On the spatial extent of localized eigenfunctions for random Schrödinger operators

Abstract

On \(\mathbb {Z}^d\), consider \(\varphi \), an \(\ell ^2\)-normalized function that decays exponentially at \(\infty \) at a rate at least \(\mu \). One can define the onset length (of the exponential decay) of \(\varphi \) as the radius of the smallest ball, say, B, such that one has the following global bound \(\displaystyle |\varphi (x)|\le \Vert \varphi \Vert _\infty e ^{-\mu \, \text {dist}(x,B)}\). The present paper is devoted to the study of the onset lengths of the localized eigenfunctions of random Schrödinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than \(\ell \) and localization center in a ball of radius L is smaller than \(C L^d\exp (-c \ell )\), for \(\ell >0\) large (for some constants \(C,c>0\)). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. For energies near the edge of the spectrum, we also provide a lower bound of the same type on the number of those eigenfunctions; in dimension 1, the upper and lower bounds only differ by a logarithmic correction. Finally, we give a number of numerical results that exemplify situations giving rise to large onset lengths, that corroborate the validity of our main result and that suggest that, up to lower order terms, the above defined cumulative distribution of onset lengths shows asymptotic exponential decay at some definite rate.