Critical points of Strichartz functional

TL;DR

This paper investigates the critical points of the Strichartz functional related to the Schrödinger equation, identifying Gaussian functions as maximizers in low dimensions through analytical and numerical methods.

Contribution

It introduces a novel analysis of two dynamical systems associated with Strichartz inequalities and proves Gaussian maximizers in dimensions one to three.

Findings

Gaussian functions are maximizers in low dimensions

Critical points and stability are characterized for the systems

A new combinatorial inequality is verified

Abstract

We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schr\"odinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.