  We study the energy landscape near the ground state of a model of a single
particle in a random potential with trivial topology. More precisely, we find
the large dimensional limit of the Hessian spectrum at the global minimum of
the Hamiltonian $X_N(x) +\frac\mu2 \|x\|^2, x\in\rz^N,$ when $\mu$ is above the
phase transition threshold so that the system has only one critical point. Here
$X_N$ is a locally isotropic Gaussian random field. The same idea is also
applied to study the more general model of elastic manifold. In the replica
symmetric regime, our results confirm the predictions of Fyodorov and Le
Doussal made in 2018 and 2020 using the replica method.
