Reflectionless Schrödinger operators, the dynamics of zeros, and the solitonic Sato formula
Jan Felipe van Diejen and Heinrich Puschmann
The zeros of the (Jost) eigenfunction of a one-dimensional Schrödinger operator with a reflectionless rapidly decreasing potential are related to the spectral data through a nonlinear algebraic system of Bethe-type equations. We show that the behavior of these zeros (with respect to translations) is governed by a rational Ruijsenaars-Schneider particle system with harmonic term. The integration of the particle system–-via an explicit construction of the action-angle transform–-then provides us with detailed information on the solution curve of the Bethe equations. As a result, we find a Wilson-type determinantal formula for the eigenfunction involving Ruijsenaars-Schneider (Lax) matrices and we furthermore recover the solitonic Sato formula (which parametrizes the eigenfunction explicitly in terms of the spectral data). The flows corresponding to the higher integrals of the rational Ruijsenaars-Schneider system with harmonic term give rise to the soliton solutions of the Korteweg–de Vries hierarchy.