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Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation

Ali Mostafazadeh
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Abstract

For a given pseudo-Hermitian Hamiltonian of the standard form: H=p^2/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator ηsatisfying H^\dagger=ηH η^{-1} to the solution of a differential equation. If the configuration space is the real line, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate ηfor the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general ηup to second order terms in the coupling constants.

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Ali Mostafazadeh. 2006-07-26. Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation. https://doi.org/10.1063/1.2212668

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