This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < ǫh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as ǫ → 0, and find the twoterm asymptotics in ǫ → 0 of ea...

It is shown that the eigenvalues of the equation −λ∆u = V u on a graph G of final total length |G|, with non-negative V ∈ L(G) and under appropriate boundary conditions, satisfy the inequality n2λn ≤ |G| ∫ G V dx, independently of geometry of a given graph. Applications and generalizations of this result are also discussed.

We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The ...