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Differential realization of pseudo-Hermiticity: a quantum mechanical analog of Einstein's field equation

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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 eta satisfying H(dagger)=eta H eta(-1) to the solution of a differential equation. If the configuration space is R, 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 eta. We apply our general results to calculate eta for 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 eta up to second-order terms in the coupling constants.

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American Institute of Physics (AIP) Publishing

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Mathematical physics

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Journal of Mathematical Physics

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10.1063/1.2212668

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