Crossed Products of C*-Algebras and Spectral Analysis of Quantum Hamiltonians

Abstract:

We study spectral properties of a hamiltonian by analyzing the structure of certain C ^*-algebras to which it is affiliated. The main tool we use for the construction of these algebras is the crossed product of abelian C ^*-algebras (generated by the classical potentials) by actions of groups. We show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians. This scheme provides a unified approach to the study of hamiltonians of anisotropic and many-body systems (including quantum fields).