On Compressed Resolvents of Schr\"odinger Operators with Complex Potentials

The compression of the resolvent of a non-self-adjoint Schr\"odinger operator $-\Delta+V$ onto a subdomain $\Omega\subset\mathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $\Omega$, the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $\Omega$ and $\mathbb R^n\setminus\overline\Omega$ are being used. In a more abstract operator theory framework this topic is closely connected and very much inspired by the so-called coupling method that has been developed for the self-adjoint case by Henk de Snoo and his coauthors.


Introduction
Let V ∈ L ∞ (R n ), n ≥ 2, be a real or complex function and consider the Schrödinger operator (1.1) in L 2 (R n ). Note that, in general, this operator is non-self-adjoint in L 2 (R n ). Assume that Ω ⊂ R n is a bounded domain with C 2 -smooth boundary, let P Ω be the orthogonal projection in L 2 (R n ) onto L 2 (Ω), and denote by ι Ω the natural embedding of L 2 (Ω) into L 2 (R n ). The aim of this note is to derive an expression for the compression of the resolvent of A onto L 2 (Ω), that is, In the special case that the potential V is real the Schrödinger operator (1.1) is self-adjoint in L 2 (R n ) and hence, from a physical point of view, the operator A can be seen as the Hamiltonian of a closed quantum system. In this situation the compressed resolvent (1.2) can be interpreted as a family of resolvents of nonselfadjoint operators in L 2 (Ω) modeling an open quantum system, and vice versa the operator A in (1.1) can be viewed as the Hamiltonian describing the natural closed extension of an open quantum system. In the self-adjoint context it also follows from abstract operator theory principles that the compressed resolvent (1.2) can be described via the Kreȋn-Naȋmark formula or can be seen as aŠtraus family of extensions of a symmetric operator in L 2 (Ω), see, e.g., [3,Chapter 2.7], the contributions [4,11,12,13,14,20,21] by Henk de Snoo and his coauthors, and also the classical works [22,23,24,25,31]. However, it is of particular interest to determine the various operators and mappings that appear in the classical abstract Kreȋn-Naȋmark formula for the present case of a Schrödinger operator; for real potentials V an explicit expression for the compressed resolvent (1.2) was given in [3,Theorem 8.6.3] and for Lipschitz subdomains of Riemann manifolds in [1,Corollary 5.5].
The main purpose of this note is to show that also in the general case of a non-selfadjoint Schrödinger operator (1.1) (that is, the values of the potential V ∈ L ∞ (R n ) are not real a.e.) the compressed resolvent (1.2) is given by where A Ω = −∆ + V Ω is the Dirichlet realization in L 2 (Ω) and V Ω ∈ L ∞ (Ω) is the restriction of V onto Ω. Furthermore, M and τ turn out to be the (minus) λdependent Dirichlet-to-Neumann maps corresponding to the differential expression −∆ + V on Ω and on R n \ Ω, respectively, and γ Ω (λ) and γ Ω (λ) are closely related solution operators, also called Poisson operators in the theory of elliptic PDEs. Our analysis is strongly inspired by the abstract coupling method and other boundary triple techniques, which were originally developed for the self-adjoint case in [11,13] (see also [27,28,29,30] for dual pairs) and some more explicit preparatory results on extension theory of non-self-adjoint Schrödinger operators from [2]. We do not make an attempt here to develop a systematic study in the non-self-adjoint context, but instead we derive (1.3) in a goal-oriented way using mostly PDE-techniques such as trace maps, the second Green identity, and well posedness of boundary value problems. We also mention that the analysis and spectral theory of non-self-adjoint Schrödinger operators has attracted a lot of attention in the recent past. In particular, eigenvalue bounds, Lieb-Thirring inequalities, and other spectral properties of Schrödinger operators with complex potentials were derived in, e.g., [5,6,9,10,15,16,17,18,19,26]. The resolvent formula in Theorem 3.2 below and the compressed resolvent (1.2)-(1.3) are intimately connected with the spectral analysis of Schrödinger operators as in (1.1). Roughly speaking, the isolated eigenvalues of A coincide with the isolated singularities of the function λ → (M (λ) + τ (λ)) −1 and also other spectral data of A can be characterized with the limit behaviour of this function; cf. [7,8,29].
Acknowledgement. Jussi Behrndt gratefully acknowledges support for the Distinguished Visiting Austrian Chair at Stanford University by the Europe Center and the Freeman Spogli Institute for International Studies.
Data Availability Statement. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Preparations
Let Ω ⊂ R n , n ≥ 2, be a bounded domain with C 2 -smooth boundary Σ = ∂Ω and denote the outward normal unit vector by ν. Furthermore, let V Ω ∈ L ∞ (Ω) be a real or complex function. We consider the differential expressions which are formally adjoint to each other. Recall that the trace mapping C ∞ (Ω) ∋ f → {f | Σ , ∂ ν f Σ } can be extended to a continuous surjective mapping and that for f, g ∈ H 2 (Ω) the second Green identity in this context reads as here H 2 (Ω) and H t (Σ), t = 1 2 , 3 2 , denote the usual L 2 -based Sobolev spaces on Ω and Σ, respectively.
In the following the Dirichlet operators , will be useful. Both operators A Ω and A Ω are closed, densely defined in L 2 (Ω), and adjoint to each other, as Ω is a bounded domain it follows from the compactness of the embedding H 2 (Ω) ֒→ L 2 (Ω) that the resolvents of A Ω and A Ω are compact operators in L 2 (Ω). Note also that (2.4) implies λ ∈ ρ(A Ω ) if and only ifλ ∈ ρ( A Ω ).
A further important object in our study will be the (minus) Dirichlet-to-Neumann map M (·) corresponding to L Ω . Let again λ ∈ ρ(A Ω ) and ϕ ∈ H 3/2 (Σ). Then the operator M (λ) is defined by is the unique solution of the boundary value problem (2.5). The Dirichlet-to-Neumann map is an operator mapping H 3/2 (Σ) into H 1/2 (Σ), but can also be viewed as a densely defined unbounded (nonclosed) operator in L 2 (Σ).
Besides the domain Ω and the operators introduced above we shall also make use of their counterparts acting on the unbounded (exterior) domain Ω ′ := R n \ Ω with C 2 -smooth boundary Σ = ∂Ω ′ . The operators will be denoted in the same way, except that we shall use the subindex Ω ′ instead of Ω, e.g., A Ω ′ stands for the Dirichlet realization of L Ω ′ = −∆ + V Ω ′ in L 2 (Ω ′ ). For the (minus) Dirichlet-to-Neumann map we will use the symbol τ (·); note that the values τ (λ) are well defined for all λ ∈ ρ(A Ω ′ ). The above statements all remain valid on the unbounded domain, with the only exception that the embedding H 2 (Ω ′ ) ֒→ L 2 (Ω ′ ) is not compact and the resolvents of A Ω ′ and A Ω ′ are not compact in L 2 (Ω ′ ).
In our considerations we shall sometimes make use of a vector notation (f, f ′ ) ⊤ : R n → C for functions f : Ω → C and f ′ : Ω → C. The following simple observation will be useful in the proof of Theorem 3.2 in the next section.
Proof. The implication (⇒) is clear from the definition of the trace maps and the implication (⇐) can be viewed as a consequence of the self-adjointness of the Laplacian H = −∆ defined on dom H = H 2 (R n ) and the second Green identity. In fact, let g ∈ dom H and assume that f ∈ H 2 (Ω) and

