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  • Open Access

Computation and Verification of Spectra for Non-Hermitian Systems

Catherine Drysdale*

Matthew Colbrook†

Michael T. M. Woodley‡

  • *Contact author: c.drysdale@lancaster.ac.uk
  • †Contact author: mjc249@cam.ac.uk
  • ‡Contact author: mw2970@bath.ac.uk
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Phys. Rev. Lett. 135, 170202 – Published 22 October, 2025

DOI: https://doi.org/10.1103/hf8s-v93b

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, we show such assumptions are necessary for spectral computation. Locally trivial pseudospectra adapt 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 HB=p2+ix3 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…. Here, truncation-induced 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.

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Phys. Rev. Lett. 135, 170202– Published 22 October, 2025
  • Received 21 October 2024
  • Revised 2 July 2025
  • Accepted 29 September 2025
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