Ground state of the Gross-Pitaevskii equation with a harmonic potential in the energy-critical case

TL;DR

This paper constructs and analyzes the ground state of the energy-critical Gross-Pitaevskii equation with a harmonic potential, revealing its behavior across an eigenvalue interval and its approximation by known special functions.

Contribution

It introduces a variational construction of the ground state, characterizes its existence interval, and employs a shooting method to relate eigenvalues to the supremum norm, connecting to classical solutions.

Findings

Ground state exists in a finite eigenvalue interval.

Supremum norm vanishes at one end and diverges at the other.

Ground state approximates Aubin-Talenti and hypergeometric functions in different regions.

Abstract

Ground state of the energy-critical Gross-Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin-Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus the supremum norm.