Variational proof of the existence of eigenvalues for star graphs

TL;DR

This paper presents a variational method to prove the existence of eigenvalues for Schrödinger operators on star graphs, applicable without restrictions on the graph's structure, and provides bounds for the lowest eigenvalue.

Contribution

It introduces a new variational proof technique for eigenvalues on star graphs that does not require assumptions on the number or arrangement of rays.

Findings

Eigenvalues exist below the essential spectrum for the considered operator.

A new upper bound for the lowest eigenvalue is established.

The method applies to general star graphs without structural restrictions.

Abstract

We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schr\"odinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.