One natural approach to sub-Riemannian geometry lies in the study of the behavior of Riemannian objects in the sub-Riemannian limit. This consists of blowing up the metric transversely to the Carnot distribution. These metric spaces converge in the Gromov-Hausdor sense to the subRiemannian ones [Gro]. However, very little is known about the convergence of some even basic and linear objects as the spectrum of the Laplacians on dierential forms. We begin here this study in the contact case. We will see that the nonblowing parts of the spectrum of the Laplacians, d + and the signature operatord d, concentrate and are described by their counterparts coming from the contact complex studied in [Ru]. In particular, an interesting innite dimensional collapsing eigenvalues phenomenon occurs on middle degree forms. It corresponds to the special second order dierential D of the contact complex. These spectrum convergences of unbounded operators are rst studied in Theorems 3.5 and 3.6 through the convergence of their resolvents. The techniques are much inspired from adiabatic limits as developed for example in [BeB], [BiL], [D], [MM]. Nevertheless, the algebraic and analytic situations here are quite dierent, in some sense opposite to the adiabatic case, where the unexploded directions need to be integrable and form a bration. Anyway, this approach, pointed out by J.-M. Bismut, relies on some formal resemblances between the problems. Mainly, we will see in section 3, that the contact complex occurs as a natural spectral sequence in the sub-Riemannian blow up, just like the Leray spectral sequence of the bration does in the adiabatic case. These algebraic structures used to predict, in a formal power series sense at rst, the dierent parts of the spectrum that blow up or collapse at dierent rates. The way to turn this into the actual convergence of the resolvents will rest here on the use of some L 2 a priori estimates. They will come from