Abstract EN:

In this thesis, we use C-star-algebraical techniques aiming for applications in spectral theory. In the first two articles, in the context of trees, we adapt the C-star-algebra methods to the study of the spectral and scattering theories of Hamiltonians of the system. We first consider a natural formulation and generalization of the problem in a Fock space context. We then get a Mourre estimate for the free Hamiltonian and its perturbations. Finally, we compute the quotient of a C-star-algebra of energy observables with respect to its ideal of compact operators. As an application, the essential spectrum of highly anisotropic Schr\"odinger operators is computed. In the third article, we give powerful critera of stability of the essential spectrum of unbounded operators. Our applications cover Dirac operators, perturbations of riemannian metrics, differential operators in divergence form. The main point of our approach is that no regularity conditions are imposed on the coefficients.