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A Dressing Method in Mathematical PhysicsApplications of dressing to linear problems

A Dressing Method in Mathematical Physics: Applications of dressing to linear problems The dressing procedures, following our “extended” understanding of this ideology, have been used for years to solve linear problems. In this chapter we concentrate on some recent results obtained along these lines. We observe considerable reciprocal influence of the nonlinear theory on linear methods, in particular via a systematic application of the Lax representation developed previously when studying nonlinear systems. We show how to dress a seed solution of a one-dimensional second-order linear differential equation when the corresponding operator allows explicit factorization. We also show how the Darboux transformation (DT) theory appears in this framework and produces the so-called integrable or solvable po- tentials entering linear differential equations. The important and far-reaching example of solvable potentials is represented by the famous regular shape- invariant potentials introduced by Gendenstein [179] (see also [412] and refer- ences therein). We could mention as well other classes of potentials, like those obtained by algebraic deformations [190], singular (pointlike of the Coulomb type or zero-range) potentials [284], and matrix potentials on the whole axis or half-axis. There is an excellent book [367] on applications of the Lax representation to classical mechanics. The integrability is established and exploited by means of the Lie algebra technique. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsApplications of dressing to linear problems

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Publisher
Springer Netherlands
Copyright
© Springer 2007
ISBN
978-1-4020-6138-7
Pages
161 –198
DOI
10.1007/1-4020-6140-4_6
Publisher site
See Chapter on Publisher Site

Abstract

The dressing procedures, following our “extended” understanding of this ideology, have been used for years to solve linear problems. In this chapter we concentrate on some recent results obtained along these lines. We observe considerable reciprocal influence of the nonlinear theory on linear methods, in particular via a systematic application of the Lax representation developed previously when studying nonlinear systems. We show how to dress a seed solution of a one-dimensional second-order linear differential equation when the corresponding operator allows explicit factorization. We also show how the Darboux transformation (DT) theory appears in this framework and produces the so-called integrable or solvable po- tentials entering linear differential equations. The important and far-reaching example of solvable potentials is represented by the famous regular shape- invariant potentials introduced by Gendenstein [179] (see also [412] and refer- ences therein). We could mention as well other classes of potentials, like those obtained by algebraic deformations [190], singular (pointlike of the Coulomb type or zero-range) potentials [284], and matrix potentials on the whole axis or half-axis. There is an excellent book [367] on applications of the Lax representation to classical mechanics. The integrability is established and exploited by means of the Lie algebra technique.

Published: Jan 1, 2007

Keywords: Green Function; Partial Wave; Linear Problem; Discrete Spectrum; Darboux Transformation

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