Counting Function of Characteristic Values and Magnetic Resonances

We consider the meromorphic operator-valued function I ? K(z) = I ? A(z)/z where A 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 I ? K, i.e. the complex numbers w ≠ 0 for which the operator I ? K(w) is not invertible, and we show that generically the characteristic values of I ? K 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 Schrödinger operator with constant magnetic field of scalar intensity b > 0, and V: ?³ → ? is the electric potential which admits a suitable decay at infinity. It is well known that the spectrum σ(H ₀) of H ₀ is purely absolutely continuous, coincides with 0, + ∞, and the so-called Landau levels 2bq with integer q ≥ 0, play the role of thresholds in σ(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.