Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

Abderemane Morame (1) , Francoise Truc (2)

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

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface $\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0. $

Mots clés: Spectral asymptotics; magnetic field; Aharanov-Bohm; hyperbolic surface.; hyperbolic surface

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Abderemane Morame, Francoise Truc. Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area. 2010. ⟨hal-00462411v1⟩