Diana Barseghyan: Spectral estimates for Schrödinger operators with unusual semiclassical behavior

In first part of talk we analyze two-dimensional Schrödinger operators with potentials unbounded from below and depended on some parameter. We show that there is a critical value such that in subcritical case of parameters the spectrum is bounded below and purely discrete, while in supercritical case it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums. The second part of talk is devoted to estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infnite cusps which are geometrically nontrivial being either curved or twisted; we show how those geometric properties enter the eigenvalue bounds.