On Shape Resonances and Darboux Transformations in Quantum Mechanics

Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed.

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GENERALIZED BÄCKLUND–DARBOUX TRANSFORMATIONS IN ONE-DIMENSIONAL QUANTUM MECHANICS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887808002989 ◽

2008 ◽

Vol 05 (04) ◽

pp. 605-640 ◽

Cited By ~ 7

Author(s):

JOSÉ F. CARIÑENA ◽

ARTURO RAMOS

Keyword(s):

Quantum Mechanics ◽

Excited State ◽

Differential Expression ◽

Transformation Group ◽

Darboux Transformations ◽

Singular Case ◽

One Dimensional ◽

First Order ◽

The Difference ◽

Usual Problem

We consider an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the set of Schrödinger equations in full similarity with the action of the group of curves in SL(2,ℝ) on the set of Riccati equations considered in previous articles. We also consider the transformations defined by a first-order differential expression which carry solutions of a Schrödinger equation into solutions of another one. We find then two non-trivial situations: transformations which can be described by the previous transformation group, generalizing previous work by us, and transformations which are singular. We show that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics. We show that the difference Bäcklund algorithm, both in the finite and confluent versions, can be understood in terms of the above mentioned transformation group, the case of two exactly equal factorization energies being an instance of the singular case. We apply the generalized theorem relating three eigenfunctions of three different Hamiltonians to the generation of new potentials with a known (excited state) eigenfunction, starting from potentials of Coulomb, Morse and Rosen–Morse type. The potentials found are new and non-trivial.

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New possibilities for supersymmetry breakdown in quantum mechanics and second-order irreducible Darboux transformations

Physics Letters A ◽

10.1016/s0375-9601(99)00736-7 ◽

1999 ◽

Vol 263 (4-6) ◽

pp. 274-280 ◽

Cited By ~ 66

Author(s):

Boris F. Samsonov

Keyword(s):

Quantum Mechanics ◽

Second Order ◽

Darboux Transformations

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GENERATING COMPLEX POTENTIALS WITH REAL EIGENVALUES IN SUPERSYMMETRIC QUANTUM MECHANICS

International Journal of Modern Physics A ◽

10.1142/s0217751x01004153 ◽

2001 ◽

Vol 16 (16) ◽

pp. 2859-2872 ◽

Cited By ~ 60

Author(s):

B. BAGCHI ◽

S. MALLIK ◽

C. QUESNE

Keyword(s):

Quantum Mechanics ◽

Lie Algebra ◽

Algebraic Approach ◽

Factorization Method ◽

Bound State ◽

Complex Potentials ◽

The Other ◽

Darboux Transformations ◽

Weierstrass Function ◽

First Time

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra [Formula: see text] . This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass ℘ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT nonsymmetric and its known eigenvalue corresponds to an unbound state.

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