Schroedinger operator, discrete spectrum, Lieb-Thirring inequality, cusp-shaped regions,geometrically induced spectrum.

In this paper we discuss several examples of Schroedinger operators describing a particle confined to a region with thin cusp-shaped "channels", given either by a potential or by a Dirichlet boundary; we focus on cases when the allowed phase space is infinite but the operator still has a discrete spectrum. First we analyze two-dimensional operators with the potential|x y|^p-\lambda(x^2+y^2)^{p/(p+2)} wherep\ge1 and \lambda\ge0.We show that there is a critical value of\lambda such that the spectrum for \lambda<\lambda_{crit}is below bounded and purely discrete, while for\lambda>\lambda_{crit}it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums. The second part of work is devoted to estimates of eigenvalue moments for Dirichlet Laplacians and Schroedinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how these geometric properties enter the eigenvalue bounds.