Bijan Bagchi

sl(2,C) as a complex lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues

The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,C) or A1. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.

Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass

Known shape-invariant potentials for the constant-mass Schrodinger equation are taken as effective potentials in a position-dependent effective mass (PDEM) one. The corresponding shape-invariance condition turns out to be deformed. Its solvability imposes the form of both the deformed superpotential and the PDEM. A lot of new exactly solvable potentials associated with a PDEM background are generated in this way. A novel and important condition restricting the existence of bound states whenever the PDEM vanishes at an end point of the interval is identified. In some cases, the bound-state spectrum results from a smooth deformation of that of the conventional shape-invariant potential used in the construction. In others, one observes a generation or suppression of bound states, depending on the mass-parameter values. The corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials.

SUPERSYMMETRY IN QUANTUM MECHANICS

A pedagogical review on supersymmetry in quantum mechanis is presented which provides a comprehensive coverage of the subject. First, the key ingredients on the quantization of the systems with anticommuting variables are discussed. The supersymmetric Hamiltotian in quantum mechanics is then constructed by emphasizing the role of partner potentials and the superpotentials. We also make explicit the mathematical formulation of the Hamiltonian by considering in detail the N=1 and N=2 supersymmetric (quantum) mechanics. Supersymmetry is then discussed in the context of one-dimensional problems and the importance of the factorization method is highlighted. We treat in detail the technique of constructing a hierarchy of Hamiltonians employing the so-called ‘shape-invariance’ of potentials. To make transparent the relationship between supersymmetry and solvable potentials, we also solve several examples. We then go over to the formulation of supersymmetry in radial problems, paying a special attention to the Co...

GENERALIZED CONTINUITY EQUATION AND MODIFIED NORMALIZATION IN PT-SYMMETRIC QUANTUM MECHANICS

The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Supersymmetry without hermiticity within PT symmetric quantum mechanics

Abstract A new model of supersymmetry between bosons and fermions is proposed. Its representation space is spanned by states with PT symmetry and real energies but the inter-related partner Hamiltonians themselves remain complex and non-Hermitian. The formalism admits vanishing Witten index.

Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry

AbstractWe develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems.

A general scheme for the effective-mass schrödinger equation and the generation of the associated potentials

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PT-symmetric sextic potentials

symmetry may facilitate reconciling our approach to the requirement that the rationally extended potentials be singularity free. Some examples are shown.