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 R of the form XN(x)+μ2‖x‖2, where X 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.