Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

Abstract

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 .

A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

“…• As already mentioned, in [19] it was supposed that V satisfies (1.1) with ρ > 1. Then the Radon transform…”

“…Recently, in [19] it was shown that if V satisfies ) as q → ∞. Moreover, in [19] the asymptotic density of the eigenvalue clusters was studied.…”

“…Recently, in [19] it was shown that if V satisfies ) as q → ∞. Moreover, in [19] the asymptotic density of the eigenvalue clusters was studied. To this end, the asymptotic behaviour of the trace Tr ϕ(λ 1/2 q (H − λ q )) with ϕ ∈ C ∞ 0 (R \ {0}) was investigated, and it was found that Tr ϕ(λ 1/2 q (H − λ q )) is of order λ 1/2 q as q → ∞, and its first asymptotic term could be written explicitly using the Radon transform of V .…”

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

“…Their perturbations have been investigated in [35,44], and the asymptotic behaviour of the eigenvalues was analysed in [37,40,41,[48][49][50]57]. The results of this paper apply for the Cauchy problem (1.1) for the operator L from (3.2).…”

The Spectral Density of the Scattering Matrix for High Energies

“…Much of the inspiration for both the content of this paper and the proofs may be found in [7], where similar asymptotics are determined for the spectrum of the Landau Hamiltonian (i.e. two-dimensional Schrödinger operator with a constant homogeneous magnetic field) perturbed by a potential which obeys the same condition (1.2).…”