ON THE SPECTRUM OF THE DIRICHLET LAPLACIAN IN A NARROW STRIP, by Leonid Friedlander and Michael Solomyak

There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière used Riemannian metrics concentrated in a small neighborhood of a graph to prove that for every manifold M of dimension greater than two and for every positive number N there exists a Riemannian metric g such that the multiplicity of the smallest positive eigenvalue of the Laplacian on (M, g) equals N . Recent interest to the problem is, in particular, motivated by possible applications to mesoscopic systems. Rubinstein and Schatzman studied in [10] eigenvalues of the Neumann Laplacian in a narrow strip surrounding an embedded planar graph. The strip has constant width ǫ everywhere except neighborhoods of vertices. Under some assumptions on the structure of the strip near vertices, they proved that eigenvalues of the Neumann Laplacian converge to eigenvalues of the Laplacian on the graph. Kuchment and Zheng extended in [6] these results to the case when the strip width is not constant. The Dirichlet boundary condition turns out to be more complicated than the Neumann condition. Eigenvalues of a domain of width ǫ are bounded from below by π/ǫ. Post studied in [9] eigenvalues λj(ǫ) of the Dirichlet Laplacian in a neighborhood of a planar graph that has constant width ǫ near the edges and that narrows down toward the vertices. He proved that λj(ǫ) − π/ǫ converge to the eigenvalues of the direct sum of certain Schrödinger operators on the edges with the Dirichlet boundary conditions. We show that this result can not be extended to neighborhoods of variable width. If the width is not constant then the spectrum of the Dirichlet Laplacian is basically determined by the points where it is the widest. In the paper, we treat a simple model case: the graph is a straight segment, and the strip is the widest in one cross-section. In this case, we derive a two-term asymptotics for λj(ǫ). We will formulate now main results of the paper. Let h(x) > 0 be a continuous function defined on a segment I = [−a, b], where a, b > 0. We assume that (i) x = 0 is the only point of global maximum of h(x) on I;