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Trace formulae for Schrödinger operators with singular interactions

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik

Abstract

Let $Σ\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $Ω_\pm$. In this note self-adjoint Schrödinger operators with $δ$ and $δ'$-interactions supported on $Σ$ are studied. For large enough $m\in\mathbb{N}$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Σ)$.

Trace formulae for Schrödinger operators with singular interactions

Abstract

Let be a -smooth closed compact hypersurface, which splits the Euclidean space into two domains . In this note self-adjoint Schrödinger operators with and -interactions supported on are studied. For large enough the difference of th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in is written in terms of Neumann-to-Dirichlet maps on the boundary space .

Paper Structure

This paper contains 12 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Traces, Neumann-to-Dirichlet maps and some operator functions
  3. Statement of the main results
  4. Preliminaries
  5. ${\mathfrak S}_{p,\infty}$-classes and the trace mapping
  6. Derivatives of holomorphic operator-valued functions
  7. Quasi boundary triples, Weyl functions and $\gamma$-fields
  8. Self-adjoint extensions and abstract Krein's resolvent formula
  9. Quasi boundary triples for coupled problems
  10. Proofs of the main results
  11. Proof of Theorem \ref{['thm1']}
  12. Proof of Theorem \ref{['thm2']}

Key Result

Theorem \oldthetheorem

Let the self-adjoint operators $\mathsf{H}_{\rm free}$ and $\mathsf{H}_{\alpha,\Sigma}$ with $\alpha \in L^\infty(\Sigma;{\mathbb R})$ be as in Definition def:Ops, and let the operator-valued function $\widetilde{M}$ be as in def:M. Then for all $m \in {\mathbb N}$ such that $m > \frac{d-2}{2}$ and belongs to the trace class, and its trace can be expressed as

Theorems & Definitions (19)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 9 more