Toda-Schr\"odinger correspondence and orthogonal polynomials
Mathematical Physics
2014-04-09 v1 math.MP
Abstract
It is known that the unrestricted Toda chain is equivalent to the Riccati equation for the Stieltjes function of the orthogonal polynomials. Under a special condition, this Riccati equation can be reduced to the Schr\"odinger equation. We show that this condition is equivalent to type B solutions of the Toda chain. We establish some nontrivial consequences arising from this Toda-Schr\"odinger correspondence. In particular, we show that the KdV densities can be identified with the moments of the corresponding orthogonal polynomials. We establish equivalence between type B solutions of the Toda molecule and the Bargmann potentials of the Schr\"odinger equation.
Keywords
Cite
@article{arxiv.1404.2012,
title = {Toda-Schr\"odinger correspondence and orthogonal polynomials},
author = {Satoshi Tsujimoto and Alexei Zhedanov},
journal= {arXiv preprint arXiv:1404.2012},
year = {2014}
}