Approximations of spectra of Schr\"odinger operators with complex potentials on $\mathbb{R}^d$

TL;DR

This paper develops a comprehensive framework for approximating the spectra of Schrödinger operators with complex potentials on unbounded and exterior domains, ensuring spectral accuracy and convergence of eigenvalues and pseudospectra.

Contribution

It introduces generalized norm resolvent convergence and spectral exactness results for complex Schrödinger operators with broad potential classes, including singular and unbounded cases.

Findings

Proves spectral approximation without pollution for complex potentials

Estimates convergence rates of eigenvalues

Demonstrates convergence of pseudospectra with numerical examples

Abstract

We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $\Omega_n$, and of boundary conditions on $\partial \Omega_n$ such as mixed Dirichlet/Robin type. In particular, ${\rm Re} \, Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for $d=1,2,3$, illustrate our results.