Hartree equation

The Hartree equation is of the form

${\displaystyle i\partial _{t}u+\Delta u=V(u)u\,}$

where

${\displaystyle V(u)=\pm |x|^{-n}*|u|^{2}\,}$

and ${\displaystyle 0. It can thus be thought of as a non-local cubic Schrodinger equation; the cubic NLS is in some sense a limit of this equation as ${\displaystyle n\rightarrow d,}$ (after suitable normalization of the kernel ${\displaystyle |x|^{-n}\,}$, which would otherwise blow up). The analysis divides into the short-range case ${\displaystyle n>1\,}$, the long-range case ${\displaystyle 0, and the borderline (or critical) case ${\displaystyle n=1\,.}$ Generally speaking, the smaller values of ${\displaystyle n\,}$ are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresponds to focusing.

The ${\displaystyle H^{1}\,}$ critical value of ${\displaystyle n\,}$ is 4, in particular the equation is always subcritical in four or fewer dimensions. For ${\displaystyle n<4\,}$ one has global existence of energy solutions. For ${\displaystyle n=4\,}$ this is only known for small energy.

In the short-range case one has scattering to solutions of the free Schrodinger equations under suitable assumptions on the data. However this is not true in the other two cases HaTs1987. For instance, in the borderline case, at large times t the solution usually resembles a free solution with initial data ${\displaystyle y}$, twisted by a Fourier multiplier with symbol ${\displaystyle e^{iV({\hat {y}})\log(t)}}$. (This can be seen formally by applying the pseudo-conformal transformation, discarding the Laplacian term, and solving the resulting ODE GiOz1993.) This creates modified wave operators instead of ordinary wave operators. A similar thing happens when ${\displaystyle 1/2n<1\,}$ but ${\displaystyle \log(t)\,}$ must be replaced by ${\displaystyle t^{n-1}/(n-1)\,.}$

The existence and mapping properties of these operators is only partly known:

• When ${\displaystyle n\geq 2\,}$ and ${\displaystyle n=1\,,}$ the wave operators map ${\displaystyle {\hat {H^{s}}}}$ to ${\displaystyle {\hat {H^{s}}}\,}$ for ${\displaystyle s>1/2\,}$ and are continuous and open Na-p3 (see also GiOz1993)
• For ${\displaystyle n>1\,}$ and ${\displaystyle n\geq 1\,}$ this is in NwOz1992
• In the defocusing case, all solutions in suitable spaces have asymptotic states in ${\displaystyle L^{2}\,}$, and one has asymptotic completeness when ${\displaystyle n>4/3\,}$ HaTs1987.
• For ${\displaystyle n<1,n\geq 3\,,}$ and ${\displaystyle 1-n/2 this is in Na-p4
• In the Gevrey and real analytic categories there are some large data results in GiVl2000, GiVl2000b, GiVl2001, covering the cases ${\displaystyle n\leq 1\,}$ and ${\displaystyle n\geq 1.\,}$
• For small decaying data one has some invertibility of the wave operators HaNm1998

A variant of the Hartree equations is the Schrodinger-Poisson system.