  We study the landscape complexity of the Hamiltonian $X_N(x) +\frac\mu2
\|x\|^2,$ where $X_{N}$ is a smooth Gaussian process with isotropic increments
on $\mathbb R^{N}$. This model describes a single particle on a random
potential in statistical physics. We derive asymptotic formulas for the mean
number of critical points of index $k$ with critical values in an open set as
the dimension $N$ goes to infinity. In a companion paper, we provide the same
analysis without the index constraint.
