The <Gravitational Hartree equation> can be written
$$
u_t=i\Delta u-i\phi u,
$$
where $\phi$ is real. Consequently
$$
\frac d{dt}\int_{\mathbb R^N}|u|^2\,dx
=2\operatorname{Re}\int_{\mathbb R^N}u_t\overline u\,dx
=2\operatorname{Re}\left(
i\int\Delta u\,\overline u-i\int\phi|u|^2
\right)=0,
$$
because both integrals multiplied by $i$ are purely imaginary after <integration by parts>. Hence <Hartree mass conservation> holds.

Solution

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

{c}
{title2=$iu_t+\Delta u-\phi u=0$}
{wiki=Hartree_equation}

The gravitational Hartree equation couples a wave function to its attractive Newtonian potential:
$$
iu_t+\Delta u-\phi u=0,
\qquad
\Delta\phi=|u|^2,
\qquad
\phi=-\frac1{C_N|x|^{N-2}}*|u|^2.
$$


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