PT Symmetric Schr\"odinger Operators: Reality of the Perturbed Eigenvalues

TL;DR

This paper proves that certain PT symmetric Hamiltonians have real eigenvalues after perturbation, using stability methods, with applications to specific 2D and 1D quantum systems.

Contribution

It demonstrates the reality of perturbed eigenvalues for PT symmetric Hamiltonians using stability techniques, including new results for specific oscillator models.

Findings

Perturbed eigenvalues remain real for studied PT symmetric systems.

Stability methods effectively prove eigenvalue reality.

Results apply to 2D harmonic oscillators and 1D polynomial potentials.

Abstract

We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^{\epsilon}$ for $-1<\epsilon<0$.