  We study amorphous solids with strong elastic disorder and find an un-jamming
instability that exists, inter alia, in an harmonic model built using Euclidean
random matrices (ERM). Employing the Zwanzig-Mori projection operator formalism
and Gaussian factorization approximations, we develop a first-principles,
self-consistent theory of transverse momentum correlations in athermal
disordered materials, extending beyond the standard Born approximation. The
vibrational anomalies in glass at low temperatures are recovered in the stable
solid limit, and floppy modes lacking restoring forces are predicted in
unstable states below the jamming transition. Near the un-jamming transition,
the speed of sound $v_0^\perp$ vanishes with $ \propto \sqrt{\epsilon}$, where
$\epsilon$ denotes the distance from the critical point. Additionally, the
density of states develops a plateau, independent of $\epsilon$ above a
frequency $\omega_*$ which vanishes at the transition, $\omega_*\propto
|\epsilon|$. We identify a characteristic length scale in the un-jammed phase,
$\lambda_-^\perp\propto1/\sqrt{\epsilon}$, indicating the distance over which
injected momentum remains correlated. We confirm the theoretical predictions
with numerical solutions of a scalar ERM model, demonstrating overall good
qualitative and partly quantitative agreement.
