TY - JOUR
T1 - The Landau Hamiltonian with δ-potentials supported on curves
AU - Behrndt, Jussi
AU - Exner, Pavel
AU - Holzmann, Markus
AU - Lotoreichik, Vladimir
PY - 2019/9/25
Y1 - 2019/9/25
N2 - The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα=(i∇+A)2+αδΣ in L2(R2) with a δ-potential supported on a finite C1,1-smooth curve Σ are studied. Here A=12B(−x2,x1)T is the vector potential, B>0 is the strength of the homogeneous magnetic field, and α∈L∞(Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ. After a general discussion of the qualitative spectral properties of Aα and its resolvent, one of the main objectives in the present paper is a local spectral analysis of Aα near the Landau levels B(2q+1), q∈N0. Under various conditions on α, 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 α. Furthermore, the use of Landau Hamiltonians with δ-perturbations as model operators for more realistic quantum systems is justified by showing that Aα can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.
AB - The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα=(i∇+A)2+αδΣ in L2(R2) with a δ-potential supported on a finite C1,1-smooth curve Σ are studied. Here A=12B(−x2,x1)T is the vector potential, B>0 is the strength of the homogeneous magnetic field, and α∈L∞(Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ. After a general discussion of the qualitative spectral properties of Aα and its resolvent, one of the main objectives in the present paper is a local spectral analysis of Aα near the Landau levels B(2q+1), q∈N0. Under various conditions on α, 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 α. Furthermore, the use of Landau Hamiltonians with δ-perturbations as model operators for more realistic quantum systems is justified by showing that Aα can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.
UR - https://doi.org/10.1142/S0129055X20500105
U2 - 10.1142/S0129055X20500105
DO - 10.1142/S0129055X20500105
M3 - Article
SN - 1793-6659
VL - 32
JO - Reviews in Mathematical Physics
JF - Reviews in Mathematical Physics
IS - 4
M1 - 20500105
ER -