# On Domains of PT Symmetric Operators Related to -y''x -1^n x^{2n}yx - Quantum Physics

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Abstract: In the recent years a generalization of Hermiticity was investigated using acomplex deformation H=p^2 +x^2ix^\epsilon of the harmonic oscillatorHamiltonian, where \epsilon is a real parameter. These complex Hamiltonians,possessing PT symmetry the product of parity and time reversal, can have realspectrum. We will consider the most simple case: \epsilon even. In this paperwe describe all self-adjoint Hermitian and at the same time PT symmetricoperators associated to H=p^2 +x^2ix^\epsilon. Surprisingly it turns out thatthere are a large class of self-adjoint operators associated to H=p^2+x^2ix^\epsilon which are not PT symmetric.

Author: ** Tomas Ya. Azizov, Carsten Trunk**

Source: https://arxiv.org/