Asymptotic Lower Bounds for a class of Schroedinger Equations

TL;DR

This paper investigates asymptotic lower bounds for solutions to Schrödinger equations with short-range potentials, deriving identities that lead to uniqueness results and precise bounds on local smoothing effects.

Contribution

It introduces a family of identities for solutions to Schrödinger equations with short-range potentials, establishing new asymptotic lower bounds and uniqueness results.

Findings

Derived identities satisfied by solutions

Established lower bounds for local smoothing effects

Proved asymptotic identities in specific cases

Abstract

We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - \Delta u + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.