ABSTRACT
In this chapter we look at some Dirac-type operators on non-compact manifolds. The example of the operator id/dx on the real line, as compared to the corresponding operator on the circle R /Z , is a helpful one to bear in mind. On the circle, id/dx has a discrete spectrum with finite-dimensional eigenspaces. On the line, Fourier series are replaced by Fourier transforms; the spectrum becomes continuous, and spectral values no longer correspond to square-integrable eigenfunctions. Nevertheless, a spectral decomposition still exists, and the functional calculus operates in the same wray as before.
