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Bounds on complex eigenvalues and resonances

A. A. Abramov, A. Aslanyan, E. B. Davies

Abstract

We obtain bounds on the complex eigenvalues of non-self-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information.

Bounds on complex eigenvalues and resonances

Abstract

We obtain bounds on the complex eigenvalues of non-self-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information.

Paper Structure

This paper contains 13 sections, 11 theorems, 98 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Bounds on eigenvalues using complex scaling
  3. Resolvent bounds on complex eigenvalues
  4. Complex scaling and resonance bounds
  5. Perturbation formulae for resonances
  6. Rapidly decaying potentials
  7. Existence and uniqueness of the resonance solution
  8. First method
  9. Second method
  10. Examples
  11. Gaussian potential
  12. A slowly decaying modified Gaussian potential
  13. The potential of Rittby et al.

Key Result

Theorem 1

Under the above assumptions on $V$ the eigenvalues $\lambda$ of $H$ which do not satisfy $-\alpha \leq \arg(\lambda)\leq 0$ all lie in the compact convex set and their only possible accumulation point is at $0$.

Figures (1)

  • Figure 1: Resonances of $H_b,\,b=10,$ and the set $S$

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 10 more