Georgi Raikov

QUASI-CLASSICAL VERSUS NON-CLASSICAL SPECTRAL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS WITH DECREASING ELECTRIC POTENTIALS

We consider the Schrodinger operator H(V) on L2 (ℝ2) or L2(ℝ3) with constant magnetic field, and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V) near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasi-classical formulas known for the case where V admits a power-like decay.

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = r θ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and r θ = r θ(x 3) is a rotation by the angle θ(x 3) depending on the longitudinal variable x 3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L 2(Ωθ). We suppose that the derivative of the rotation angle can be written as (x 3) = β − ϵ(x 3) with a positive constant β and ϵ(x 3) ∼ L|x 3|−α, |x 3| → ∞. We show that if L > 0 and α ∈ (0,2), or if L > L 0 > 0 and α = 2, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.

QUANTIZATION OF EDGE CURRENTS ALONG MAGNETIC BARRIERS AND MAGNETIC GUIDES

We investigate the edge conductance of particles submitted to an Iwatsuka magnetic field, playing the role of a purely magnetic barrier. We also consider magnetic guides generated by generalized Iwatsuka potentials. In both cases, we prove quantization of the edge conductance. Next, we consider magnetic perturbations of such magnetic barriers or guides and prove stability of the quantized value of the edge conductance. Further, we establish a sum rule for edge conductances. Regularization within the context of disordered systems is discussed as well.

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 ⊂ ℂ, 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: ℝ3 → ℝ 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.

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schrödinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.

Low Energy Asymptotics of the Spectral Shift Function for Pauli Operators with Nonconstant Magnetic Fields

We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V, we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.

Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

Let

Discrete spectrum of quantum Hall effect Hamiltonians I. Monotone edge potentials

H_{0, D}

Spectral shift function for magnetic Schrödinger operators

(resp.,