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

5 pages; compared to the published version in Physics Letters A, a factor $2^{7/2}$ has been corrected into $2^3$ in the Proposition. The improved bound is not only better than the originally reported one, it is strong enough to prove concavity of the monomial pre-factor

References in corpus (1)

• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

Cited by in corpus (2)

• Scaling of highly excited Schrödinger-Poisson eigenstates and universality of their rotation curves
• From Klein-Gordon-Wave to Schrödinger-Wave: a Normal Form Approach