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

We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to $\mathbf{X}\times ]1,+\mathrm{\infty }[$ with metric $d{s}^2=(h+d{y}^2)\mathrm{/}{y}^{2\delta }\mathrm{.}$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h\mathrm{.}$ 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 $-{\mathrm{\Delta }}_A=(id+A{)}^{\star }(id+A)$ 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 $-\mathrm{\Delta }=-{\mathrm{\Delta }}_0\mathrm{.}$