Laplace operator in networks of thin fibers: Spectrum near the threshold

Abstract

Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in [21]. The present paper contains a detailed review of [21] under some assumptions which make the results much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation.

Graph-like models for thin waveguides with Robin boundary conditions

“…For these reasons even in the simpler example of a waveguide in most of the cases the operator on the corresponding graph is defined by the decoupling condition in the vertex and in general, for fixed α, one can expect coupling at most in one transverse mode. This is in agreement with previous results derived in [3,7,15,32].…”

“…This was already argued by Molchanov and Vainberg, see [32]. In these works, for compact networks, the spectral convergence of the Laplacian on the network to the operator on the graph is studied; the approach is based on the analysis of the scattering problem associated to the network of tubes and makes use of the analytic properties of the resolvent of the Laplacian on the manifold.…”

“…The analysis of the case δ ε = ε would be of great interest because it reproduces the scaling of [15] and [32]. The failing of the adiabatic approach makes this case much more complicated and we guess that in this setting the uniform resolvent convergence could be too demanding.…”

Effective Schrödinger dynamics on ε-thin Dirichlet waveguides via quantum graphs: I. Star-shaped graphs

“…It is to be expected that a resonance (or its counterpart, the resonant sequence) is not present for a generic waveguide, accordingly to the general wisdom that decoupling generically occurs for thin Dirichlet waveguides (Molchanov & Vainberg 2007) (Molchanov & Vainberg 2008) (Grieser 2008a).…”

Asymptotics of eigenfunctions on plane domains

“…The Dirichlet problem is more difficult to handle since the dependence on the counting parameter k appears in a lower order term; compare with (1-2). The Dirichlet and mixed boundary value problems are studied in [Grieser 2008a] and [Molchanov and Vainberg 2007]. The existence of asymptotic expansions for the eigenvalues and eigenfunctions in terms of ε = N −1 can also be proved by the methods of these papers; however, Theorem 1 contains more precise information that we need here, which is specific to the situation of domains (1-1).…”