Serguei Naboko

Moment analysis for localization in random Schrödinger operators

We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

The Spectral Shift Operator

We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the Birman-Solomyak spectral averaging formula are discussed.

Infinite Jacobi Matrices with Unbounded Entries: Asymptotics of Eigenvalues and the Transformation Operator Approach

In this paper the exactasymptotics of eigenvalues

LOCALIZATION NEAR FLUCTUATION BOUNDARIES VIA FRACTIONAL MOMENTS AND APPLICATIONS

\lambda_n (J), \ n \to \infty,

Semiclassical approach to Regge poles trajectories calculations for nonsingular potentials: Thomas–Fermi type

of a class of unbounded self-adjoint Jacobi matrices J with discrete spectrum are given. Their calculation is based on a successive diagonalization approach---a new version of the classical transformation operator method. The approximations of the transformation operator are constructed step by step using a successive diagonalization procedure, which results in higher order approximations of the

On the essential spectrum of a class of singular matrix differential operators. I: Quasiregularity conditions and essential self-adjointness

\lambda_n (J).

On Regge pole trajectories for a rational function approximation of Thomas–Fermi potentials

We present a short, new, self-contained proof of localization properties of multi-dimensional continuum random Schödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in [2] but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.

Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lust operator

A simple semiclassical approach, based on the investigation of anti-Stokes line topology, is presented for calculating Regge poles for nonsingular (Thomas–Fermi type) potentials, namely potentials with singularities at the origin weaker than order −2. The anti-Stokes lines for Thomas–Fermi potentials have a more complicated structure than those of singular potentials and require careful application of complex analysis. The explicit solution of the Bohr–Sommerfeld quantization condition is used to obtain approximate Regge poles. We introduce and employ three hypotheses to obtain several terms of the Regge pole approximation.Please note that the pdf of this article was replaced with a corrected version on 29 June 2004. Minor changes have been made to pages 6950–6953.

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

AbstractThe essential spectrum of singular matrix differential operator determined by the operator matrix