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Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

Antonio Arnal, Jussi Behrndt, Markus Holzmann, Petr Siegl

TL;DR

This work extends generalized boundary triples to adjoint pairs, enabling Krein-type resolvent formulas for non-self-adjoint Schrödinger operators with complex potentials on Lipschitz domains. By defining $\gamma$-fields and Weyl functions for adjoint-pair triples, it provides Birman–Schwinger criteria for eigenvalues and explicit resolvent formulas for Robin-type boundary conditions. The abstract framework is then applied to $-\Delta+V$ with $V\in L^p(\Omega)$, showing Robin realizations are closed with nonempty resolvent and establishing decay and regularity properties of the Weyl function, yielding practical Krein formulas. These results offer a systematic approach to non-self-adjoint boundary value problems on irregular domains, with explicit operator and form-theoretic characterizations in terms of $H^{3/2}_\Delta(\Omega)$-regularity and boundary traces.

Abstract

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex $L^p$-potentials on bounded and unbounded Lipschitz domains with compact boundaries.

Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

TL;DR

This work extends generalized boundary triples to adjoint pairs, enabling Krein-type resolvent formulas for non-self-adjoint Schrödinger operators with complex potentials on Lipschitz domains. By defining -fields and Weyl functions for adjoint-pair triples, it provides Birman–Schwinger criteria for eigenvalues and explicit resolvent formulas for Robin-type boundary conditions. The abstract framework is then applied to with , showing Robin realizations are closed with nonempty resolvent and establishing decay and regularity properties of the Weyl function, yielding practical Krein formulas. These results offer a systematic approach to non-self-adjoint boundary value problems on irregular domains, with explicit operator and form-theoretic characterizations in terms of -regularity and boundary traces.

Abstract

We extend the notion of generalized boundary triples and their Weyl functions from extension theory of symmetric operators to adjoint pairs of operators, and we provide criteria on the boundary parameters to induce closed operators with a nonempty resolvent set. The abstract results are applied to Schrödinger operators with complex -potentials on bounded and unbounded Lipschitz domains with compact boundaries.

Paper Structure

This paper contains 7 sections, 15 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Generalized boundary triples for adjoint pairs
  3. Generalized boundary triples
  4. $\gamma$-fields and Weyl functions
  5. Closed extensions and their resolvents
  6. Schrödinger operators with unbounded complex potentials on Lipschitz domains with compact boundaries
  7. Dirichlet and Neumann trace maps on Lipschitz domains with compact boundary

Key Result

Theorem 2.2

Let ${\mathcal{H}}$ and ${\mathcal{G}}$ be Hilbert spaces, let $T$ and $\widetilde{T}$ be operators in ${\mathcal{H}}$, and assume that there are linear mappings such that the following holds: Then are closed operators in ${\mathcal{H}}$ and form an adjoint pair $\{S,\widetilde{S}\}$ such that $\{T,\widetilde{T}\}$ is a core of $\{S^*,\widetilde{S}^*\}$, and $\{{\mathcal{G}},(\Gamma_0,\Gamma_1)

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • proof : Sketch of the proof of (i)
  • Lemma 3.2
  • ...and 14 more