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arXiv:hep-th/0404145 · Journal of Physics A Mathematical and General

Script PScript T-symmetric regularizations in supersymmetric quantum mechanics

Miloslav Znojil

Published Oct 15, 2004• 14 citations
arXivHTMLPDF

Abstract

Within supersymmetric quantum mechanics the necessary regularization of the poles of the superpotentials on the real line of coordinates x may be most easily mediated by a small constant shift of this axis into the complex plane. Detailed attention is paid here to the resulting recipe which works, in effect, with non-Hermitian (a.k.a. -symmetric or pseudo-Hermitian) Hamiltonians. Besides an exhaustive discussion of the role of the complex spike in harmonic oscillator, we mention also some applications concerning the regularized versions of the Smorodinsky–Winternitz and Calogero models and of the relativistic Klein–Gordon equation.

DOI
10.1088/0305-4470/37/43/013

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Within supersymmetric quantum mechanics the necessary regularization of the poles of the superpotentials on the real line of coordinates x may be most easily mediated by a small constant shift of this axis into the complex plane. Detailed attention is paid here to the resulting recipe which works, in effect, with non-Hermitian (a.k.a. -symmetric or pseudo-Hermitian) Hamiltonians. Besides an exhaustive discussion of the role of the complex spike in harmonic oscillator, we mention also some applications concerning the regularized versions of the Smorodinsky–Winternitz and Calogero models and of the relativistic Klein–Gordon equation.
A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are P T -symmetric and can be either periodic or regular (of the Pöschl–Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.
We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.
Extended quantum mechanics using non-Hermitian, pseudo-Hermitian Hamiltonians is briefly reviewed. Supersymmetric regularizations, solvable simulations and large-N expansion techniques are recollected as suitable means for the study of non-local quasi-particles. Their many-particle or even, perhaps, unstable and decaying generalizations are also considered. Picking up the Klein-Gordon equation and its Feshbach-Villars' Hamiltonian for definiteness, we argue that the PT symmetry might help to clarify experimental aspects of relativistic quantum mechanics.
The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.