The purpose of this book is to give a self contained presentation of the spectral theory for globally elliptic differential operators on $\mathbb R^{n}$ (generalizing the harmonic oscillator).
First we recall the theory of global pseudo-differential operators and the classical spectral theory. Then, we study classes of global Fourier-integral operators and present in a particular case results obtained in collaboration with D. Robert which are analogous for globally elliptic operators to the well known results for the elliptic operators on a compact manifold.