Two-body threshold spectral analysis, the critical case

TL;DR

This paper analyzes the low-energy spectral and scattering behavior of two-body Schrödinger operators with inverse-square potentials in higher dimensions, revealing oscillatory resolvent behavior and deriving phase shift asymptotics.

Contribution

It provides a detailed spectral analysis of Schrödinger operators with inverse-square potentials, including asymptotics of the resolvent and phase shifts in all angular momentum sectors.

Findings

Resolvent exhibits oscillatory behavior near zero energy.

Infinite negative eigenvalues accumulate at zero.

Derived asymptotic formula for phase shift.

Abstract

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schr\"odinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular momentum sectors, labelled by $l=0,1,\dots$, for which $\gamma>(l+d/2-1)^2$. In each such sector the reduced Schr\"odinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.