Journal of the European Mathematical Society

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Volume 11, Issue 6, 2009, pp. 1365–1383

DOI: 10.4171/JEMS/184

Pólya's conjecture in the presence of a constant magnetic field

Rupert L. Frank (1), Michael Loss (2) and Timo Weidl (3)

(1) Department of Mathematics, Princeton University, Fine Hall, NJ 08544, PRINCETON, UNITED STATES
(2) School of Mathematics, Georgia Institute of Technology, GA 30332-0160, ATLANTA, UNITED STATES
(3) Mathematisches Institut A, 5. Lehrstuhl , Universität Stuttgart, Pfaffenwaldring 57, 70569, STUTTGART, GERMANY

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.

Keywords: Eigenvalue bounds, semi-classical estimates, Pólya's conjecture, Laplace operator, magnetic Schrödinger operators.