NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2015, 6 (6), P. 751-756
From “fat” graphs to metric graphs: the problem of boundary conditions
G. F. Dell’Antonio – Sapienza, Rome, Italy and SISSA, Via Bonomea 265, 34136, Trieste, Italy; email@example.com
A. Michelangeli – SISSA, Via Bonomea 265, 34136, Trieste, Italy and Center for Advanced Studies, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz, 1, 80539, Munich, Germany
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.
Keywords: quantum billiards, Schrödinger equation.
PACS 03.65.Ge, 02.30.Jr