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Weakly coupled states on branching graphs
We consider a Schr\"odinger particle on a graph consisting of links
joined at a single point. Each link supports a real locally integrable
potential ; the self--adjointness is ensured by the type
boundary condition at the vertex. If all the links are semiinfinite and ideally
coupled, the potential decays as 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
coupling constant may be interpreted in terms of a family of
squeezed potentials.Comment: LaTeX file, 7 pages, no figure
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