  In this paper we study resonances and eigenvalues for the nonlinear constant
mean curvature eqn. linearized around the bubbles found by Brezis-Coron. This
nonlinear eqn. is also called a H-system eqn. For degree one bubbles(the degree
relates to a certain winding number) we only find resonances. For higher degree
we prove eigenvalues do occur. Our goal is to eventually obtain dispersive
estimates for the wave eqn. associated to the H-systems eqn. in its linear and
non-linear form, a study of which was initiated by Chanillo-Yung.
