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

TL;DR

This paper develops a new approach to analyzing oscillatory integrals using Gaussian regularization and iterative integration by parts, applying these results to solve the one-dimensional time-dependent Schrödinger equation with polynomially growing initial data.

Contribution

It introduces a novel method for interpreting oscillatory integrals as limits of regularized Lebesgue integrals, improving convergence analysis and applying it to quantum mechanics.

Findings

Established convergence of regularized oscillatory integrals

Derived explicit solutions for Schrödinger equation with polynomial growth initial data

Enhanced integrability properties through iterative integration by parts

Abstract

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 C^n-regular initial conditions with polynomial growth at infinity via the Green's function as an oscillatory integral.