Variational Problems¶on Multiply Connected Thin Strips I:¶Basic Estimates and Convergence¶of the Laplacian Spectrum

Abstract

Let M be a planar embedded graph whose arcs meet transversally at the vertices. Let ?(ɛ) be a strip-shaped domain around M, of width ɛ except in a neighborhood of the singular points. Assume that the boundary of ?(ɛ) is smooth. We define comparison operators between functions on ?(ɛ) and on M, and we derive energy estimates for the compared functions. We define a Laplace operator on M which is in a certain sense the limit of the Laplace operator on ?(ɛ) with Neumann boundary conditions. In particular, we show that the p-th eigenvalue of the Laplacian on ?(ɛ) converges to the p-th eigenvalue of the Laplacian on M as ɛ tends to 0. A similar result holds for the magnetic Schrödinger operator.