From ``fat'' graphs to metric graphs: the problem of boundary conditions

G. F. Dell'Antonio ; A. Michelangeli

Nanosistemy: fizika, himiâ, matematika, Tome 6 (2015) no. 6, pp. 751-756.

Résumé

We discuss how the vertex boundary conditions for the dynamics of a quantum particle on a metric graph emerge when the dynamics is regarded as a limit of the dynamics in a tubular region around the graph. We give evidence for the fact that the boundary conditions are determined by the possible presence of a zero-energy resonance. Therefore, the boundary conditions depend on the shape of the fat graph near the vertex. We also give evidence, by studying the case of the half-line, for the fact that on the contrary, in general, adding on a graph a shrinking support potentials at the vertex either does not alter the boundary condition or does not produce a self-adjoint dynamics. Convergence, throughout, is meant in the sense of strongly resolvent convergence.