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 $\mu$, the solution ${\alpha }_n\left(t\right)$, $b_n\left(t\right)$ of the Toda lattice is exactly determined and by taking $t=0$, the solution ${\alpha }_n\left(0\right)$, $b_n\left(0\right)$ 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 ${\alpha }_n\left(0\right)>0$ and $b_n\left(0\right)\in ℝ$ such that the semi-infinite Toda lattice is not integrable in the sense that the functions ${\alpha }_n\left(t\right)$ and $b_n\left(t\right)$ are not finite for every $t>0$.