus of the self-adjoint operator $A_{0}$ one can conclude from (2.5) that

The values $M(\lambda)$ of the Weyl function satisfy $M(\lambda)\subset M(\overline{\lambda})^{*}$ and, in particular, the operators $M(\lambda)$ are closable. In general, the operators $M(\lambda)$ and their closures $\overline{M(\lambda)}$ are not bounded. However, if $M(\lambda_{0})$ is bounded for some $\lambda_{0}\in\rho(A_{0})$, then $M(\lambda)$ is bounded for all $\lambda\in\rho(A_{0})$; see Lemma 2.5 below. The function $\lambda\mapsto M(\lambda)$ is holomorphic in the sense that for any fixed $\mu\in\rho(A_{0})$ it can be written as the sum of the possibly unbounded operator $\operatorname{Re}M(\mu)$ and a ${\mathcal{B}}({\mathcal{G}})$-valued holomorphic function,

for all $\lambda\in\rho(A_{0})$. In particular, $\operatorname{Im}M(\lambda)$ is a bounded operator for each $\lambda\in\rho(A_{0})$.

Further, for every $x\in\operatorname{ran}\Gamma_{0}$ we have

for all $\lambda\in\rho(A_{0})$ and all $n\in\mathbb{N}$, and hence the $n$th strong derivative $M^{(n)}(\lambda)$ (viewed as an operator defined on $\operatorname{ran}\Gamma_{0}$) admits a continuous extension $\overline{M^{(n)}(\lambda)}\in{\mathcal{B}}({\mathcal{G}})$. It satisfies

see [28, Lemma 2.4 (iii)].

The Weyl function also satisfies (see [22, Proposition 2.6 (v)])

and with $\mu=\overline{\lambda}$ and the relation $M(\overline{\lambda})\subset M(\lambda)^{*}$ it follows that

In the case when the values of $M$ are bounded operators we provide a simple bound for the norms $\|\overline{M(\lambda)}\|$ in the next lemma.

Decay properties of the Weyl function play an important role in this paper. The next lemma shows that a decay of the Weyl function along a non-real ray implies a uniform decay in certain sectors.

In the context of the previous lemma we remark that $\lambda\mapsto\|\overline{M(\lambda)}\|$ decays at most as $|\lambda|^{-1}$ since $\lambda\mapsto-(M(\lambda)x,x)^{-1}$ grows at most linearly as it is a Nevanlinna function for every $x\in\operatorname{ran}\Gamma_{0}$. We also recall from [29, Lemma 2.3] that for $x\in\operatorname{ran}\Gamma_{0}\setminus\{0\}$ the function

is strictly increasing on each interval in $\rho(A_{0})\cap\mathbb{R}$; moreover, if $A_{0}$ is bounded from below and

for all $x\in\operatorname{ran}\Gamma_{0}$, then

In the next proposition the case when the self-adjoint operator $A_{0}=T\upharpoonright\ker\Gamma_{0}$ is bounded from below and $\|\overline{M(\lambda)}\|\to 0$ as $\lambda\to-\infty$ is considered. Here the extension

is investigated. Observe that the abstract Green identity (2.2) yields that $A_{1}$ is symmetric in ${\mathcal{H}}$, but in the setting of quasi boundary triples or generalized boundary triples $A_{1}$ is not necessarily self-adjoint (in contrast to the case of ordinary boundary triples).

In the next theorem it is shown that under additional assumptions on $B$ and the Weyl function $M$ that corresponds to $\{{\mathcal{G}},\Gamma_{0},\Gamma_{1}\}$ the operator $A_{[B]}$ is sectorial. Recall first that the numerical range, $W(A)$, of a linear operator $A$ is defined as

and that $A$ is called sectorial if $W(A)$ is contained in a sector of the form

for some $\eta\in\mathbb{R}$ and $\kappa>0$. An operator $A$ is called m-sectorial if $W(A)$ is contained in a sector (3.2) and the complement of (3.2) has a non-trivial intersection with $\rho(A)$. In this case the spectrum of $A$ is contained in the closure of $W(A)$; see, e.g. [125, Propositions 2.8 and 3.19]. Note that if $A$ is m-sectorial, then $-A$ generates an analytic semigroup; see, e.g. [95, Theorem IX.1.24].