# Difference between revisions of "Hartree equation"

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− | + | The '''Hartree equation''' is of the form | |

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− | The Hartree equation is of the form | ||

<center>i u<sub>t</sub> + <font face="Symbol">D</font> u = V(u) u</center> | <center>i u<sub>t</sub> + <font face="Symbol">D</font> u = V(u) u</center> | ||

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<center>V(u) = <u>+</u> |x|^{-<font face="Symbol">n</font>} * |u|<sup>2</sup></center> | <center>V(u) = <u>+</u> |x|^{-<font face="Symbol">n</font>} * |u|<sup>2</sup></center> | ||

− | 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. | + | 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. |

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> | 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 Gevrey and real analytic categories there are some large data results in [[Bibliography#GiVl2000|GiVl2000]], [[Bibliography#GiVl2000b|GiVl2000b]], [[Bibliography#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> | ** In the Gevrey and real analytic categories there are some large data results in [[Bibliography#GiVl2000|GiVl2000]], [[Bibliography#GiVl2000b|GiVl2000b]], [[Bibliography#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> | ||

** For small decaying data one has some invertibility of the wave operators [[Bibliography#HaNm1998|HaNm1998]] | ** For small decaying data one has some invertibility of the wave operators [[Bibliography#HaNm1998|HaNm1998]] | ||

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[[Category:Equations]] | [[Category:Equations]] | ||

+ | [[Category:Schrodinger]] |

## Revision as of 04:06, 29 July 2006

The **Hartree equation** is of the form

_{t}+ D u = V(u) u

where

__+__|x|^{-n} * |u|

^{2}

and 0 < n < 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 n -> n (perhaps after suitable normalization of the kernel |x|^{-n}, which would otherwise blow up). The analysis divides into the *short-range case* n > 1, the *long-range case* 0 < n < 1, and the *borderline (or critical) case* n=1. Generally speaking, the smaller values of n are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresopnds to focusing.

The H^{1} critical value of n is 4, in particular the equation is always subcritical in four or fewer dimensions. For n<4 one has global existence of energy solutions. For 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 y, twisted by a Fourier multiplier with symbol exp(i V(hat{y}) ln 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 1/2 < n < 1 but ln t must be replaced by t^{n-1}/(n-1).

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

- When n
__>__2 and n=1, the wave operators map \hat{H^{s}} to \hat{H^{s}} for s > 1/2 and are continuous and open [Nak-p3] (see also GiOz1993)- For n>1 and n
__>__1 this is in NwOz1992- In the defocusing case, all solutions in suitable spaces have asymptotic states in L
^{2}, and one has asymptotic completeness when n > 4/3 HaTs1987.

- In the defocusing case, all solutions in suitable spaces have asymptotic states in L
- For n < 1, n ³3, and 1 - n/2 < s < 1 this is in [Nak-p4]
- Many earlier results in HaKakNm1998, HaKaiNm1998, HaNm2001, HaNm1998b

- In the Gevrey and real analytic categories there are some large data results in GiVl2000, GiVl2000b, GiVl2001, covering the cases n
__<__1 and n__>__1. - For small decaying data one has some invertibility of the wave operators HaNm1998

- For n>1 and n