Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions

TL;DR

This paper generalizes the construction of metric operators for non-Hermitian quantum systems with multiple parameters, applying it to a complex scattering potential with point interactions, and explores the physical implications of PT-symmetry and nonlocality.

Contribution

It introduces a perturbative method for constructing metric operators for multi-parameter non-Hermitian Hamiltonians and applies it to a complex delta-function scattering potential.

Findings

Constructed a positive-definite metric operator for the model.

Identified conditions where the metric operator remains bounded.

Analyzed energy expectation values showing insensitivity to PT-symmetry.

Abstract

We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian: H=p^2/2m+\zeta_-\delta(x+a)+\zeta_+\delta(x-a), where \zeta_\pm and a are respectively complex and real parameters and \delta(x) is the Dirac delta function. For regions in the space of coupling constants \zeta_\pm where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator \eta and the corresponding equivalent Hermitian Hamiltonian h. \eta turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, \Im(\zeta_+) = -\Im(\zeta_-). This in particular contains the PT-symmetric case: \zeta_+ = \zeta_-^*. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of $\rh$ or equivalently the non-Hermitian nature of $\rH$. We show that these physical quantities are not directly sensitive to the presence of PT-symmetry.