 95522 Pavel Exner
 Weakly coupled states on branching graphs
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Dec 11, 95

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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 selfadjointness 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 nonrepulsive 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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