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A Dressing Method in Mathematical PhysicsMathematical preliminaries

A Dressing Method in Mathematical Physics: Mathematical preliminaries In this chapter we sketch the basic mathematical notions used in this book, starting from general relations and illustrating them by the simplest exam- ples. We also briefly review the ideas of the dressing from the viewpoint of intertwining relations under the scope of Lie algebras [151]. There is a long history of the applications of (semisimple) Lie algebras for determination of operator spectra. One line dating back to Weyl [450] relates to the explicit algebraic solution of an eigenvalue problem; an overview has been given by Joseph and Coulson [223, 224, 225]. Perhaps the best known example of such a construction is the quantum theory of angular momentum, including its development for many-particle systems (from three particles to aggregates) in terms of hyperspherical harmonics [154, 456]. The good old geometry of surfaces and conjugate nets uses the Laplace equations and transformations as a starting point [138]. The challenging problem of the Laplace operator factorization, perhaps first addressed by Laplace, created something like an “undressing” procedure which, being cut at some step, leads to the complete integrability. The direct attempt to extend the technique of the Laplace trans- formations and invariants to higher-order operators was made in [264]. In http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsMathematical preliminaries

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Publisher
Springer Netherlands
Copyright
© Springer 2007
ISBN
978-1-4020-6138-7
Pages
1 –30
DOI
10.1007/1-4020-6140-4_1
Publisher site
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Abstract

In this chapter we sketch the basic mathematical notions used in this book, starting from general relations and illustrating them by the simplest exam- ples. We also briefly review the ideas of the dressing from the viewpoint of intertwining relations under the scope of Lie algebras [151]. There is a long history of the applications of (semisimple) Lie algebras for determination of operator spectra. One line dating back to Weyl [450] relates to the explicit algebraic solution of an eigenvalue problem; an overview has been given by Joseph and Coulson [223, 224, 225]. Perhaps the best known example of such a construction is the quantum theory of angular momentum, including its development for many-particle systems (from three particles to aggregates) in terms of hyperspherical harmonics [154, 456]. The good old geometry of surfaces and conjugate nets uses the Laplace equations and transformations as a starting point [138]. The challenging problem of the Laplace operator factorization, perhaps first addressed by Laplace, created something like an “undressing” procedure which, being cut at some step, leads to the complete integrability. The direct attempt to extend the technique of the Laplace trans- formations and invariants to higher-order operators was made in [264]. In

Published: Jan 1, 2007

Keywords: Laplace Transformation; Hilbert Problem; Toda Lattice; Toda Chain; Hyperspherical Harmonic

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