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Higher-order supersymmetric quantum mechanics
We review the higher-order supersymmetric quantum mechanics (H-SUSY QM),
which involves differential intertwining operators of order greater than one.
The iterations of first-order SUSY transformations are used to derive in a
simple way the higher-order case. The second order technique is addressed
directly, and through this approach unexpected possibilities for designing
spectra are uncovered. The formalism is applied to the harmonic oscillator: the
corresponding H-SUSY partner Hamiltonians are ruled by polynomial Heisenberg
algebras which allow a straight construction of the coherent states.Comment: 42 pages, 12 eps figure
SUSUSY quantum mechanics
The exactly solvable eigenproblems in Schr\"odinger quantum mechanics
typically involve the differential "shift operators". In the standard
supersymmetric (SUSY) case, the shift operator turns out to be of first order.
In this work, I discuss a technique to generate exactly solvable eigenproblems
by using second order shift operators. The links between this method and SUSY
are analysed. As an example, we show the existence of a two-parametric family
of exactly solvable Hamiltonians, which contains the Abraham-Moses potentials
as a particular case.Comment: 7 pages, 2 encapsulated postscript figures, uses epsf.sty talk given
at the II International Workshop on Classical and Quantum Integrable Systems,
Dubna (Russia), 8-12 July (1996) to be published in Int. J. Mod. Phys.
Supersymmetric Quantum Mechanics
Supersymmetric quantum mechanics (SUSY QM) is a powerful tool for generating
new potentials with known spectra departing from an initial solvable one. In
these lecture notes we will present some general formulas concerning SUSY QM of
first and second order for one-dimensional arbitrary systems, and we will
illustrate the method through the trigonometric Poschl-Teller potentials. Some
intrinsically related subjects, as the algebraic structure inherited by the new
Hamiltonians and the corresponding coherent states will be analyzed. The
technique will be as well implemented for periodic potentials, for which the
corresponding spectrum is composed of allowed bands separated by energy gaps.Comment: 36 pages, 8 figures, lectures delivered at the Advanced Summer School
2009, Cinvestav (Mexico City), July 200
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