Hydrogen-like Schr\"odinger Operators at Low Energies

TL;DR

This paper analyzes the low-energy behavior of Schr"odinger operators with Coulomb-like potentials on asymptotically Euclidean manifolds, revealing oscillatory asymptotics and constructing detailed asymptotic expansions for solutions.

Contribution

It introduces a novel microlocal approach to understand the low-energy resolvent of Coulomb-like Schr"odinger operators, capturing oscillatory asymptotics uniformly down to zero energy.

Findings

Low-energy resolvent exhibits oscillatory asymptotics distinct from short-range cases.

The resolvent remains smooth down to zero energy.

Constructs a compactification where the resolvent is expressed by a complex oscillatory function.

Abstract

Consider a Schr\"odinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. Using Vasy's second 2nd-microlocal approach, "the Lagrangian approach," we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\pm i 0) = \lim_{\epsilon \to 0^+} R(E\pm i \epsilon)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmou, Hassell, Sikora, and (more recently) Hintz and Vasy. Specifically, the compound asymptotics at low energy and large spatial scales are more delicate, and the resolvent output is smooth all the way down to $E=0$. In fact, we will construct a compactification of $(0,1]_E\times X$ on which the resolvent output is given by a specified (and relatively complicated) function that oscillates as $r\to\infty$ times something polyhomogeneous. As a corollary, we get complete and compatible asymptotic expansions for solutions to the scattering problem as functions of both position and energy, with a transitional regime.