Convergent Quantum Normal Forms, ${\mathcal P}{\mathcal T}$-symmetry and reality of the spectrum

TL;DR

This paper develops a convergent quantum normal form method for PT-symmetric operators, demonstrating their spectrum's reality and deriving an exact eigenvalue quantization formula.

Contribution

It introduces a convergent quantum normal form approach to analyze PT-symmetric operators, establishing their similarity to self-adjoint operators and ensuring real spectra.

Findings

Explicit similarity transformation constructed

Proved uniform convergence of the quantum normal form

Derived an exact quantization formula for eigenvalues

Abstract

A class of non-selfadjoint, $\PT$-symmetric operators is identified similar to a self-adjoint one, thus entailing the reality of the spectrum. The similarity transformation is explicitly constructed through the method of the quantum normal form, whose convergence (uniform with respect to the Planck constant) is proved. Further consequences of the uniform convergence of the quantum normal form are the establishment of an exact quantization formula for the eigenvalues.