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Spectral estimates for resolvent differences of self-adjoint elliptic operators

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik

Abstract

The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with $δ$ and $δ'$-potentials supported on hypersurfaces.

Spectral estimates for resolvent differences of self-adjoint elliptic operators

Abstract

The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with and -potentials supported on hypersurfaces.

Paper Structure

This paper contains 13 sections, 39 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and linear relations
  4. Operator ideals and singular values
  5. Quasi boundary triples and Krein's formula
  6. Quasi boundary triples, $\gamma$-fields and Weyl functions
  7. Krein's formula and self-adjoint extensions
  8. Resolvent differences in operator ideals
  9. Dissipative and accumulative realizations
  10. Self-adjoint elliptic operators and spectral estimates for resolvent differences
  11. Quasi boundary triples and Weyl functions for second order elliptic differential expressions
  12. Spectral estimates for resolvent differences of self-adjoint elliptic operators on bounded and exterior domains
  13. Elliptic operators with $\delta$ and $\delta'$-interactions on smooth hypersurfaces

Key Result

Lemma \oldthetheorem

Let ${\mathfrak A}$ be a class of operator ideals. Moreover, let $H$ and $K$ be closed linear relations in a separable Hilbert space ${\mathcal{H}}$. If for some $\lambda\in\rho(H)\cap\rho(K)$, then resdiffideal holds for all $\lambda\in\rho(H)\cap\rho(K)$.

Theorems & Definitions (80)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • ...and 70 more