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

Abstract: The asymptotics of the ground state

u (r)

of the Schr"odinger--Newton equation in

m a t h b b R^{3}

was determined by V. Moroz and J. van Schaftingen to be

u (r) s i m A e^{- r} / r^{1 - | u |_{2}^{2} / 8 p i}

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 p i^{2} l e q | u |_{2}^{2} l e q 2^{3} p i^{3 / 2}

. It is reported that numerically

| u |_{2}^{2} a p p r o x 14.03 p i

, 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

s i m - K / r

potential, and for the related Hartree equation of a bosonic atom or ion.

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