Difference between revisions of "Hartree equation"

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The '''Hartree equation''' is of the form
 
The '''Hartree equation''' is of the form
  
<center>i u<sub>t</sub> + <font face="Symbol">D</font> u = V(u) u</center>
+
<center><math>\partial_tu + Du= V(u)u\,</math></center>
  
 
where
 
where
  
<center>V(u) = <u>+</u> |x|^{-<font face="Symbol">n</font>} * |u|<sup>2</sup></center>
+
<center><math>V(u)= \pm |x|^{-n}|u|^2\,</math></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 <math>0 < n < d\,</math>. 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 <math>n \rightarrow n\,</math> (perhaps after suitable normalization of the kernel <math>|x|^{-n}\,</math>, which would otherwise blow up). The analysis divides into the ''short-range case'' <math>n > 1\,</math>, the ''long-range case'' <math>0 < n < 1\,</math>, and the ''borderline (or critical) case'' <math>n=1\,.</math> Generally speaking, the smaller values of <math>n\,</math> 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 <math>H^1\,</math> critical value of <math>n\,</math> is 4, in particular the equation is always subcritical in four or fewer dimensions. For <math>n<4\,</math> one has global existence of energy solutions. For <math>n=4\,</math> 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 [[Bibliography#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 [[Bibliography#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).
+
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 [[Bibliography#HaTs1987|HaTs1987]]. For instance, in the borderline case, at large times t the solution usually resembles a free solution with initial data <math>y\,,</math> twisted by a Fourier multiplier with symbol <math>e^{i V(\hat{y}) ln (t)}\,.</math> (This can be seen formally by applying the pseudo-conformal transformation, discarding the Laplacian term, and solving the resulting ODE [[Bibliography#GiOz1993|GiOz1993]]). This creates modified wave operators instead of ordinary wave operators. A similar thing happens when <math>1/2 n < 1\,</math> but <math>ln (t)\,</math> must be replaced by <math>t^{n-1}/(n-1)\,.</math>
  
 
The existence and mapping properties of these operators is only partly known: <br />
 
The existence and mapping properties of these operators is only partly known: <br />
  
* 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 [[Bibliography#GiOz1993|GiOz1993]])
+
* When <math>n \ge 2\,</math> and <math>n=1\,,</math> the wave operators map \hat{H^s} to <math>\hat{H^s}\,</math> for <math>s > 1/2\,</math> and are continuous and open [Nak-p3] (see also [[Bibliography#GiOz1993|GiOz1993]])
** For <font face="Symbol">n</font>>1 and n <u><font face="Symbol">></font></u> 1 this is in [[Bibliography#NwOz1992|NwOz1992]]
+
** For <math>n>1\,</math> and <math>n \ge 1\,</math> this is in [[Bibliography#NwOz1992|NwOz1992]]
*** 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 [[Bibliography#HaTs1987|HaTs1987]].
+
*** In the defocusing case, all solutions in suitable spaces have asymptotic states in <math>L^2\,</math>, and one has asymptotic completeness when <math>n > 4/3\,</math> [[Bibliography#HaTs1987|HaTs1987]].
** 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]
+
** For <math>n < 1, n \ge 3\,,</math> and <math>1 - n/2 < s < 1\,</math> this is in [Nak-p4]
 
*** Many earlier results in [[Bibliography#HaKakNm1998|HaKakNm1998]], [[Bibliography#HaKaiNm1998|HaKaiNm1998]], [[Bibliography#HaNm2001|HaNm2001]], [[Bibliography#HaNm1998b|HaNm1998b]]
 
*** Many earlier results in [[Bibliography#HaKakNm1998|HaKakNm1998]], [[Bibliography#HaKaiNm1998|HaKaiNm1998]], [[Bibliography#HaNm2001|HaNm2001]], [[Bibliography#HaNm1998b|HaNm1998b]]
** 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 <math>n \le 1\,</math> and <math>n \ge 1.\,</math>
 
** 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]]
  

Revision as of 18:31, 7 August 2006

The Hartree equation is of the form

where

and . 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 (perhaps after suitable normalization of the kernel , which would otherwise blow up). The analysis divides into the short-range case , the long-range case , and the borderline (or critical) case Generally speaking, the smaller values of are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresopnds to focusing.

The critical value of is 4, in particular the equation is always subcritical in four or fewer dimensions. For one has global existence of energy solutions. For 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 twisted by a Fourier multiplier with symbol (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 but must be replaced by

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

  • When and the wave operators map \hat{H^s} to for and are continuous and open [Nak-p3] (see also GiOz1993)
    • For and this is in NwOz1992
      • In the defocusing case, all solutions in suitable spaces have asymptotic states in , and one has asymptotic completeness when HaTs1987.
    • For and this is in [Nak-p4]
    • In the Gevrey and real analytic categories there are some large data results in GiVl2000, GiVl2000b, GiVl2001, covering the cases and
    • For small decaying data one has some invertibility of the wave operators HaNm1998

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