Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

Abstract

In this paper we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schrödinger 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.

22 pages

References in corpus (2)

• Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation
• Some recent progress on sharp Fourier restriction theory

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• Sharp mixed norm spherical restriction
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• Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere
• Band-limited maximizers for a Fourier extension inequality on the circle
• Strichartz Estimates with Broken Symmetries
• Extremals for $\alpha$-Strichartz inequalities
• On profile decomposition for Airy type equation