We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Pólya's conjecture is not true in the presence of a magnetic field.
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Timo Weidl, Rupert L. Frank, Michael Loss, Pólya's conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc. 11 (2009), no. 6, pp. 1365–1383DOI 10.4171/JEMS/184