Spectral estimates for a class of Schrödinger operators with infinite phase space and potential unbounded from below

We analyse two-dimensional Schrödinger operators with the potential |xy|p − λ(x2 + y2)p/(p+2) where p ⩾ 1 and λ ⩾ 0. We show that there is a critical value of λ such that the spectrum for λ < λcrit is bounded below and purely discrete, while for λ > λcrit it is unbounded from below. In the subcritical case, we prove upper and lower bounds for the eigenvalue sums. (paper)