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

Abstract

In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^εof the harmonic oscillator Hamiltonian, where εis 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: εeven. 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)^ε. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^εwhich are not PT symmetric.