Uniform resolvent estimates and absence of eigenvalues for Lamé operators with complex potentials

We consider the 0-order perturbed Lamé operator −Δ⁎+V(x). It is well known that if one considers the free case, namely V=0, the spectrum of −Δ⁎ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about V.