Sharp stability near sums of ground states for fractional Schrödinger equations

Abstract

We study quantitative stability for the fractional Schrödinger equation $(-\Delta {)}^{s}u+u-|u{|}^{\alpha }u=0$ near finite sums of widely separated positive ground states. For every $n\ge 1$, $0<s<1$, and Sobolev-subcritical exponent $\alpha >0$, we estimate the $H^s$ distance to the family of such sums in terms of the $H^{-s}$ norm of the equation's residual. The optimal rate changes at $\alpha =n/[2(n+2s)]$, with a logarithmic correction at the threshold. Above the threshold the rate is $t^{(n+2s)/(n+2s+1)}$; below it the exponent is $[(1+\alpha )(n+2s)-n/2]/(n+2s+1)$. The proof combines uniform invertibility away from the translation modes with precise interaction estimates and a weighted correction of the approximate configuration. This correction resolves the translation interactions even when the nonlinearity has a small exponent. We also construct nonnegative configurations that attain these rates. The estimates quantify the effect of the algebraic decay of fractional ground states on stability.