02-140 M. Melgaard and G. Rozenblum
Eigenvalue asymptotics for even-dimensional perturbed Dirac and Schr\"{o}dinger operators with constant magnetic fields (129K, LaTeX 2e) Mar 21, 02
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Abstract. The even-dimensional Dirac and Schr\"{o}dinger operators with a constant magnetic field have purely essential spectrum consisting of isolated eigenvalues, so-called Landau levels. For a sign-definite electric potential $V$ which tends to zero at infinity, {\em not too fast}, it is known for the Schr\"{o}dinger operator that the number of eigenvalues near each Landau level is infinite and their leading (quasi-classical) asymptotics depends on the rate of decay for $V$. We show, both for Schr\"{o}dinger and Dirac operators, that, for {\em any} sign-definite, bounded $V$ which tends to zero at infinity, there still are an infinite number of eigenvalues near each Landau level. For compactly supported $V$ we establish the {\em non-classical} formula, not depending on $V$, describing how the eigenvalues converge to the Landau levels asymptotically.

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