On the essential spectrum of Nadirashvili-Martin-Morales minimal   surfaces
  We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau. 
G. Pacelli Bessa, Luquesio P. Jorge, J. Fabio Montenegro