Eigenvalue estimates for Schroedinger operators on metric trees

TL;DR

This paper establishes eigenvalue bounds for Schroedinger operators on regular metric trees, demonstrating how these bounds depend on the trees' volume growth and are valid across different coupling regimes.

Contribution

It provides new Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities tailored for Schroedinger operators on metric trees, highlighting the influence of volume growth.

Findings

Bounds depend on volume growth of the tree

Inequalities hold at endpoint cases

Bounds reflect correct order in coupling limits

Abstract

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.