On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)

Details On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x) On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)

2009-11-06

arxiv.org/abs/0911.1284v1

Physics

Quantum Physics

Scientific paper

In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator Hamiltonian, where \epsilon 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: \epsilon even. In this paper we describe all self-adjoint (Hermitian) and at the same time PT symmetric operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that there are a large class of self-adjoint operators associated to H=p^2 +x^2(ix)^\epsilon which are not PT symmetric.

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