Computation and Verification of Spectra for Non-Hermitian Systems

Area of Science:

• Quantum Mechanics
• Computational Theory
• Numerical Analysis

Background:

• Spectral computations in quantum mechanics are essential but face challenges, particularly for non-Hermitian systems.
• Existing algorithms often struggle with accuracy and spurious modes, especially when dealing with complex Hamiltonians.

Purpose of the Study:

• To establish a rigorous connection between quantum mechanics and computational theory for spectral analysis.
• To introduce a novel framework for accurate spectral computations in challenging non-Hermitian settings.
• To overcome limitations of current algorithms and provide error-controlled results.

Main Methods:

• Introduction of the concept of locally trivial pseudospectra, which adapt dynamically to system energies.
• Development of a computational framework based on locally trivial pseudospectra for spectral analysis.
• Application of the framework to compute eigenvalues and eigenfunctions of the imaginary cubic oscillator (H_{B}=p^{2}+ix^{3}).

Main Results:

• Demonstration that locally trivial pseudospectra are necessary for accurate spectral computation.
• Successful computation of eigenvalues and eigenfunctions for the imaginary cubic oscillator with error bounds and no spurious modes.
• Identification of truncation-induced PT-symmetry breaking as a source of spurious eigenvalues, a pitfall avoided by the new method.

Conclusions:

• The developed framework provides a precise tool for spectral calculations with error bounds, linking computational theory and quantum mechanics.
• The method overcomes a longstanding obstacle in computing spectra for non-Hermitian systems.
• The approach is general and applicable to a range of physically relevant operators.