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Generalizations of Three-Term Relations in Solvable Models of Mathematical Physics

thesis
posted on 2011-04-14, 00:00 authored by Olena Korovnichenko
We study various aspects of three-term relations and their generalizations in solvable problem of mathematical physics. First, we consider Leonard pairs, pairs of matrices with the property of mutual tri-diagonality. We introduce and study a classical analogue of Leonard pairs and show that functions forming Leonard pairs satisfy non-linear relations of the AW- type with respect to Poisson brackets. Continuing this approach we introduce Leonard triples and describe an algorithm which leads to a chain of Leonard triples. Second, we introduce a family of non-Abelian nonlinear Kostant-Toda lattices in $GL_n$. We introduce some orbits with special parametrization, present evolution equation on these orbits and show that matrix Weyl functions can be used to encode the Hamiltonian structure of these lattices, to establish their complete integrability and to explicitly solve them via the matrix generalization of the inverse moment problem.

History

Date Modified

2017-06-05

Defense Date

2011-04-07

Research Director(s)

Mary Ann McDowell

Committee Members

Michael Gekhtman Sam Evens Leonid Faybusovich J. Arlo Caine

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

etd-04142011-145154

Publisher

University of Notre Dame

Program Name

  • Mathematics

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