# Orthogonal Polynomials and Sharp Estimates for the Schr\"odinger Equation

## TL;DR

This paper develops sharp estimates for the Schrödinger equation using orthogonal polynomial frameworks, including spherical harmonics and Gegenbauer polynomials, leading to new inequalities and connections with combinatorial problems.

## Contribution

It introduces novel weighted inequalities and sharp Strichartz estimates for the Schrödinger equation leveraging orthogonal polynomial expansions.

## Key findings

- New weighted inequality maximized by radial functions
- Sharp Strichartz estimates for even exponents
- Connection between Strichartz norm and combinatorial word problem

## Abstract

In this paper we study sharp estimates for the Schr\"odinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schr\"odinger equation that is maximized by radial functions. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension 2, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters.