Weakly coupled states on branching graphs

TL;DR

This paper analyzes the spectral properties of a Schrödinger operator on a star graph with weak, decaying potentials, revealing conditions for a single negative eigenvalue and bounds on the number of bound states.

Contribution

It provides asymptotic behavior of the negative eigenvalue and bounds on the number of bound states for Schrödinger operators on star graphs with weak, decaying potentials.

Findings

Single negative eigenvalue under certain decay and coupling conditions

Asymptotic behavior of the negative eigenvalue derived

Bound on the number of bound states established

Abstract

We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $\,x^{-1-\epsilon}$ along each of them, is non--repulsive in the mean and weak enough, the corresponding Schr\"odinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the $\,\delta\,$ coupling constant may be interpreted in terms of a family of squeezed potentials.