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

Abderemane Morame (1) , Francoise Truc (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IF - Institut Fourier

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 $-\Delta =-\Delta_0 .$

Mots clés: cuspidal manifold; spectral asymptotics; magnetic Laplacian; embedded eigenvalues; cuspidal manifold.

Domaines: Physique mathématique [math-ph]

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⟩