Asymptotics of reflectionless potentials.

Abstract

Analytic reflectionless potentials ${\mathrm{\ensuremath{\omega}}}^{2}$(\ensuremath{\tau}) are constructed for the one-dimensional equation ${\mathrm{\ensuremath{\epsilon}}}^{2}$${\mathit{d}}^{2}$q/d${\mathrm{\ensuremath{\tau}}}^{2}$+${\mathrm{\ensuremath{\omega}}}^{2}$(\ensuremath{\tau})q=0. Unlike generic potentials which reflect waves with amplitudes of order exp(-1/\ensuremath{\epsilon}) as \ensuremath{\epsilon}\ensuremath{\rightarrow}0, these potentials have reflection coefficients which are identically 0. It is shown that in the reflectionless case the adiabatic perturbation or iteration does not converge absolutely or terminate at some order. Since exact integrability is less restrictive than having a reflectionless potential, the case studied also shows that integrability does not imply convergence of the approximation methods used.