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Non-Hermitian Hamiltonian; $\mathcal{PT}-symmetry; closed-form solutions.

Making use of an ansatz for the eigenfunction, we investigate closed-form solutions of the Schrödinger equation for an even power complex deictic potential and its variant in one dimension. For this purpose, extended complex phase-space approach is utilized and nature of the eigenvalue and the corresponding eigenfunction is determined by the analyticity property of the eigenfunction. The imaginary part of the energy eigenvalue exists only if the potential parameters are complex, whereas it reduces to zero for real coupling parameters and the result coincides with those derived from the invariance of Hamiltonian under $\mathcal{PT}$ operations. Thus, a non-Hermitian Hamiltonian possesses real eigenvalue, if it is $\mathcal{PT}$-symmetric.

Ram Mehar Singh¹

1. Department of Physics, Ch. Devi Lal University, Sirsa 125 055, India

Published

September 2014

Manuscript received

17 September 2013

Manuscript revised

5 January 2014

Accepted

28 January 2014

Early published

18 June 2014