  We present a novel mechanism for the anomalous behaviour of the specific heat
in low-temperature amorphous solids. The analytic solution of a mean-field
model belonging to the same universality class as high-dimensional glasses, the
spherical perceptron, suggests that there exists a crossover temperature above
which the specific heat scales linearly with temperature while below it a cubic
scaling is displayed. This relies on two crucial features of the phase diagram:
(i) The marginal stability of the free-energy landscape, which induces a
gapless phase responsible for the emergence of a power-law scaling (ii) The
vicinity of the classical jamming critical point, as the crossover temperature
gets lowered when approaching it. This scenario arises from a direct study of
the thermodynamics of the system in the quantum regime, where we show that,
contrary to crystals, the Debye approximation does not hold.
