Weakly coupled states on branching graphs

Details Weakly coupled states on branching graphs Weakly coupled states on branching graphs

1995-12-11

arxiv.org/abs/funct-an/9512001v1

Lett.Math.Phys. 38 (1996) 313-320; Erratum-ibid. 42 (1997) 193

Physics

Mathematical Physics

LaTeX file, 7 pages, no figures

Scientific paper

10.1007/BF00398355

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.

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