Pavel Exner
Weakly coupled states on branching graphs
(21K, LaTeX)
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.