Multifractal scaling of quantum eigenfunctions in the thermodynamic limit

Abstract

We prove multifractal scaling of measures related to the eigenfunctions of Schrödinger operators with point-like bump potentials, in the thermodynamic limit, in dimension $d\geq 5$. This is consistent with a longstanding expectation in the physics literature in these dimensions. The fractal exponents are computed explicitly as a function of the parameter defining the moments. Our result requires a weak disorder condition on the location of the scatterers and a strong coupling condition on the sequence of eigenvalues.