  Using the replica method we calculate the mean spectral density of the
Hessian matrix at the global minimum of a random $N \gg 1$ dimensional
isotropic, translationally invariant Gaussian random landscape confined by a
parabolic potential with fixed curvature $\mu>0$. Simple landscapes with
generically a single minimum are typical for $\mu>\mu_{c}$, and we show that
the Hessian at the global minimum is always {\it gapped}, with the low spectral
edge being strictly positive. When approaching from above the transitional
point $\mu= \mu_{c}$ separating simple landscapes from 'glassy' ones, with
exponentially abundant minima, the spectral gap vanishes as $(\mu-\mu_c)^2$.
For $\mu<\mu_c$ the Hessian spectrum is qualitatively different for 'moderately
complex' and 'genuinely complex' landscapes. The former are typical for
short-range correlated random potentials and correspond to 1-step
replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again
gapped, with the gap vanishing on approaching $\mu_c$ from below with a larger
critical exponent, as $(\mu_c-\mu)^4$. At the same time in the 'most complex'
landscapes with long-ranged power-law correlations the replica symmetry is
completely broken. We show that in that case the Hessian remains gapless for
all values of $\mu<\mu_c$, indicating the presence of 'marginally stable'
spatial directions. Finally, the potentials with {\it logarithmic} correlations
share both 1RSB nature and gapless spectrum. The spectral density of the
Hessian always takes the semi-circular form, up to a shift and an amplitude
that we explicitly calculate.
