Spectral transition, Schroedinger operators, discrete spectrum, essential spectrum

We analyze two-dimensional Schroedinger operators -\Delta-\lambda (x^2 + y^2), \lambda\ge0 on a region with four hyperbolic "horns", D=\{(x, y)\in \mathbb{R}^2:|x y|<l\} with the Dirichlet boundary conditions on \partial D. We study the spectrum for different regimes of coupling constant \lambda.