Leonid Pastur, B.I.Verkin: Analogs of Szegö’s Theorem for Ergodic Operators

We present a setting generalizing that for Szegö’s theorem on the Töplitz (or discrete convolution) operators. Viewing the theorem as an asymptotic trace formula determined by a certain underlying operator and by two functions (the symbol and the test function), we replace the Töplitz operator by an ergodic operator (e.g. random or quasiperiodic), in particular, by the discrete Schrödinger operator with ergodic potential.
In the framework of this setting we discuss a variety of asymptotic formulas different from those given by Szegö’s theorem and including certain Central Limit Theorems in spectral context and large block asymptotic formulas for the entanglement entropy of free disordered fermions.