02-145 B. Helffer, A. Morame

Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case) (215K, LaTeX 2e) Mar 25, 02

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Abstract. We consider the first eigenvalue e(h) of the Neumann operator associated to the Laplace operator with constant magnetic field B, $(ih\bigtriangledown +B\wedge x/2)^2$, on a bounded domain. We show that, as h go to zero, e(h) has an asymptotic expansion $e(h)\sim h |B|\Theta_0 +h^{4/3}|B|^{2/3}\Theta_1$. The first constant, $\Theta_0$, is independent of the domain and the second one, $\Theta_1$, is related to the curvature of the boundary, on the curve where the magnetic field is tangent to the boundary. This paper is an extension of the previous preprint mp_arc 01-362 and gives a complete proof in the general generic case.

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