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

Details On the counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian On the counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

2011-09-09

arxiv.org/abs/1109.1995v1

Physics

Mathematical Physics

Scientific paper

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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