Schr\"odinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds

TL;DR

This paper analyzes Schr"odinger operators on the half line, providing resolvent expansions at zero energy and applying these to eigenvalue perturbations and the Fermi golden rule.

Contribution

It offers explicit resolvent asymptotics at zero energy for Schr"odinger operators with various potentials, and extends the Fermi golden rule to embedded eigenvalues at zero.

Findings

Resolved the classification of zero in the spectrum.

Derived explicit resolvent expansions at zero energy.

Applied results to eigenvalue perturbations and Fermi golden rule.

Abstract

We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is classified, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coefficients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modified form of the Fermi golden rule.