Toda-Schrödinger correspondence and orthogonal polynomials

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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