= Gravitational Hartree equation
{c}
{title2=$iu_t+\Delta u-\phi u=0$}
The <Gravitational Hartree equation> couples a complex <wave function> to the attractive <Newtonian gravitational potential> generated by its own squared modulus. In dimension $N\ge3$, a dimensionless convention is
$$
iu_t+\Delta u-\phi u=0,\qquad \Delta\phi=|u|^2,\qquad
\phi=-\frac1{C_N|x|^{N-2}}*|u|^2,
$$
where $C_N=(N-2)|S^{N-1}|$ and $|S^{N-1}|$ is the area of the <unit sphere>. The minus sign in the fundamental solution makes $\phi\le0$ for nonnegative density, with the potential selected to decay at infinity when the convolution permits it.
The <convolution> makes the nonlinearity nonlocal: changing the density at one point changes the potential elsewhere. The Newtonian kernel and its dimension differ from a local cubic <nonlinear Schrödinger equation>. In three dimensions, the <Hardy–Littlewood–Sobolev inequality> controls the interaction energy and leads to <Global H1 solutions of the three-dimensional gravitational Hartree equation>.
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