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Title: Shape invariance through Crum transformation

Abstract

We show in a rigorous way that Crum's result regarding the equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darboux-transformed potentials of higher order, u{sub k} and u{sub k+1}, satisfy the condition of shape invariance provided the original potential u does so. Based on this result, we prove that under the condition of shape invariance, the nth iteration of the original Sturm-Liouville problem defined solely through the shape invariance is equal to the nth Crum transformation.

Authors:
; ;  [1];  [2]
  1. Departamento de Fisica, Universidad de los Andes, Cra.1E No. 18A-10, Santafe de Bogota (Colombia)
  2. (Mexico)
Publication Date:
OSTI Identifier:
20861550
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2397556; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVALUES; ITERATIVE METHODS; POTENTIALS; STURM-LIOUVILLE EQUATION; TRANSFORMATIONS

Citation Formats

Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Shape invariance through Crum transformation. United States: N. p., 2006. Web. doi:10.1063/1.2397556.
Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., & Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Shape invariance through Crum transformation. United States. doi:10.1063/1.2397556.
Organista, Jose Orlando, Nowakowski, Marek, Rosu, H. C., and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi. Fri . "Shape invariance through Crum transformation". United States. doi:10.1063/1.2397556.
@article{osti_20861550,
title = {Shape invariance through Crum transformation},
author = {Organista, Jose Orlando and Nowakowski, Marek and Rosu, H. C. and Potosinian Institute of Science and Technology, Apartado Postal 3-74 Tangamanga, 78231 San Luis Potosi},
abstractNote = {We show in a rigorous way that Crum's result regarding the equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darboux-transformed potentials of higher order, u{sub k} and u{sub k+1}, satisfy the condition of shape invariance provided the original potential u does so. Based on this result, we prove that under the condition of shape invariance, the nth iteration of the original Sturm-Liouville problem defined solely through the shape invariance is equal to the nth Crum transformation.},
doi = {10.1063/1.2397556},
journal = {Journal of Mathematical Physics},
number = 12,
volume = 47,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}
  • The present paper contains an investigation of the mechanical energy associated with the transformation of the stress-induced martensite, ..beta..', to the parent phase, ..beta.., during the shape recovery (SR) of a deformed shape-memory (SM) material. We describe a heat-mechanical energy converter, or solid-state engine, which operates by this SR phenomenon. The energy output of such an engine is expressed in terms of a fraction ..cap alpha.. of the latent heat ..delta..H of the martensitic reaction. This ..cap alpha.. is found to depend on two parameters. One is the difference between the ..delta..H of the ..beta..' ..-->.. ..beta.. reaction and themore » ..delta..H of the transformation of the quench-induced martensite, ..gamma..', to ..beta.., the other is the fraction of ..gamma..' which can be transformed via the channel ..gamma..' ..-->.. ..beta..' ..-->.. ..beta.. instead of the direct channel ..gamma..' ..-->.. ..beta... Moreover, it is shown that within certain ranges of temperature T and applied strain epsilon, the heat-mechanical energy balance equation leads to an expression identical in form to the Clapeyron-Clausius equation, which is usually valid for a first-order transition. Within these epsilon and T ranges the coefficient ..cap alpha.. is also found to be equal to log (T/sub csigma//T/sub c/) where T/sub csigma/ and T/sub c/ are the SR critical temperatures with and without the presence of an applied stress sigma, respectively. We discuss the role of the ..gamma..' martensite in this process and explain the so-called two-way SR phenomenon. In addition, the parameters that limit the output of the SR energy are evaluated. This output depends sensitively on both ..cap alpha.. and the material characteristic temperature h = C/sup -1/..delta..H, where C is the specific heat. For a solid-state engine made with the Ni-Ti SM alloy, the efficiency is found to be limited to about 5%.« less
  • The present paper contains an investigation of the mechanical energy associated with the transformation of the stress-induced martensite, ..beta..', to the parent phase, ..beta.., during the shape recovery (SR) of a deformed shape-memory (SM) material. We describe a heat-mechanical energy converter, or solid-state engine, which operates by this SR phenomenon. The energy output of such an engine is expressed in terms of a fraction ..cap alpha.. of the latent heat ..delta..H of the martensitic reaction. This ..cap alpha.. is found to depend on two parameters. One is the difference between the ..delta..H of the ..beta..' ..-->.. ..beta.. reaction and themore » ..delta..H of the transformation of the quench-induced martensite, ..gamma..', to ..beta.., the other is the fraction of ..gamma..' which can be transformed via the channel ..gamma..' ..-->.. ..beta..' ..-->.. ..beta.. instead of the direct channel ..gamma..' ..-->.. ..beta... Moreover, it is shown that within certain ranges of temperature T and applied strain epsilon, the heat-mechanical energy balance equation leads to an expression identical in form to the Clapeyron-Clausius equation, which is usually valid for a first-order transition. Within these epsilon and T ranges the coefficient ..cap alpha.. is also found to be equal to log(T/sub c sigma//T/sub c/) where T/sub c sigma/ and T/sub c/ are the SR critical temperatures with and without the presence of an applied stress sigma, respectively. We discuss the role of the ..gamma..' martensite in this process and explain the so-called two-way SR phenomenon. In addition, the parameters that limit the output of the SR energy are evaluated. This output depends sensitively on both ..cap alpha.. and the material characteristic temperature h = C/sup -1/..delta..H, where C is the specific heat. For a solid-state engine made with the Ni-Ti SM alloy, the efficiency is found to be limited to about 5%.« less
  • We discuss in some detail the self-similar potentials of Shabat [Inverse Prob. 8, 303 (1992)] and Spiridonov [Phys. Rev. Lett. 69, 298 (1992)] which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape-invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling [ital Ansatz] for the change of parameters, we obtain a large class of new, reflectionless, shape-invariant potentials of which the Shabat-Spiridonov ones are a special case. These new potentials can be viewed as [ital q] deformations of the single-soliton solution corresponding to themore » Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions, and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape-invariant double-well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling and (ii) extending the usual concept of shape invariance in one step to a multistep situation. These extensions can be viewed as [ital q] deformations of the harmonic oscillator or multisoliton solutions corresponding to the Rosen-Morse potential.« less
  • The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as strength and range. Shape-invariance algebras, in general, are shown to be infinite dimensional. The conditions under which they become finite dimensional are explored. {copyright} {ital 1998} {ital The American Physical Society}