On the asymptotic decay of the Schr\"odinger--Newton ground state

TL;DR

This paper rigorously determines bounds for the squared L2 norm of the ground state of the Schrödinger--Newton equation, clarifying its asymptotic decay and extending results to related equations with external potentials.

Contribution

It provides rigorous bounds for the L2 norm of the ground state and explores asymptotic behaviors, advancing understanding of the Schrödinger--Newton ground state properties.

Findings

Bounds for  of ground state: ^{1/3}3\u03c0^2    2^{3}pi^{3/2}

Numerical estimate of     14.03pi

Asymptotic behaviors proposed for related equations with external potentials

Abstract

The asymptotics of the ground state $u(r)$ of the Schr\"odinger--Newton equation in $\mathbb{R}^3$ was determined by V. Moroz and J. van Schaftingen to be $u(r) \sim A e^{-r}/ r^{1 - \|u\|_2^2/8\pi}$ for some $A>0$, in units that fix the exponential rate to unity. They left open the value of $\|u\|_2^2$, the squared $L^2$ norm of $u$. Here it is rigorously shown that $2^{1/3}3\pi^2\leq \|u\|_2^2\leq 2^{3}\pi^{3/2}$. It is reported that numerically $\|u\|_2^2\approx 14.03\pi$, revealing that the monomial prefactor of $e^{-r}$ increases with $r$ in a concave manner. Asymptotic results are proposed for the Schr\"odinger--Newton equation with external $\sim - K/r$ potential, and for the related Hartree equation of a bosonic atom or ion.