Abstract and Applied Analysis
Volume 2004 (2004), Issue 5, Pages 435-451

Exact solutions of the semi-infinite Toda lattice with applications to the inverse spectral problem

E. K. Ifantis and K. N. Vlachou

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.


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) 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.