An exact effective Hamiltonian for a perturbed Landau level. (English) Zbl 0639.47010

The two-dimensional problem of the influence of a scalar potential V(x,y) on a Landau level is represented by an exact one-dimensional effective Hamiltonian, valid provided the gaps between the n th Landau levels remain open and the matrix of the projection operator for the n th perturbed level is nonsingular. The effective Hamiltonian can be expressed as a power series in \(V/E_ c\), where \(E_ c\) is the cyclotron energy. The phase space function \(H^{(x)}_{eff}(x,p)\) resembles the potential V(x,y) and when the area of a flux quantum is much smaller than the characteristic length scale of V, then \(H\to V\). The effective Hamiltonian preserves the translational and rotational symmetries of V exactly, but reflection symmetries are retained only in the lowest order of the perturbative expansion.
Reviewer: O.Gherman


47A55 Perturbation theory of linear operators
46N99 Miscellaneous applications of functional analysis
81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text: DOI Link