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 δ type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x −1−∈ along each of them, is nonrepulsive 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 δ coupling constant may be interpreted in terms of a family of squeezed potentials.
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Exner, P. Weakly coupled states on branching graphs. Lett Math Phys 38, 313–320 (1996). https://doi.org/10.1007/BF00398355
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DOI: https://doi.org/10.1007/BF00398355