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

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+ζ_-δ(x+a)+ζ_+δ(x-a), where ζ_\pm and a are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ_\pm where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator ηand the corresponding equivalent Hermitian Hamiltonian h. ηturns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, \Im(ζ_+) = -\Im(ζ_-). This in particular contains the PT-symmetric case: ζ_+ = ζ_-^*. 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.

22 pages, 4 figures

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• Vortices in Bose-Einstein condensates with $\mathcal{P}\mathcal{T}$-symmetric gain and loss
• Numerical calculation of the complex berry phase in non-Hermitian systems
• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
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• Simple models of three coupled $\mathcal{P}\mathcal{T}$-symmetric wave guides allowing for third-order exceptional points
• PT-symmetric quantum systems with positive P
• Stability of Bose-Einstein condensates in a $\mathcal{P}\mathcal{T}$ symmetric double-$\delta$ potential close to branch points
• Purity oscillations in coupled Bose-Einstein condensates
• Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method