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

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 α n ( t ) , b n ( t ) of the Toda lattice is exactly determined and by taking t = 0 , the solution α n ( 0 ) , b n ( 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 α n ( 0 ) > 0 and b n ( 0 ) ∈ ℝ such that the semi-infinite Toda lattice is not integrable in the sense that the functions α n ( t ) and b n ( t ) are not finite for every t > 0 .