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arXiv · 1404.2012

Toda-Schrödinger correspondence and orthogonal polynomials

Satoshi Tsujimoto·Alexei Zhedanov
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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 Schrödinger 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-Schrödinger 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 Schrödinger equation.

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Satoshi Tsujimoto, Alexei Zhedanov. 2014-04-08. Toda-Schrödinger correspondence and orthogonal polynomials. https://arxiv.org/abs/1404.2012

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