An exact effective Hamiltonian for a perturbed Landau level

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Abstract
Considers the effect of a scalar potential V (x, y) on a Landau level in two dimensions. An exact effective Hamiltonian is derived which describes the effect of the potential on a single Landau level, expressed as a power series in V/Ec, where Ec is the cyclotron energy. The effective Hamiltonian can be represented as a function H (x, p) in a one-dimensional phase space. The function H (x, p) resembles the potential V (x, y): when the area of a flux quantum is much smaller than the square of the characteristic length scale of V, then H approximately=V. Also H (x, p) retains the translational and rotational symmetries of V(x, y) exactly, but reflection symmetries are not retained beyond the lowest order of the perturbation expansion.