The Landau Hamiltonian with δ-potentials supported on curves

Abstract

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian [Formula: see text] in [Formula: see text] with a [Formula: see text]-potential supported on a finite [Formula: see text]-smooth curve [Formula: see text] are studied. Here [Formula: see text] is the vector potential, [Formula: see text] is the strength of the homogeneous magnetic field, and [Formula: see text] is a position-dependent real coefficient modeling the strength of the singular interaction on the curve [Formula: see text]. After a general discussion of the qualitative spectral properties of [Formula: see text] and its resolvent, one of the main objectives in the present paper is a local spectral analysis of [Formula: see text] near the Landau levels [Formula: see text], [Formula: see text]. Under various conditions on [Formula: see text], it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of [Formula: see text]. Furthermore, the use of Landau Hamiltonians with [Formula: see text]-perturbations as model operators for more realistic quantum systems is justified by showing that [Formula: see text] can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.