## 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

### MSC:

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