Finite Approximations of Quantum Systems in a Non-Archimedean and Archimedean Setting

Abstract

Approximation of quantum systems by nite dimensional quantum systems goes back to the foundation of quantum mechanics. Finite dimensional quantum systems were considered by Hermann Weyl, and were considered in much detail by Julian Schwinger. Our main interest is to approximate the spectrum of Hamiltonians by the spectrum of nite dimensional Hamiltonians. In a paper from 1994 by Digernes, Varadarajan and Varadhan, an approximation theorem was proved for a wide class of Hamiltonians. The main goal of this thesis is to generalize these results to di erent settings. One of the cases we investigate is the Hamiltonian with Coulomb potential. We will also generalize these results to the more unconventional setting of non-Archimedean quantum mechanics. Quantum mechanics over p-adic numbers was introduced by Volovich in 1987. Quantum mechanics in the p-adic setting is the most studied non-Archimedean model in quantum mechanics, and it has been generalized to local elds which will be our setting for non-Archimedean physics.