On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)

TL;DR

This paper explores the self-adjoint operators related to PT symmetric Hamiltonians, revealing many such operators are not PT symmetric, thus expanding understanding of their spectral properties.

Contribution

It characterizes all self-adjoint operators associated with a class of PT symmetric Hamiltonians, highlighting the existence of non-PT symmetric self-adjoint operators.

Findings

Many self-adjoint operators are not PT symmetric.

Complete description of self-adjoint extensions for the given Hamiltonian.

Insight into spectral properties of PT symmetric operators.

Abstract

In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians, possessing PT symmetry (the product of parity and time reversal), can have real spectrum. We will consider the most simple case: \epsilon even. In this paper we describe all self-adjoint (Hermitian) and at the same time PT symmetric operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^\epsilon which are not PT symmetric.