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The Spectral Shift Function for Non-Self-Adjoint Perturbations

Vincent Bruneau, Nicolas Frantz, François Nicoleau

Abstract

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{ö}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.

The Spectral Shift Function for Non-Self-Adjoint Perturbations

Abstract

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{ö}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.
Paper Structure (36 sections, 27 theorems, 228 equations)

This paper contains 36 sections, 27 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Abstract setting
  4. Assumptions
  5. Main results
  6. Functional Calculus
  7. Helffer-Sjöstrand formula
  8. Decomposition
  9. Definition and properties
  10. Comparison with Frantz-Faupin calculus
  11. Spectral changing of variables
  12. SSF for Trace class perturbations
  13. Existence
  14. A first representation formula for the SSF
  15. The SSF for non-selfadjoint relatively trace class perturbations
  16. ...and 21 more sections

Key Result

Theorem 2.2

Assume that Hypotheses 1, 2, and 3 hold on an open interval $I$ with $m=-1$. Then for every $f \in \mathcal{D}(I)$ the operator difference $f(H)-f(H_0)$ belongs to $\mathcal{L}_1(\mathcal{H})$ and the map defines a distribution on $I$. The spectral shift function $\xi(\cdot; H, H_0)$ is defined, up to an additive constant, by Moreover, it satisfies where for $z \in \rho(H) \cap \rho(H_0)$,

Theorems & Definitions (63)

  • Remark 2.1
  • Theorem 2.2: Existence of the SSF
  • Theorem 2.3: Relatively trace-class perturbations
  • Definition 3.1: Almost-analytic extension
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Example 3.5
  • ...and 53 more