  We study the energy landscape of a model of a single particle on a random
potential, that is, we investigate the topology of level sets of smooth random
fields on $\mathbb R^{N}$ of the form $X_N(x) +\frac\mu2 \|x\|^2,$ where
$X_{N}$ is a Gaussian process with isotropic increments. We derive asymptotic
formulas for the mean number of critical points with critical values in an open
set as the dimension $N$ goes to infinity. In a companion paper, we provide the
same analysis for the number of critical points with a given index.
