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Supersymmetric η operators

B. F. Samsonov

Published Aug 15, 2016• 0 citations
OpenAlex

Abstract

Being chosen as a differential operator of a special form, metric $\eta$ operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.

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Being chosen as a differential operator of a special form, metric η\etaη operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.
Being chosen as a differential operator of a special form, the metric η operator has become unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits us to determine both the metric operator and the corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it has become a superpartner of another Hermitian Hamiltonian. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form H = ωJ 3 + αJ -+ βJ + , α = β, is analyzed.The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established.A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed.They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators.We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method.The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space.The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.
One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result, the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of a spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point, the equivalent Hermitian Hamiltonian becomes undetermined.