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In the recent years a generalization $H=p^2 +x^2(ix)^\epsilon$ of theharmonic 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 threedifferent classes of operators associated with the differential expression H:The class of all self-adjoint (Hermitian) operators, the class of all PTsymmetric operators and the class of all P-self-adjoint operators.Surprisingly, some of the PT symmetric operators associated to this expressionhave no resolvent set.

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