Contribution of poles to the large parameter asymptotics of the solution to the free-Schrödinger equation

Abstract

We obtain the contribution of a simple pole to the large asymptotic expansion of the solution to the free-Schrödinger equation. The specific solution that is considered is a Stieltjes transform of a separable solution of the same Schrödinger equation. In large parameter asymptotics, the moments for the free-Schrödinger equation generally do not exist, so that the classical theory of asymptotics fails to evaluate the resulting divergent infinite series. Therefore, a special treatment is necessary for the moments to exist. This involves rotating the contour of integration within the fourth quadrant of the complex plane. Moreover, we demonstrate that the contribution of a simple pole is more significant than the remainder term in the large parameter regime.