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

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.