Born-oppenheimer approximation applied to some singular systems

In this thesis, we investigate the spectral properties of a Schrödinger operator H0, modified by the introduction of δ interactions. Our analysis explicitly illustrates the reconfiguration of poles in the new Green's function concerning those found in the original Green's function of H0, showcasing that the energies of new bound states are interleaved among the existing ones. Notably, the ground state energy consistently decreases when the δ interaction is attractive. We devise an alternative perturbative approach to determine bound state energies and wave functions, assuming a small coupling constant. We extend our findings to cases requiring renormalization. Additionally, our study explores the applicability of these results to more intricate scenarios, including multi-center configurations, δ interactions localized along curves, and the motion of particles within a compact two-dimensional manifold influenced by δ interactions. We explicitly delve into the semi-relativistic extension of the problem.Subsequently, we introduce a heavy particle to the classical kink system in one dimension. Within the Born-Oppenheimer approach, we observe that this heavy particle behaves similarly to a repulsive delta potential within the kink equation of motion. By understanding how this delta potential affects the wave function within the system, we derive a new phase and a change in the kink's mass. This change can also be determined through perturbation. Finally, we analyze the alterations in the energy of the heavy particle. The concluding section focuses on a fundamental singular system in a two-dimensional space involving two heavy particles interacting with a light particle through an attractive contact interaction. With a careful application of the Born-Oppenheimer approximation, we demonstrate its relevance to this specific scenario. Consequently, we identify the dominant term within the approximation and offer insights into obtaining higher-order corrections.