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.

http://arxiv.org/abs/1108.5923
Quantum Physics (quant-ph)