Oscillatory Integrals and Applications

Abstract

This dissertation aims to study oscillatory integrals; in particular it deals with the three principles governing the behaviour of these integrals for large values of the variable λ, localization, scaling and asymptotics. Through analysis and mathematical techniques, this dissertation provides insight into the fundamental principles governing the asymptotic behaviour of these integrals. It focuses on proving key theorems such as the Van der Corput lemma and the stationary phase theorem. The behaviour study of the oscillatory integrals is based on the existence and the nature of critical points for the phase function. The dissertation starts with the case where the phase function has no critical point. In this case, the non–stationary phase theorem, a powerful tool for the behaviour study of oscillatory integrals, it takes centre stage in estimating these integrals. The Van der Corput lemma and its consequence give significant and valuable information about the oscillatory integral’s behaviour for large values of the variable λ. In this part, an estimate decay is given showcasing its versatility and wide-ranging applicability. The final segment of this chapter investigates the stationary phase theorem. In this case, the dissertation provides a mechanism for extracting complete asymptotic approximations by focusing on the critical points of the oscillating phase. The exposition explores only a single critical point. The last task in dissertation completes the theoretical discussions by exploring two illustrative examples of applications, the asymptotic behaviour of Bessel functions and the dispersive estimate for the solution of the linear Schrödinger equation.