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On the discrete spectrum of non-selfadjoint operators with applications to Schrödinger operators with complex potentials

Sabine Bögli, Sukrid Petpradittha

Abstract

For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial trace of the real part of the Birman--Schwinger operator, or an appropriate rotation thereof. As an application to Schrödinger operators, we generalise the Cwikel--Lieb--Rozenblum inequality to complex potentials and derive new Lieb--Thirring type inequalities.

On the discrete spectrum of non-selfadjoint operators with applications to Schrödinger operators with complex potentials

Abstract

For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial trace of the real part of the Birman--Schwinger operator, or an appropriate rotation thereof. As an application to Schrödinger operators, we generalise the Cwikel--Lieb--Rozenblum inequality to complex potentials and derive new Lieb--Thirring type inequalities.
Paper Structure (8 sections, 8 theorems, 85 equations)

This paper contains 8 sections, 8 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Abstract eigenvalue counting estimate
  3. Operators acting on antisymmetric tensor products
  4. Perturbation result to break up Jordan chains
  5. Proof of the abstract eigenvalue counting estimate
  6. Applications to non-selfadjoint Schrödinger operators
  7. CLR inequalities for complex potentials
  8. LT inequalities for complex potentials

Key Result

Theorem 2

Let $\varepsilon>0$ and $\alpha\in{\mathbb{R}}$. Then, for any $N\in\mathbb{N}$ such that there are $\lambda_{1},\dots,\lambda_{N}\in\sigma_{\rm d}(H)$ (repeated according to their algebraic multiplicities) with $\mathop\mathrm{Re}\nolimits\lambda_{j} + \alpha\mathop\mathrm{Im}\nolimits\lambda_{j}<- where $S:=(W(H_{0}+\varepsilon)^{-1/2})^{*}(W_{0}(H_{0}+\varepsilon)^{-1/2})$ is a compact operator

Theorems & Definitions (23)

  • Remark 1
  • Theorem 2
  • Lemma 3
  • proof
  • Remark 4
  • Lemma 5
  • Remark 6
  • proof : Proof of Lemma \ref{['lem:geometric bd']}
  • Lemma 7
  • proof
  • ...and 13 more