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

2025/05/16 by Sameh Gana, Gana, Sameh
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Brake Systems and Friction Analysis #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.11338

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper aims to investigate the pseudo-modes of the one-dimensional Schrödinger 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_η= bη+ cη^p where b > 0, (1)/(3)< p <3 independent of η> 0. The present work narrows the focus to the case where p = (1)/(3). Numerical computations of pseudo-eigenvalues are performed to verify spectral instability.

Citations

• Numerical computation of spectral solutions for Sturm-Liouville eigenvalue problems
• Classes of second order nonlinear partial differential equations reducible to first order
• A MATLAB differentiation matrix suite
• Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra

Related

• Schrödinger operator with a complex steplike potential
• Resolvent estimates for one-dimensional Schrödinger operators with complex potentials
• The Pseudospectral Properties of non-self-adjoint Schrödinger Operators in the semi-classical limit
• Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra
• Spectral Instability for Some Schroedinger Operators