PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics

TL;DR

This paper characterizes Hermitian, PT symmetric, and P-self-adjoint operators related to a specific quantum mechanical potential, revealing some PT symmetric operators lack a resolvent set, thus advancing understanding of non-Hermitian quantum operators.

Contribution

It provides a complete classification of operators associated with a PT symmetric potential for even epsilon, including the surprising finding of PT operators without a resolvent set.

Findings

Some PT symmetric operators have no resolvent set.

Complete characterization of Hermitian, PT symmetric, and P-self-adjoint operators.

Advances understanding of operator classes in PT quantum mechanics.

Abstract

In the recent years a generalization $H=p^2 +x^2(ix)^\epsilon$ of the harmonic oscillator using a complex deformation was investigated, where \epsilon\ is a real parameter. Here, we will consider the most simple case: \epsilon even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.