Schroedinger operator, eigenvalue estimates, spectral transition

We analyze two-dimensional Schrooedinger operators with the potential |xy|^p -\lambda(x^2 +y^2)^{p/ (p+2)} where p\ge1 and \lambda\ge0 which exhibit an abrupt change of spectral properties at a critical value of the coupling constant \lambda. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for \lambda below the critical value the spectrum is purely discrete and we establish a Lieb? Thirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.