# Strichartz Estimates for the Schr\"odinger Equation with a Measure-Valued Potential

## TL;DR

This paper establishes Strichartz estimates for the Schr"odinger equation with measure-valued potentials supported on sets of certain dimensions, extending the understanding of dispersive properties in singular settings.

## Contribution

It proves Strichartz estimates for Schr"odinger equations with measure-valued potentials supported on sets with specific dimensional constraints, a novel extension of classical results.

## Key findings

- Strichartz estimates hold for measure-supported potentials with dimension > n - (1 + 1/(n-1))
- Local decay estimates in L^2(μ) are established for both free and perturbed evolutions
- The results extend dispersive analysis to singular measures with controlled dimensions

## Abstract

We prove Strichartz estimates for the Schr\"odinger equation in $\mathbb R^n$, $n\geq 3$, with a Hamiltonian $H = -\Delta + \mu$. The perturbation $\mu$ is a compactly supported measure in $\mathbb R^n$ with dimension $\alpha > n-(1+\frac{1}{n-1})$. The main intermediate step is a local decay estimate in $L^2(\mu)$ for both the free and perturbed Schr\"odinger evolution.