  Atomic vibrations play a vital role in the functions of various physical,
chemical, and biological systems. The vibrational properties and the specific
heat of crystalline bulk materials are well described by Debye theory, which
successfully predicts the quadratic $\omega^{2}$ low-frequency scaling of the
vibrational density of states (VDOS) in bulk ordered solids from few
fundamental assumptions. However, the analogous framework for nanoconfined
materials with fewer degrees of freedom has been far less well explored. Using
inelastic neutron scattering, we characterize the VDOS of amorphous ice
confined to a thickness of $\approx 1$ nm inside graphene oxide membranes and
we observe a crossover from the Debye $\omega^2$ scaling to an anomalous
$\omega^3$ behaviour upon reducing the confinement size $L$. Additionally,
using molecular dynamics simulations, we confirm the experimental findings and
also prove that such a scaling of the VDOS appears in both crystalline and
amorphous solids under slab-confinement. We theoretically demonstrate that this
low-frequency $\omega^3$ law results from the geometric constraints on the
momentum phase space induced by confinement along one spatial direction.
Finally, we predict that the Debye scaling reappears at a characteristic
frequency $\omega_\times= v L/2\pi$, with $v$ the speed of sound of the
material, and we confirm this quantitative estimate with simulations. This new
physical phenomenon, revealed by combining theoretical, experimental and
simulations results, is relevant to a myriad of systems both in synthetic and
biological contexts and it could impact various technological applications for
systems under confinement such as nano-devices or thin films.

Materials Science;Mesoscale and Nanoscale Physics;Soft Condensed Matter;Chemical Physics