The distribution of negative eigenvalues of Schr\"odinger operators on asymptotically hyperbolic manifolds

TL;DR

This paper investigates how the negative eigenvalues of Schr"odinger operators behave on asymptotically hyperbolic manifolds, establishing conditions for their finiteness and asymptotic distribution near zero.

Contribution

It provides new criteria for the finiteness of negative eigenvalues and describes their asymptotic distribution near zero on asymptotically hyperbolic manifolds.

Findings

Conditions for finite or infinite negative eigenvalues

Asymptotic behavior of eigenvalue counting function near zero

Eigenvalues can only accumulate at zero

Abstract

We study the asymptotic behavior of the counting function of negative eigenvalues of Schr\"odinger operators with real valued potentials on asymptotically hyperbolic manifolds. We establish conditions on the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval $(-\infty,-E)$ as $E\rightarrow 0$.