A formula for the resolvent of the operator A
In this section we obtain a Kreȋn type formula for the resolvent of the nonself-adjoint Schrödinger operator in (1.1), and as an immediate consequence we conclude the form (1.2) of the compressed resolvent. The construction is based on the abstract coupling method developed in [11,13], but is made more explicit here in the context of differential operators.
In the following let A be the non-self-adjoint Schrödinger operator in (1.1), let A Ω and A Ω ′ be the Dirichlet realizations of L = −∆ + V in L 2 (Ω) and L 2 (Ω ′ ), respectively, and denote by M (·) and τ (·) the (minus) Dirichlet-to-Neumann maps on Ω and Ω ′ . The Dirichlet realizations of L = −∆ + V in L 2 (Ω) and L 2 (Ω ′ ) are denoted by A Ω and A Ω ′ , respectively. The next lemma is needed in the proof of our resolvent formula in Theorem 3.2 below. In the self-adjoint context this lemma was shown in [3, Lemma 8.6.1]. As the proof remains the same in the general non-self-adjoint situation we do not repeat it here. is bijective.
For our purposes it is convenient to use the notation A Ω,Ω ′ := A Ω × A Ω ′ and we regard A Ω,Ω ′ as a closed operator in L 2 (R n ) = L 2 (Ω) × L 2 (Ω ′ ). Note that ρ(A Ω,Ω ′ ) = ρ(A Ω ) ∩ ρ(A Ω ′ ) and Furthermore, we set and The next theorem is the main result of this note. We express the resolvent of the Schrödinger operator A in terms of the resolvent of the orthogonal sum A Ω,Ω ′ of the Dirichlet realizations and a perturbation term, which contains the Dirichletto-Neumann maps M (·) and τ (·), the solution operators γ Ω (·) and γ Ω ′ (·), and their adjoints. In particular, since the solutions operators are analytic on the resolvent sets ρ(A Ω ) and ρ(A Ω ′ ), respectively, it follows that the poles of the resolvent of A (and hence also the isolated eigenvalues) in ρ(A Ω ) ∩ ρ(A Ω ′ ) coincide with the isolated singularities of the function Θ(·) in (3.3).