2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60713We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so far that uses information about the spin structure. As a corollary we obtain lower bounds for the Willmore functional of a torus embedded into S_3. In the final section we compare Dirac spectra for two different spin structures on an arbitrary Riemannian spin manifold.uses pstricks package for figuresDifferential GeometrySpectral Theory53C27, 58J50, 53C80text