Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

Abstract : We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-\Delta =-\Delta_0 .$
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Submitted on : Wednesday, December 5, 2012 - 6:05:56 PM
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• HAL Id : hal-00621070, version 2
• ARXIV : 1109.1995

Citation

Abderemane Morame, Francoise Truc. Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian. Mathematical Review Letters, 2012, 19 ((2)), pp 417-429. ⟨hal-00621070v2⟩

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