Microlocal Analysis and Spectral Theory of Elliptic Operators on Non-compact Manifolds

In this thesis we will deal with SG-pseudodifferential operators, a class of global pseudodifferential operators introduced independently by H. O. Cordes and C. Parenti in the '70s. A main feature of this pseudodifferential operators class is the possibility to transfer the calculus to a rather wide set of non-compact manifolds, the so-called S-manifolds.
In Chapter 1 we will recall the basic definitions and results about the SG-calculus on R^n. Here we will mainly focus on three aspects: the problem of the existence of a parametrix for elliptic operators in this class, some properties of the weighted Sobolev spaces, and the analysis of a version of the calculus on the polycylinder.
The description of the S-structures will be given in Chapter 2. Here we will describe how the SG-calculus can be transferred to manifolds which admit a S-structure and show that the good properties of the calculus hold also on S-manifolds. We will also discuss the Laplace-Beltrami operator on such manifolds, and show that it is not md-elliptic.
In Chapter 3 we will then fix the attention to the spectral properties of elliptic SG-operators of positive order on S-manifolds. Finally, in some concluding remarks, we briefly discuss how the theory could be used to build explicit examples of estimates for the corresponding spectral counting functions.