Counting function of characteristic values and magnetic resonances

TL;DR

This paper analyzes the distribution of characteristic values of a meromorphic operator function near zero and applies these results to study the asymptotic distribution of resonances near Landau levels in a magnetic Schrödinger operator.

Contribution

It provides a new abstract framework for understanding the distribution of characteristic values and applies it to magnetic Schrödinger operators to analyze resonance asymptotics.

Findings

Characteristic values converge to zero at the same rate as eigenvalues of A(0)

Resonance distribution near Landau levels is characterized asymptotically

Main asymptotic term of resonance counting function is explicitly derived

Abstract

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of 1-K(z), i.e. the complex numbers w for which the operator 1-K(w) is not invertible, and we show that generically the characteristic values of 1-K(z) converge to 0 with the same rate as the eigenvalues of A(0). We apply our abstract results to the investigation of the resonances of the operator H = H_0 + V where H_0 is the shifted 3D Schr\"odinger operator with constant magnetic field of scalar intensity b>0, and V is a real electric potential which admits a suitable decay at infinity. It is well known that the spectrum of H_0 is purely absolutely continuous, coincides with [0,+\infty[, and the so-called Landau levels 2bq with integer q, play the role of thresholds in the spectrum of H_0. We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.