  We consider an elastic manifold of internal dimension $d$ and length $L$
pinned in a $N$ dimensional random potential and confined by an additional
parabolic potential of curvature $\mu$. We are interested in the mean spectral
density $\rho(\lambda)$ of the Hessian matrix $K$ at the absolute minimum of
the total energy. We use the replica approach to derive the system of equations
for $\rho(\lambda)$ for a fixed $L^d$ in the $N \to \infty$ limit extending
$d=0$ results of our previous work. A particular attention is devoted to
analyzing the limit of extended lattice systems by letting $L\to \infty$. In
all cases we show that for a confinement curvature $\mu$ exceeding a critical
value $\mu_c$, the so-called "Larkin mass", the system is replica-symmetric and
the Hessian spectrum is always gapped (from zero). The gap vanishes
quadratically at $\mu\to \mu_c$. For $\mu<\mu_c$ the replica symmetry breaking
(RSB) occurs and the Hessian spectrum is either gapped or extends down to zero,
depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes
in all $d$ as $(\mu_c-\mu)^4$ near the transition. In the full RSB case the gap
is identically zero. A set of specific landscapes realize the so-called
"marginal cases" in $d=1,2$ which share both feature of the 1-step and the full
RSB solution, and exhibit some scale invariance. We also obtain the average
Green function associated to the Hessian and find that at the edge of the
spectrum it decays exponentially in the distance within the internal space of
the manifold with a length scale equal in all cases to the Larkin length
introduced in the theory of pinning.
