Regularity of the invariant measure and of the density of states in the one-dimensional Anderson model
The one dimensional random Schrödinger operator H acts on a sequence \(\{\psi (n)\}\in \ell^ 2({\mathbb{Z}})\) by the formula \[ [H\psi](n)=-\psi (n+1)-\psi (n-1)+V(n)\psi (n). \] Assuming that \(\{\) V(n); \(n\in {\mathbb{Z}}\}\) is an i.i.d. sequence of real random variables with distribution \(\mu\) not concentrated on one point and such that \({\mathbb{E}}\{\log (1+| V(0)| \}<\infty\) it is known that for any value of the energy \(\lambda\) there exists a unique probability measure \(\nu_{\lambda}\) on the real line satisfying the invariance equation \[ \int {\mathbb{E}}\{f(V(0)-\lambda -1/x)\}d\nu_{\lambda}(x)=\int f(x)d\nu_{\lambda}(x) \] for any bounded Borel function f. Moreover one can construct a probability distribution on the real line, say N, called the distribution of states. These two probability distributions are continuous and a challenging question is to determine whether they are absolutely continuous or not and to study the smoothness of their densities. It has been proved by \textit{E. Le Page} [Probability measures on groups VII, Proc. Conf., Oberwolfach 1983, Lect. Notes Math. 1064, 309-367 (1984; Zbl 0538.60031)] that if \(\mu\) is absolutely continuous with moments of all orders, then \(\nu_{\lambda}\) and N also are absolutely continuous with \(C^{\infty}\) densities. Here the authors assume a weaker property of the distribution of potential \(\mu\), namely that its Fourier transform and its derivatives up to the order n are bounded functions, going to 0 at infinity. As a consequence they obtain that \(\nu_{\lambda}\) and N are smooth distributions up to the order n for N and (n-1)/2-1 for \(\nu_{\lambda}.\) An essential step in the proof is to perform a Fourier transform on the invariance equation given above and to study the spectral properties of the operators associated with the new equation. A careful choice of the underlying Banach spaces allows to prove that these operators are compact with 1 as the only eigenvalue of modulus one and moreover with algebraic multiplicity one. Then routine perturbation arguments yield the desired results.
scientific article; zbMATH DE number 4145059