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

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In the recent years a generalization H=p² +x²(ix)ε of the harmonic oscillator using a complex deformation was investigated, where ε is a real parameter. Here, we will consider the most simple case: εeven 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.