## Abstract and Applied Analysis

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

#### Article information

Source
Abstr. Appl. Anal., Volume 2004, Number 5 (2004), 435-451.

Dates
First available in Project Euclid: 1 June 2004

https://projecteuclid.org/euclid.aaa/1086103974

Digital Object Identifier
doi:10.1155/S1085337504306135

Mathematical Reviews number (MathSciNet)
MR2063337

Zentralblatt MATH identifier
1070.37054

Subjects
Primary: 34A55: Inverse problems 37K10
Secondary: 37L60

#### Citation

Ifantis, E. K.; Vlachou, K. N. Exact solutions of the semi-infinite Toda lattice with applications to the inverse spectral problem. Abstr. Appl. Anal. 2004 (2004), no. 5, 435--451. doi:10.1155/S1085337504306135. https://projecteuclid.org/euclid.aaa/1086103974