Abstract and Applied Analysis
Volume 2004 (2004), Issue 5, Pages 435-451
doi:10.1155/S1085337504306135
Exact solutions of the semi-infinite Toda lattice with applications to the inverse spectral problem
Department of Mathematics, University of Patras, Patras 26500, Greece
Received 30 July 2002
Copyright © 2004 E. K. Ifantis and K. N. Vlachou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Several inverse spectral problems are solved by a method which is
based on exact solutions of the semi-infinite Toda lattice. In
fact, starting with a well-known and appropriate probability
measure μ, the solution αn(t), bn(t) of
the Toda lattice is exactly determined and by taking t=0,
the solution αn(0), bn(0) of the inverse
spectral problem is obtained. The solutions of the Toda lattice
which are found in this way are finite for every t>0 and can
also be obtained from the solutions of a simple differential
equation. Many other exact solutions obtained from this
differential equation show that there exist initial conditions
αn(0)>0 and bn(0)∈ℝ such that the
semi-infinite Toda lattice is not integrable in the sense that
the functions αn(t) and bn(t) are not finite for every
t>0.