Exploring the Pseudo-modes of Schr\"odinger Operators with Complex Potentials: A Focus on Resolvent Norm Estimates and Spectral Stability

TL;DR

This paper investigates the pseudo-modes and spectral stability of one-dimensional Schr"odinger operators with complex potentials, focusing on resolvent norm behavior along specific complex plane curves and confirming spectral instability through numerical pseudo-eigenvalue computations.

Contribution

It narrows the analysis to the case where p=1/3, providing new insights into resolvent norm estimates and spectral stability for complex Schr"odinger operators.

Findings

Resolvent norm exhibits specific growth behavior along chosen curves.

Numerical pseudo-eigenvalues confirm spectral instability.

Focus on the case p=1/3 advances understanding of spectral properties.

Abstract

This paper aims to investigate the pseudo-modes of the one-dimensional Schr\"odinger operator with complex potentials, focusing on the behavior of the resolvent norm along specific curves in the complex plane and assessing the stability of the spectrum under small perturbations. The study builds upon previous work of E.B. Davies, L.S. Boulton, and N. Trefethen, specifically examining the resolvent norm of the complex harmonic oscillator along curves of the form $z_{\eta }= b\eta + c\eta ^{p} $ where $ b > 0$, $ \frac{1}{3}< p <3 $ independent of $\eta> 0$. The present work narrows the focus to the case where $p = \frac{1}{3}$. Numerical computations of pseudo-eigenvalues are performed to verify spectral instability.