On a class of oscillatory integrals and their application to the time dependent Schr\"odinger equation

In this paper a class of oscillatory integrals is interpreted as a limit of Lebesgue integrals with Gaussian regularizers. The convergence of the regularized integrals is shown with an improved version of iterative integration by parts that generates additional decaying factors and hence leads to better integrability properties. The general abstract results are then applied to the Cauchy problem for the one dimensional time dependent Schr\"odinger equation, where the solution is expressed for Cⁿ-regular initial conditions with polynomial growth at infinity via the Green's function as an oscillatory integral.