Instanton methods and the Dyon atom

In quantum mechanics, perturbation theories have crucial importance for almost any practical calculation. However, perturbative expansions are generally asymptotic series due to the singularities on the complex couping constant plane. In these cases, there is a non-zero error for the perturbative series. The divergent behaviour of an perturbative expansion can also be examined by using the non-perturbative techniques, such as WKB approximation and instanton methods. Futhermore, the non-perturbative methods can also be used to improve the error. In this thesis, instanton methods are applied to the perturbed Dyon atom. After a detailed discussion about the divergences of perturbation theory, the divergent behaviour of the energy coefficients of the one dimensional anharmonic oscillator is estimated by instanton methods. Then, the divergent behaviour of the perturbation series of the Dyon atom is calculated by a map from the estimation of $4$ dimensional anharmonic oscillator, which is again done by instanton methods.