Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph

Let M be a finite graph in the plane and let Mε be a domain that looks like the ε-fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on Mε converges when ε → 0 to the spectrum of an ODE problem on M. The presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman.