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:where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, ℑ(ζ + ) = −ℑ(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 h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of PT -symmetry.A Hamiltonian operator H is called PT -symmetric if it has parity-time reversal symmetry, i.e., [H, PT ] = 0. Since the publication of the pioneering work of Bender and Boettecher [1] non-Hermitian PT -symmetric Hamiltonians have received much attention. This has led to the discovery of a number of interesting theoretical [2,3,4,5,6,7] as well as experimental [8] implications of PT -symmetric Hamiltonians. For extensive reviews see [9,10] and references therein.Perhaps the most prominent feature of a non-Hermitian PT -symmetric Hamiltonian H is that its spectrum is symmetric about the real axis of the complex plane. In particular, if H has a discrete spectrum, either it is purely real or the nonreal eigenvalues come in complex-conjugate pairs [1,7]. It turns out that this is a characteristic property of a wider class of non-Hermitian Hamiltonian operators called the pseudo-Hermitian operators [11,12,13]. A Hamiltonian H is said to be pseudo-Hermitian if its adjoint H † satisfies

Pseudo-Hermitian Representation of Quantum Mechanics

“…A generalization of the delta-function potential (143) that allows for a similar analysis is the double-delta function potential: v(x) = z − δ(x + a) + z + δ(x − a), where z ± and a are complex and real parameters, respectively [177,138]. Depending on the values of the coupling constants z ± , this potential may develop spectral singularities [184,132].…”

“…The general N = 2 case, that depending on the choice of ζ k may or may not posses PT -symmetry, has been examined in [177,138]. The main difficulty with the approaches presented in this section (and its subsections) is that they may lead to a "metric" operator that is unbounded or non-invertible.…”

Spectral singularities in ${\mathcal P}{\mathcal T}$-symmetric Bose–Einstein condensates

“…The influence of a pair of PT -symmetric delta-functions on the bound and scattering states of a single real delta-function has been investigated by Jones [27]. The spectral properties, in particular bound states and spectral singularities, in non-Hermitian delta-potentials have been studied by Mostafazadeh et al [28][29][30].…”

Supersymmetric Model of a Bose-Einstein Condensate in a 𝓟𝓣-Symmetric Double-delta Trap

“…To do so, we apply the SUSY formalism to the case of vanishing Gross-Pitaevskii nonlinearity. This potential has often been used to gain deeper insight with analytically accessible energies or wave functions [30][31][32][33][34][35]. We show that the SUSY scheme can indeed be used to remove arbitrary PT -symmetric and PT -broken eigenstates.…”