The Landau Hamiltonian with delta-potentials supported on curves

0524523 - ÚJF 2021 RIV SG eng J - Journal Article
Behrndt, J. - Exner, Pavel - Holzmann, M. - Lotoreichik, Vladimir
The Landau Hamiltonian with delta-potentials supported on curves.
Reviews in Mathematical Physics. Roč. 32, č. 4 (2020), č. článku 2050010. ISSN 0129-055X. E-ISSN 1793-6659
R&D Projects: GA MŠMT 7AMB17AT022; GA ČR GA17-01706S
Institutional support: RVO:61389005
Keywords : Landau Hamiltonian * magnetic Schrodinger operator * singular perturbation * delta-potential * eigenvalue clustering * spectral asymptotics * Toeplitz operator * approximation by regular potentials
OECD category: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)
Impact factor: 1.361, year: 2020
Method of publishing: Limited access
https://doi.org/10.1142/S0129055X20500105

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian A(alpha) = (i del + A)(2) + alpha delta(Sigma) in L-2(R-2) with a delta-potential supported on a finite C-1,C-1-smooth curve Sigma are studied. Here A = 1/2 B(-x(2), x(1))(T) is the vector potential, B > 0 is the strength of the homogeneous magnetic field, and alpha is an element of L-infinity(Sigma) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Sigma. After a general discussion of the qualitative spectral properties of A(alpha) and its resolvent, one of the main objectives in the present paper is a local spectral analysis of A(alpha) near the Landau levels B(2q + 1), q is an element of N-0. Under various conditions on alpha, 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 alpha. Furthermore, the use of Landau Hamiltonians with delta-perturbations as model operators for more realistic quantum systems is justified by showing that A(alpha) can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.
Permanent Link: http://hdl.handle.net/11104/0308871