  The multifractal superconducting state originates from the interplay of
Anderson localization and interaction effects. In this article we overview the
recent theory of the superconductivity enhancement by multifractality and
extend it to describe the spectral properties of superconductors on the scales
of the order of the superconducting gap. Specifically, using the approach based
on renormalization group within the nonlinear sigma model, we develop the
theory of a multifractal superconducting state in thin films. We derive a
modified Usadel equation that incorporates the interplay of disorder and
interactions at energy scales larger than the spectral gap and study the effect
of such an interplay on the low-energy physics. We determine the spectral gap
at zero temperature which occurs to be proportional to the multifracally
enhanced superconducting transition temperature. The modified Usadel equation
results in the disorder-averaged density of states that, near the spectral gap,
resembles the one obtained in the model of a spatially random superconducting
order parameter. We reveal strong mesoscopic fluctuations of the local density
of states in the superconducting state. Such strong mesoscopic fluctuations
imply that the interval of energies in which the superconducting gap
establishes is parametrically large in systems with multifractally-enhanced
superconductivity.
