On the Schr\"odinger equations with time-dependent potentials growing polynomially in the spatial direction

TL;DR

This paper investigates the existence, uniqueness, and parameter dependence of solutions to Schrödinger equations with polynomially growing time-dependent potentials, using weighted Sobolev spaces.

Contribution

It establishes solution existence and uniqueness, and analyzes how solutions vary continuously and differentiably with parameters in such Schrödinger equations.

Findings

Solutions exist and are unique in weighted Sobolev spaces.

Solutions depend continuously on parameters.

Solutions depend differentiably on parameters.

Abstract

The Cauchy problem for the Schr\"odinger equations is studied with time-dependent potentials growing polynomially in the spatial direction. First the existence and the uniqueness of solutions are shown in the weighted Sobolev spaces. In addition, we suppose that our potentials are depending on a parameter. Secondly it is shown that if potentials depend continuously and differentiably on the parameter, the solutions to the Schr\"odinger equations respectively become continuous and differentiable with respect to its parameter.