Spectrum of the Lichnerowicz Laplacian on Asymptotically Hyperbolic Surfaces

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian Δ _L contains the ray $\big\frac{1}{4},+\infty\big$ . If moreover the scalar curvature is constant then ??2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $\langle \Delta u, u\rangle_{L^2}\geqslant \frac{1}{4}||u||^2_{L^2}$ holds for all smooth compactly supported function u, then there is no other value in the spectrum.