On domains of PT symmetric operators related to −y′′(x) + (−1)nx2ny(x)
In the recent years a generalization of Hermiticity was investigated using a complex deformation H = p2 + x2(ix)? of the harmonic oscillator Hamiltonian, where ? 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: ? even. In this paper we describe all self-adjoint (Hermitian) and at the same time PT symmetric operators associated to H = p2 + x2(ix)?. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H = p2 + x2(ix)? which are not PT symmetric.
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