Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

TL;DR

This paper establishes a detailed asymptotic analysis of the resolvent at zero energy for certain long-range Schrödinger operators, using advanced spectral and microlocal techniques.

Contribution

It provides the first comprehensive asymptotic expansion of the resolvent at zero energy for potentials with positive virial at infinity, extending previous results to long-range cases.

Findings

Proved a limiting absorption principle at zero energy.

Derived a complete asymptotic expansion of the resolvent.

Established absence of eigenvalues at zero energy.

Abstract

We prove a limiting absorption principle at zero energy for two-body Schr\"odinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter varies in cones; one of the two branches of boundary for the cones being given by the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.