Friday, December 7, 2012

1109.1995 (Abderemane Morame et al.)

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

Abderemane Morame, Francoise Truc
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 $-\Delta =-\Delta_0.$
View original: http://arxiv.org/abs/1109.1995

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