Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition

We analyze two-dimensional Schrödinger operators with the potential jxyjp 􀀀 (x2 + y2)p=(p+2) where p 1 and 0, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive hal ine while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.