Abderemane Morame, Francoise Truc
Counting function of the embedded eigenvalues
for some manifold with cusps, and magnetic Laplacian
(395K, PDF)
ABSTRACT. We consider a non compact, complete manifold M of finite area with cuspidal ends. The generic cusp is isomorphic to $X imes (1,+\infty )$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ X is a compact manifold equipped with the metric h. For a one-form A on 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 $-\Delta =-\Delta_0 .$