Computation and Verification of Spectra for Non-Hermitian Systems

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 H_{B}=p^{2}+ix^{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…. 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.