5 research outputs found
Lifshitz Tails in Constant Magnetic Fields
We consider the 2D Landau Hamiltonian perturbed by a random alloy-type
potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of
the corresponding integrated density of states (IDS) near the edges in the
spectrum of . If a given edge coincides with a Landau level, we obtain
different asymptotic formulae for power-like, exponential sub-Gaussian, and
super-Gaussian decay of the one-site potential. If the edge is away from the
Landau levels, we impose a rational-flux assumption on the magnetic field,
consider compactly supported one-site potentials, and formulate a theorem which
is analogous to a result obtained in the case of a vanishing magnetic field
-approximation of the integrated density of states for Schr\"odinger operators with finite local complexity
We study spectral properties of Schr\"odinger operators on \RR^d. The
electromagnetic potential is assumed to be determined locally by a colouring of
the lattice points in \ZZ^d, with the property that frequencies of finite
patterns are well defined. We prove that the integrated density of states
(spectral distribution function) is approximated by its finite volume
analogues, i.e.the normalised eigenvalue counting functions. The convergence
holds in the space where is any finite energy interval and is arbitrary.Comment: 15 pages; v2 has minor fixe
Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian
We consider the Landau Hamiltonian (i.e. the 2D Schroedinger 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.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) -
(ii) indicated. Typos corrected. To appear in Communications in Mathematical
Physic