Borel summability in the disorder parameter of the averaged Green's function for Gaussian disorder
In this note we consider a tight-binding model \(H_ v=-\Delta +V\). Here -\(\Delta\) is the lattice Laplacian on \(Z^{\nu}\) and \(\{V_ i\}_{i\in Z^{\nu}}\) is a collection of independent identically distributed random variables with common distribution \(d\lambda (V)=(2\pi \gamma)^{- 1/2}\exp (-V^ 2/2\gamma)dV.\) Averages are performed with respect to the product measure. It is shown that for \(| E|\) large enough the averaged Green's function \(<G_{00}(E,\gamma)>\) has two branches in the complex E-plane which are obtained by starting in the upper and lower half-plane resp. Both branches can be continued beyond the real axis establishing analyticity of the density of states \(\rho \sim <G_{00}(E+io)>-<G_{00}(E-io)>\) in the large \(| E|\) region. Furthermore for each of the branches we give an asymptotic series in powers of \(\gamma\) which is proved to be Borel summable to \(<G_{00}(E,\gamma)>.\) For further applications see: \textit{F. Constantinescu} and \textit{U. Scharffenberger}: ''Borel transform, resurgence and the density of states: lattice Schrödinger operators with exponential disorder.'' J. Phys. A. 18, L 999 - L 1002 (1985).
scientific article; zbMATH DE number 3932166
scientific article