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

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.