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| ===Hartree equation===
| | #REDIRECT [[Hartree equation]] |
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| [Sketchy! More to come later. Contributions are of course very welcome and will be acknowledged. - Ed.] | |
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| The Hartree equation is of the form
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| <center>i u<sub>t</sub> + <font face="Symbol">D</font> u = V(u) u</center>
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| where
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| <center>V(u) = <u>+</u> |x|^{-<font face="Symbol">n</font>} * |u|<sup>2</sup></center>
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| and 0 < <font face="Symbol">n</font> < d. 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 <font face="Symbol">n</font> -> n (perhaps after suitable normalization of the kernel |x|^{-<font face="Symbol">n</font>}, which would otherwise blow up). The analysis divides into the ''short-range case'' <font face="Symbol">n</font> > 1, the ''long-range case'' 0 < <font face="Symbol">n</font> < 1, and the ''borderline (or critical) case'' <font face="Symbol">n</font><nowiki>=1. Generally speaking, the smaller values of </nowiki><font face="Symbol">n</font> are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresopnds to focusing.
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| The H<sup>1</sup> critical value of <font face="Symbol">n</font> is 4, in particular the equation is always subcritical in four or fewer dimensions. For <font face="Symbol">n</font><4 one has global existence of energy solutions. For <font face="Symbol">n</font><nowiki>=4 this is only known for small energy. </nowiki>
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| 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 [[references:HaTs1987 HaTs1987]]. For instance, in the borderline case, at large times t the solution usually resembles a free solution with initial data <font face="Symbol">y</font>, twisted by a Fourier multiplier with symbol exp(i V(hat{<font face="Symbol">y</font>}) ln t). (This can be seen formally by applying the pseudo-conformal transformation, discarding the Laplacian term, and solving the resulting ODE [[references#GiOz1993 GiOz1993]]). This creates modified wave operators instead of ordinary wave operators. A similar thing happens when 1/2 < <font face="Symbol">n</font> < 1 but ln t must be replaced by t^{<font face="Symbol">n</font>-1}/(<font face="Symbol">n</font>-1).
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| The existence and mapping properties of these operators is only partly known: <br />
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| * When n <u><font face="Symbol">></font></u> 2 and <font face="Symbol">n</font><nowiki>=1, the wave operators map \hat{H</nowiki><sup>s</sup>} to \hat{H<sup>s</sup>} for s > 1/2 and are continuous and open [Nak-p3] (see also [[references:GiOz1993 GiOz1993]])
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| ** For <font face="Symbol">n</font>>1 and n <u><font face="Symbol">></font></u> 1 this is in [[references:NwOz1992 NwOz1992]]
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| *** In the defocusing case, all solutions in suitable spaces have asymptotic states in L<sup>2</sup>, and one has asymptotic completeness when <font face="Symbol">n</font> > 4/3 [[references:HaTs1987 HaTs1987]].
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| ** For <font face="Symbol">n</font> < 1, n <font face="Symbol">³</font>3, and 1 - <font face="Symbol">n</font>/2 < s < 1 this is in [Nak-p4]
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| *** Many earlier results in [[references:HaKakNm1998 HaKakNm1998]], [[references:HaKaiNm1998 HaKaiNm1998]], [[references:HaNm2001 HaNm2001]], [[references:HaNm1998b HaNm1998b]]
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| ** In the Gevrey and real analytic categories there are some large data results in [[references:GiVl2000 GiVl2000]], [[references:GiVl2000b GiVl2000b]], [[references:GiVl2001 GiVl2001]], covering the cases <font face="Symbol">n<u><</u> 1</font> and n <u><font face="Symbol">></font></u><font face="Symbol"> 1.</font>
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| ** For small decaying data one has some invertibility of the wave operators [[references:HaNm1998 HaNm1998]]
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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