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

Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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.

Files: 02-140.src( 02-140.keywords , manus1.tex )