  To test the stability of the Parisi solution near T=0, we study the spectrum
of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues
are the masses of the bare propagators in the expansion around the mean-field
solution. In the limit $T\ll 1$ two regions can be identified. In the first
region, for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking
scheme parameter, the spectrum of the Hessian is not trivial and maintains the
structure of the full replica symmetry breaking state found at higher
temperatures. In the second region $T\ll x \leq 1$ as $T\to 0$, the components
of the Hessian become insensitive to changes of the overlaps and the bands
typical of the full replica symmetry breaking state collapse. In this region
only two eigenvalues are found: a null one and a positive one, ensuring
stability for $T\ll 1$. In the limit $T\to 0$ the width of the first region
shrinks to zero and only the positive and null eigenvalues survive. As
byproduct we enlighten the close analogy between the static Parisi replica
symmetry breaking scheme and the multiple time-scales approach of dynamics, and
compute the static susceptibility showing that it equals the static limit of
the dynamic susceptibility computed via the modified fluctuation dissipation
theorem.
