Dispersive Estimates for Schr\"odinger Operators with Measure-Valued Potentials in R^3

TL;DR

This paper establishes dispersive estimates for the Schr"odinger equation with measure-valued potentials in three-dimensional space, extending understanding to fractal measures with significant dimensionality.

Contribution

It provides the first dispersive estimates for Schr"odinger operators with measure-valued potentials of fractal dimension at least 3/2.

Findings

Dispersive estimates hold for potentials with fractal dimension ≥ 3/2.

Extends classical results to measure-valued potentials beyond regular functions.

Enhances understanding of quantum evolution with fractal or singular potentials.

Abstract

We prove dispersive estimates for the linear Schr\"odinger evolution associated to an operator -\Delta + V, where the potential is a signed measure of fractal dimension at least 3/2.