  We analyze two-dimensional Schr\"odinger operators with the potential $|xy|^p
- \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$. We show that
there is a critical value of $\lambda$ such that the spectrum for
$\lambda<\lambda_\mathrm{crit}$ is below bounded and purely discrete, while for
$\lambda>\lambda_\mathrm{crit}$ it is unbounded from below. In the subcritical
case we prove upper and lower bounds for the eigenvalue sums.
