Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide. (English) Zbl 1136.81442

Cacciapuoti, Claudio; Exner, Pavel

J. Phys. A, Math. Theor. 40, No. 26, F511-G523 (2007).

Summary: In distinction to the Neumann case, the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of \(\delta\)-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.

MSC:

81V99	Applications of quantum theory to specific physical systems
35J10	Schrödinger operator, Schrödinger equation
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35Q40	PDEs in connection with quantum mechanics
82D37	Statistical mechanics of semiconductors