Sharp eigenvalue bounds on quantum star graphs

TL;DR

This paper investigates eigenvalue bounds on quantum star graphs, showing that for even edges the bounds match those on the real line, and for odd edges, they match under radial symmetry or are nearly optimal generally.

Contribution

It establishes the optimal Lieb-Thirring constant on star graphs, revealing parity-dependent behavior and extending bounds to non-radial potentials.

Findings

Optimal constant matches that on  for even N

Radial potentials satisfy the same bounds for odd N

Nearly optimal bounds are achieved for general potentials

Abstract

We prove that the optimal constant in the Lieb--Thirring inequality on a star graph with $N$ edges coincides with that on $\mathbb R$ if $N$ is even. For odd $N$ we show that this property holds when restricting to radial potentials and we prove an almost optimal bound for general potentials.