On the asymptotic decay of the Schrödinger–Newton ground state

Abstract

The asymptotics of the ground state u(r) of the Schrödinger–Newton equation in R3 was determined by V. Moroz and J. van Schaftingen to be u(r)∼Ae−r/r1−‖u‖22/8π for some A>0, in units in which the ground state energy is −1. They left open the value of ‖u‖22, the squared L2 norm of u. Here it is rigorously shown that 21/33π2⩽‖u‖22⩽27/2π3/2. It is reported that numerically ‖u‖22≈14.03π, revealing that the monomial prefactor of e−r increases with r in a concave manner. Asymptotic results are proposed for the Schrödinger–Newton equation with external ∼−K/r potential, and for the related Hartree equation of a bosonic atom or ion.