Titolo della tesi: Two-Anyon Schrodinger Operators
We introduce anyons, we discuss the physical evidencies of their existence and present the formulation of the N-anyon problem in the magnetic gauge. We then illustrate the main phenomena linked to anyons, as the Fractional Quantum Hall Effect. We pass to the magnetic gauge. The main results obtained in the magnetic gauge include the classi cation of all self-adjoint extensions and the associated quadratic forms for the 2-anyon symmetric hamiltonian and of the operators associated
with these forms. We generalize the results to the case of trapping or interaction potentials, which are regular at the coincidence points . As a special case, we study anyons immersed in a harmonic trapping potential. We study some singular perturbation, namely the Coulomb, the Log-Coulomb and the Hardy potentials. In the second half of the thesis, we apply a transformation to the relative coordinate of the 2-anyon phace space. Through this mapping, the kinetic energy is mapped onto a Dirichlet form on a weighted Hilbert space. We gather a series of results such Hardy inequalities and embeddings between Sobolev spaces. In this new setting we start by classifying all the closed forms that extend the Dirichlet form in the weighted Sobolev space. We apply both the Krein and Von Neumann theory and we precisely identify the form and operator domain of each extension. We do the spectral theory: we diagonalize the operator and write explicitly the resolvent kernel of the Friedrichs extension. Then, we use the Krein formula and write the resolvents of all the other operators. We conclude by showing that all the self-adjoint operators realizing the formal kinetic energy operator can be obtained as norm resolvent limits of suitable scaled potentials, similarly as for delta extensions. We see the implication of this results on the extended anyons approximation.