Matris kuantum mekaniği ve entegre edilebilir sistemler.

In this thesis we improve and extend an algebraic technique pioneered by M. Gaudin. The technique is based on an infinite dimensional Lie algebra and a related family of mutually commuting Hamiltonians. In order to find energy eigenvalues of such Hamiltonians one has to solve the equations of Bethe ansatz. However, in most cases analytical solutions are not available. In this study we examine a special case for which analytical solutions of Bethe ansatz equations are not needed. Instead, some special properties of these equations are utilized to evaluate the energy eigenvalues. We use this method to find exact expressions for the energy eigenvalues of a class of interacting boson models. In addition to that, we also introduce a q-deformation of the algebra of Gaudin. This deformation leads us to another family of mutually commuting Hamiltonians which we diagonalize using algebraic Bethe ansatz technique. The motivation for this deformation comes from a relationship between Gaudin algebra and a spin extension of the integrable model of F. Calogero. Observing this relation, we then consider a well known periodic version of Calogero's model which is due to B. Sutherland. The search for a Gaudin-like algebraic structure which is in a similar relationship with the spin extension of Sutherland's model naturally leads to the above mentioned q-deformation of Gaudin algebra. The deformation parameter q and the periodicity d of the Sutherland model are related by the formula q=i{Ph.D. - Doctoral Progra