Difference between revisions of "Schrodinger:Hartree equation"

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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>
 
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>
  
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).
+
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).
  
 
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 [[references:GiOz1993 GiOz1993]])
+
* 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]])
** For <font face="Symbol">n</font>>1 and n <u><font face="Symbol">></font></u> 1 this is in [[references:NwOz1992 NwOz1992]]
+
** For <font face="Symbol">n</font>>1 and n <u><font face="Symbol">></font></u> 1 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 [[references:HaTs1987 HaTs1987]].
+
*** 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]].
 
** 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 <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]
*** Many earlier results in [[references:HaKakNm1998 HaKakNm1998]], [[references:HaKaiNm1998 HaKaiNm1998]], [[references:HaNm2001 HaNm2001]], [[references: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 [[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>
+
** 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 [[references:HaNm1998 HaNm1998]]
+
** For small decaying data one has some invertibility of the wave operators [[Bibliography#HaNm1998|HaNm1998]]
  
 
<div class="MsoNormal" style="text-align: center"><center>
 
<div class="MsoNormal" style="text-align: center"><center>

Revision as of 20:26, 28 July 2006

Hartree equation

[Sketchy! More to come later. Contributions are of course very welcome and will be acknowledged. - Ed.]

The Hartree equation is of the form

i ut + D u = V(u) u

where

V(u) = + |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 H1 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{Hs} to \hat{Hs} 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 L2, and one has asymptotic completeness when n > 4/3 HaTs1987.
    • For n < 1, n ³3, and 1 - n/2 < s < 1 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 n< 1 and n > 1.
    • For small decaying data one has some invertibility of the wave operators HaNm1998