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Journal of Physics A Mathematical and Theoretical

Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop

Petr Siegl

Published Jun 3, 2008• 16 citations
Publisher

Abstract

We explore some aspects of -symmetric Hamiltonians with two point interactions. We determine classes of point interactions for which the Hamiltonians are supersymmetric. We prove that these Hamiltonians are quasi-Hermitian and find a very simple formula for the metric operator Θ and its square root ϱ as well. Further, we present the quasi-Hermitian Hamiltonian (with one-point interaction) with a continuous spectrum.

DOI
10.1088/1751-8113/41/24/244025

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We explore some aspects of -symmetric Hamiltonians with two point interactions. We determine classes of point interactions for which the Hamiltonians are supersymmetric. We prove that these Hamiltonians are quasi-Hermitian and find a very simple formula for the metric operator Θ and its square root ϱ as well. Further, we present the quasi-Hermitian Hamiltonian (with one-point interaction) with a continuous spectrum.
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.