Computation and Verification of Spectra for Non-Hermitian Systems

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 $627.6947122484365113526737029011536\dots$. Here, truncation-induced $\mathcal{P}\mathcal{T}$-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.

Published in Physical Review Letters

References in corpus (45)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Complex Extension of Quantum Mechanics
• Exceptional Topology of Non-Hermitian Systems
• PT-Symmetric Quantum Mechanics
• Topological Origin of Non-Hermitian Skin Effects
• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Experimental Study of Active LRC Circuits with PT-Symmetries
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Observation of mesoscopic crystalline structures in a two-dimensional Rydberg gas
• Metrology with $\mathcal{P}\mathcal{T}$-symmetric cavities: Enhanced sensitivity near the $\mathcal{P}\mathcal{T}$-phase transition
• Must a Hamiltonian be Hermitian?
• Observation of PT phase transition in a simple mechanical system
• Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices
• PT symmetry and spontaneous symmetry breaking in a microwave billiard
• Observation of Asymmetric Transport in Structures with Active Nonlinearities
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Observation of non-Bloch parity-time symmetry and exceptional points
• Mechanisms of Spatiotemporal Mode-Locking
• Synthetic parity-time symmetry breaking in a single microcavity
• PT-symmetric scattering in flow duct acoustics
• Pseudospectra in non-Hermitian quantum mechanics
• Models for the modern power grid
• Real-time wavefront-shaping through scattering media by all optical feedback
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• On the metric operator for the imaginary cubic oscillator
• Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
• Stimulation of the fluctuation superconductivity by the PT-symmetry
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• Topological classification of defects in non-Hermitian systems
• Model of supersymmetric quantum field theory with broken parity symmetry
• Variational Ansatz for PT-Symmetric Quantum Mechanics
• Fast State Transfer and Entanglement Renormalization Using Long-Range Interactions
• Non-Hermitian Topological Phases: Principles and Prospects
• Geometric Origin of Non-Bloch PT Symmetry Breaking
• Experimental Demonstration of Controllable PT and anti-PT Coupling in a non-Hermitian Metamaterial
• PT-Symmetric Quantum Electrodynamics
• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Linear waves in sheared flows. Lower bound of the vorticity growth and propagation discontinuities in the parameters space
• Spectral projections of the complex cubic oscillator
• Non-hermitian topology in multiterminal superconducting junctions
• Experimentally-realizable $\mathcal{P}\mathcal{T}$ phase transitions in reflectionless quantum scattering
• Mimicking quantum correlation of a long-range Hamiltonian by finite-range interactions
• A Bound on the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential