## Abstract and Applied Analysis

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

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

E. K. Ifantis and K. N. Vlachou

#### 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 ${b}_{n}\left(0\right)\in \mathbb{R}$ 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 $t>0$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 1 June 2004

**Permanent link to this document**

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