Magnetic bottles on geometrically finite hyperbolic surfaces

We consider a magnetic Laplacian −ΔA=(id+A)⋆(id+A)−ΔA=(id+A)⋆(id+A) on a hyperbolic surface M, when the magnetic field dA is infinite at the boundary at infinity. We prove that the counting function of the eigenvalues has a particular asymptotic behavior when M has an infinite area.