Weakly coupled states on branching graphs

Abstract

We consider a Schrödinger 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 Schrödinger 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.

LaTeX file, 7 pages, no figures

References in corpus (3)

• Contact interactions on graph superlattices
• Band spectra of rectangular graph superlattices
• Quantum Mechanics of Multi-Prong Potentials

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