PT-symmetric operators and metastable states of the 1D relativistic oscillators

TL;DR

This paper explores PT-symmetric operators in 1D relativistic oscillators, revealing their role in defining metastable states and resonances, with numerical evidence showing convergence to non-relativistic energy levels as the speed of light increases.

Contribution

It introduces a novel application of PT-symmetric operators to relativistic oscillators, linking their spectra to metastable states and resonances, supported by numerical validation.

Findings

PT-symmetric operators define metastable states in relativistic oscillators.

Energy levels of these operators converge to Schrödinger levels as c increases.

Numerical results confirm the correspondence between eigenvalues and resonances.

Abstract

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT-symmetric operators give metastable states for each dynamics.