Spectral Analysis by Algebraic and Topological Methods
by
Marius Mantoiu
IMAR

Vladimir Georgescu has developped recently an algebraic formalism with implications in the spectral analysis of pseudodifferential operators of certain types, inclouding Schrodinger operators. We intend to contributeto this topic at the abstract level and with some interesting new examples. The key point in the algebraic approach is to show that the operators we study are affiliated to certain C*-algebras and to express the quotient of these algebras by the ideal of compact operators in a relevant manner. We consider the case in which the configuration space is a locally compact, abelian group (the main case being that of an euclidean space). Our C*-algebras will be crossed products. They are associated to a topological dynamical system, the locally compact group acting on a suiteble compactification of itself. Invariant decompositions of the boundary give efficient decompositions of the quotient C*-algebra. As immediate consequences we get formulae for the essential spectrum and propagation properties of the affiliated operators. Some examples of pseudodifferential operators with very anisotropic symbols are considered.

Date received: June 19, 2002