  The concept of vibrational density of states in glasses has been mirrored in
liquids by the instantaneous-normal-mode spectrum. While in glasses
instantaneous configurations correspond to minima of the potential-energy
hypersurface and all eigenvalues of the associated Hessian matrix are therefore
positive, in liquids this is no longer true, and modes corresponding to both
positive and negative eigenvalues exist. The instantaneous-normal-mode spectrum
has been numerically investigated in the past, and it has been demonstrated to
bring important information on the liquid dynamics. A systematic deeper
theoretical understanding is now needed. Heterogeneous-elasticity theory has
proven to be successful in explaining many details of the low-frequency
excitations in glasses, ranging from the thoroughly studied boson peak, down to
the more elusive non-phononic excitations observed in numerical simulations at
the lowest frequencies. Here we present an extension of
heterogeneous-elasticity theory to the liquid state, and show that the outcome
of the theory agrees well to the results of extensive molecular-dynamics
simulations of a model liquid at different temperatures. We show that the
spectral shape strongly depends on temperature, being symmetric at high
temperatures and becoming rather asymmetric at low temperatures, close to the
dynamical critical temperature. Most importantly, we demonstrate that the
theory naturally reproduces a surprising phenomenon, a zero-energy spectral
singularity with a cusp-like character developing in the vibrational spectra
upon cooling. This feature, known from a few previous numerical studies, has
been generally overlooked in the past due to a misleading representation of the
data. We provide a thorough analysis of this issue, based on both very accurate
predictions of our theory, and computational studies of model liquid systems
with extended size.
