Computation and Verification of Spectra for Non-Hermitian Systems

TL;DR

This paper introduces a new framework for computing spectra of non-Hermitian quantum systems with error bounds, overcoming previous obstacles and avoiding spurious solutions, thereby bridging computational theory and quantum physics.

Contribution

The paper develops the concept of locally trivial pseudospectra (LTP) and applies it to compute spectra of challenging non-Hermitian operators with rigorous error control.

Findings

Successfully computed eigenvalues of the imaginary cubic oscillator with error bounds

Avoided spurious eigenvalues caused by truncation-induced symmetry breaking

Demonstrated the method's applicability to various physically relevant operators

Abstract

We establish a connection between quantum mechanics and computation, revealing fundamental limitations for algorithms computing spectra, especially in non-Hermitian settings. Introducing the concept of locally trivial pseudospectra (LTP), we show such assumptions are necessary for spectral computation. LTP adapts dynamically to system energies, enabling spectral analysis across a broad class of challenging non-Hermitian problems. Exploiting this framework, we overcome a longstanding obstacle by computing the eigenvalues and eigenfunctions of the imaginary cubic oscillator $H_{\mathrm{B}} = p^2 + i x^3$ with error bounds and no spurious modes -- yielding, to our knowledge, the first such error-controlled result. We confirm, for instance, the 100th eigenvalue as $627.6947122484365113526737029011536\ldots$. Here, truncation-induced $\mathcal{PT}$-symmetry breaking causes spurious eigenvalues -- a pitfall our method avoids, highlighting the link between truncation and physics. Finally, we illustrate the approach's generality via spectral computations for a range of physically relevant operators. This letter provides a rigorous framework linking computational theory to quantum mechanics and offers a precise tool for spectral calculations with error bounds.