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Spectral Instability for Some Schroedinger Operators

A. Aslanyan, E. B. Davies

TL;DR

This work defines and analyzes the instability index $\kappa(\lambda)$ for isolated eigenvalues of non-self-adjoint Schrödinger operators, proving that it equals the norm of the spectral projection and connecting it to pseudospectral properties. It then develops a numerically stable framework for computing $\kappa(\lambda)$ in 1D, using boundary-transfer transfer functions to locate eigenvalues and auxiliary ODEs to evaluate the index without oscillatory integrals. The authors apply the method to the complex harmonic oscillator and to dilation-analytic resonances, reporting that the instability indices grow exponentially with the eigenvalue index, and that large eigenvalues and resonances present intrinsic computational challenges. The results illuminate the limits of pseudospectral reasoning for large non-self-adjoint problems and provide practical, robust tools for studying complex resonances and spectral instability in quantum-type operators.

Abstract

We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schroedinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential.

Spectral Instability for Some Schroedinger Operators

TL;DR

This work defines and analyzes the instability index for isolated eigenvalues of non-self-adjoint Schrödinger operators, proving that it equals the norm of the spectral projection and connecting it to pseudospectral properties. It then develops a numerically stable framework for computing in 1D, using boundary-transfer transfer functions to locate eigenvalues and auxiliary ODEs to evaluate the index without oscillatory integrals. The authors apply the method to the complex harmonic oscillator and to dilation-analytic resonances, reporting that the instability indices grow exponentially with the eigenvalue index, and that large eigenvalues and resonances present intrinsic computational challenges. The results illuminate the limits of pseudospectral reasoning for large non-self-adjoint problems and provide practical, robust tools for studying complex resonances and spectral instability in quantum-type operators.

Abstract

We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schroedinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential.

Paper Structure

This paper contains 16 sections, 6 theorems, 76 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Instability Index
  3. The Computational Procedures
  4. Finding Eigenvalues
  5. Calculating the Instability Index
  6. Possible Difficulties
  7. The Harmonic Oscillator
  8. Basic Facts
  9. Perturbations of the Operator $H_o$
  10. Another Approach. Instability Index
  11. Complex Resonances
  12. Definitions
  13. Location of Resonances
  14. Numerical Range and Complex Resonances
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Lemma 1

If $f$ and $f^{\ast}$ are the normalised eigenvectors of $A$ and $A^{\ast}$ associated with the eigenvalues $\lambda$ and $\overline{\lambda}$ respectively, and if $P$ has rank $1$, then ${\langle} f,f^{\ast}{\rangle} \not= 0$ and the instability index of $\lambda$ is equal to $\Vert P \Vert$.

Figures (6)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Theorem 3