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arXiv:hep-th/0201056 · Journal of Physics A Mathematical and General

Non-Hermitian supersymmetry and singular, 𝒫𝒯-symmetrized oscillators

Miloslav Znojil

Published Feb 25, 2002• 55 citations
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Abstract

Hermitian supersymmetric partnership between singular potentials V ( q ) = q 2 + G / q 2 breaks down and can only be restored on certain ad hoc subspaces (Das A and Pernice S 1999 Nucl. Phys. B 561 357). We show that within extended, -symmetric quantum mechanics, the supersymmetry between singular oscillators can be completely re-established in a way which is continuous near G = 0 and leads to a new form of the bosonic creation and annihilation operators.

DOI
10.1088/0305-4470/35/9/320

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Hermitian supersymmetric partnership between singular potentials V ( q ) = q 2 + G / q 2 breaks down and can only be restored on certain ad hoc subspaces (Das A and Pernice S 1999 Nucl. Phys. B 561 357). We show that within extended, -symmetric quantum mechanics, the supersymmetry between singular oscillators can be completely re-established in a way which is continuous near G = 0 and leads to a new form of the bosonic creation and annihilation operators.
Re-establishing supersymmetry between harmonic oscillators in D≠1D \neq 1D=1 dimensions
Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wavefunctions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q = ± 1 in 𝒫𝒯 symmetric (non-Hermitian) quantum mechanical potential models, the relation of 𝒫𝒯 symmetry and supersymmetry (SUSY) is studied. As an illustrative example, the 𝒫𝒯 invariant version of the Scarf II potential is presented, and it is shown that the ‘bosonic’ Hamiltonian has two different ‘fermionic’ SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q = 1 and q = −1. It is shown that the ‘fermionic’ potentials cease to be 𝒫𝒯 invariant when the 𝒫𝒯 symmetry of the ‘bosonic’ potential is spontaneously broken. A modified 𝒫𝒯 symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator 𝒯. It is shown that in this scheme the ‘fermionic’ Hamiltonians are just the complex conjugates of the original ‘fermionic’ Hamiltonians, and thus possess the same energy eigenvalues.
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.