Energy Stableness for Schr\"{o}dinger Operators with Time-Dependent Potentials

TL;DR

This paper establishes the energy stability of Schrödinger operators with time-dependent potentials using Hardy inequality, which is crucial for quantum scattering and nonlinear problems, and also derives a Sobolev type inequality.

Contribution

It introduces a novel proof of energy stability for time-dependent Schrödinger operators leveraging Hardy inequality, with applications to Sobolev inequalities.

Findings

Proves energy stability for Schrödinger operators with time-dependent potentials.

Derives a Sobolev type inequality as an application.

Highlights importance in quantum scattering and nonlinear analysis.

Abstract

In this paper, we prove the energy stable property for time-dependent (generalized) Schr\"{o}dinger operators by using Hardy inequality. Such property acts very important roles in quantum scattering theory and nonlinear problem. As an application, we prove Sobolev type inequality.