Pt-Symmetric Nonpolynomial Oscillators and Hyperbolic Potential With Two Known Real Eigenvalues in a Susy Framework

Abstract

Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasiexactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.Running head: SUSY and PT-symmetric potentials PACS: 03.65.Fd, 03.65.Ge

Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians

“…This is the main result of this article. It implies, in particular, that all the attempts in the literature [5,6,7,8,9,10,11,12,13,14] to generate non-Hermitian Hamiltonians with a real spectrum by intertwining a Hermitian Hamiltonian fit to the framework provided by pseudo-supersymmetry. 7 Obviously this applies when zero belongs to their common spectrum.…”

“…In order to reveal the mathematical meaning of the ∆, we first observe that according to Equations (13),…”

“…The purpose of this article is to explore various properties and applications of pseudo-supersymmetry. In particular, we establish the fact that all the available methods of generating non-Hermitian Hamiltonians with a real spectrum by intertwining the latter with Hermitian Hamiltonians [5,6,7,8,9,10,11,12,13,14] fit to the framework provided by pseudo-supersymmetry. We also point out the implications of pseudo-supersymmetry for ordinary unitary quantum mechanics.…”

A General Approach of Quasi-Exactly Solvable Schrödinger Equations With Three Known Eigenstates

“…potential, with real eigenvalues. This potential is new compared to the ones already studied through other developments [10].…”