Global Uniform in N Estimates for Solutions of a System of Hartree–Fock–Bogoliubov Type in the Case $\beta <1$

Abstract

We extend the results of [Commun. Partial. Differ. Equ. 44(12), 1431–1465 (2019)] by the third and fourth author globally in time. More precisely, we prove uniform-in-N Strichartz estimates for the solutions $\phi $, $\Lambda $ and $\Gamma $ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta }v(N^{\beta }(x-y))$ for $\beta <1$. The potential v satisfies some technical conditions, but is not small. The initial conditions have finite energy and the “pair correlation” part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda $ develop dynamically in time. The estimates are expected to improve the Fock space bounds of [Ann. Henri Poincaré 23(2), 615–673 (2021)] by the first and fifth author. This will be addressed in a subsequent paper.

Keywords

The Hartree–Fock–Bogoliubov System / Mean-field equations

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