On generalized Bohr-Sommerfeld quantization rules for operators with PT symmetry

TL;DR

This paper develops Bohr-Sommerfeld quantization rules for one-dimensional PT-symmetric pseudodifferential operators, providing a framework to understand their quasi-eigenvalues in quantum mechanics.

Contribution

It introduces generalized quantization rules specifically for PT-symmetric operators with real principal symbols, extending classical methods to this class.

Findings

Derived Bohr-Sommerfeld rules for PT-symmetric operators

Established correspondence between quasi-eigenvalues and classical orbits

Extended quantization conditions to non-Hermitian operators with PT symmetry

Abstract

We give Bohr-Sommerfeld quantization rules corresponding to quasi-eigenvalues for a 1-D h-Pseudodifferential operator with real principal symbol and verifying PT symmetry.