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

Sameh Gana

TL;DR

The paper analyzes the ε-pseudospectrum of the non-self-adjoint complex harmonic oscillator $H_c=-\dfrac{d^{2}}{dx^{2}}+cx^{2}$ with $Re(c)>0$, $Im(c)\neq0$, focusing on resolvent-norm behavior along curves $z_{\eta}=b\eta+c\eta^{p}$ and, in particular, the border case $p=\tfrac{1}{3}$. Employing semiclassical scaling and microlocal factorization, it proves that for $p=\tfrac{1}{3}$ the resolvent norm satisfies $\|(H_c-zI)^{-1}\|=O(|z|^{-1/3})$ as $|z|\to\infty$ along $z_{\eta}$, indicating spectral stability in this regime; this contrasts with known blow-up results for $\tfrac{1}{3}<p<3$. The numerical portion, using Chebyshev collocation and pseudo-spectral techniques, corroborates the theoretical predictions and demonstrates pronounced pseudospectral growth far from the stability line, signaling high-energy instability under perturbations. Overall, the work clarifies the border between stability and instability in the pseudospectrum of complex potentials and provides a methodology for validating spectral sensitivity via numerical pseudo-spectra.

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$, $ \frac{1}{3}< p <3 $ independent of $η> 0$. 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.

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

TL;DR

The paper analyzes the ε-pseudospectrum of the non-self-adjoint complex harmonic oscillator with , , focusing on resolvent-norm behavior along curves and, in particular, the border case . Employing semiclassical scaling and microlocal factorization, it proves that for the resolvent norm satisfies as along , indicating spectral stability in this regime; this contrasts with known blow-up results for . The numerical portion, using Chebyshev collocation and pseudo-spectral techniques, corroborates the theoretical predictions and demonstrates pronounced pseudospectral growth far from the stability line, signaling high-energy instability under perturbations. Overall, the work clarifies the border between stability and instability in the pseudospectrum of complex potentials and provides a methodology for validating spectral sensitivity via numerical pseudo-spectra.

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 where , independent of . The present work narrows the focus to the case where . Numerical computations of pseudo-eigenvalues are performed to verify spectral instability.
Paper Structure (11 sections, 5 theorems, 110 equations, 3 figures)

This paper contains 11 sections, 5 theorems, 110 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some results for the complex harmonic oscillator
  3. Study of the resolvent estimates in the case $p = \frac{1}{3}$
  4. First case: if $(x,\xi) \in V_{(0,1)}$
  5. If $\|(x,y)\| \leq c_{1}\mu ^{1/2}$ for small positive constant $c_{1}$ independent of $\mu$.
  6. If $\| (x,y) \| \geq c_{2} \mu ^{1/2}$ for some constant $c_{2} > 0$.
  7. Second case: if $(x,\xi)\in V_{(0,-1)}$
  8. Numerical results
  9. Spectral discretization of the complex harmonic oscillator
  10. Numerical Computation of pseudo-spectra
  11. Conclusion

Key Result

Theorem 3.1

Let $H_{c}$ be the complex harmonic oscillator defined by where $Re(c)>0$ and $Im(c)$$\neq 0.$ We suppose that there exists C$_{0}>0$ such that Then: when $| z| \rightarrow +\infty$.

Figures (3)

  • Figure 1: The first twenty eigenvalues of $H_{c}$ for $c=1+5i$.
  • Figure 2: The eigenfunctions of $H_{c}$ for $c=1+5i$ associated to the eigenvalues $\lambda_{\text{1}}$, $\lambda_{\text{5}}$, $\lambda_{\text{10}}$, and $\lambda _{\text{20}}$.
  • Figure 3: Eigenvalues and $\varepsilon$-pseudo-spectra of the matrix $H_{c}^{N}$ for $N = 200$, $c=1+5i$. The left column gives the corresponding values of $\varepsilon$.

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 1 more