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 Incoming links: Gravitational Hartree equation

Hartree equations 2026-10-13

The Hartree equations are self-consistent one-particle equations obtained by approximating a many-body wave function by a tensor product of orbital wave functions. Each orbital sees the potential averaged over the other particles' probability densities; attractive Newtonian mean-field variants include the Gravitational Hartree equation.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 2 / 1 / Solution 2026-09-28

The Gravitational Hartree equation can be written

u_{t} = i Δ u - i ϕ u,

where

ϕ

is real. Consequently

\frac{d}{d t} \int_{R^{N}} ∣ u ∣^{2} d x = 2 Re \int_{R^{N}} u_{t} \overline{u} d x = 2 Re (i \int Δ u \overline{u} - i \int ϕ ∣ u ∣^{2}) = 0,

because both integrals multiplied by

i

are purely imaginary after integration by parts. Hence Hartree mass conservation holds.

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