Weakly coupled Schroedinger operators on regular metric trees

TL;DR

This paper investigates how the spectral properties of a Schrödinger operator on regular metric trees change with the coupling parameter, revealing that the behavior of negative eigenvalues depends on the tree's global structure as the coupling tends to zero.

Contribution

It provides new insights into the spectral behavior of Schrödinger operators on regular metric trees, emphasizing the influence of the tree's global structure on eigenvalue asymptotics.

Findings

Negative eigenvalues' behavior depends on the global structure of the tree as coupling approaches zero.

Spectral properties are characterized for regular metric trees.

The asymptotic behavior of eigenvalues is linked to the geometry of the underlying tree.

Abstract

Spectral properties of the Schroedinger operator $A_{\lambda} = -\Delta +\lambda V$ on regular metric trees are studied. It is shown that as $\lambda$ goes to zero the behavior of the negative eigenvalues of $A_{\lambda}$ depends on the global structure of the tree.