  We consider the $2$-spin spherical Sherrington--Kirkpatrick model whose
disorder is given by a deformed Wigner matrix of the form $W+\lambda V$, where
$W$ is a Wigner matrix and $V$ is a random diagonal matrix with i.i.d. entries.
Assuming that the density function of the entries of $V$ decays faster than a
certain rate near the edges of its spectrum, we prove the sharp phase
transition of the limiting free energy and its fluctuation. In the high
temperature regime, the fluctuation of $F_N$ converges in distribution to a
Gaussian distribution, whereas it converges to a Weibull distribution in the
low temperature regime. We also prove several results for deformed Wigner
matrices, including a local law for the resolvent entries, a central limit
theorem of the linear spectral statistics, and a theorem on the rigidity of
eigenvalues.
