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><math>\partial_tu + Du= V(u)u\,</math></center>
+
<center><math>i\partial_tu + \Delta u= V(u)u\,</math></center>
  
 
where
 
where
  
<center><math>V(u)= \pm |x|^{-n}|u|^2\,</math></center>
+
<center><math>V(u)= \pm |x|^{-n} * |u|^2\,</math></center>
  
 
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 d,</math> (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 corresponds 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 d,</math> (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 corresponds to focusing.
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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.  
 
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 [[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 [[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>
+
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 <math>y</math>, twisted by a Fourier multiplier with symbol <math>e^{i V(\hat{y}) \log(t)}</math>. (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 <math>1/2 n < 1\,</math> but <math>\log (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 <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 [[Na-p3]] (see also [[GiOz1993]])
+
* When <math>n \ge 2\,</math> and <math>n=1\,,</math> the wave operators map <math>\hat{H^s}</math> to <math>\hat{H^s}\,</math> for <math>s > 1/2\,</math> and are continuous and open [[Na-p3]] (see also [[GiOz1993]])
 
** For <math>n>1\,</math> and <math>n \ge 1\,</math> this is in [[NwOz1992]]
 
** For <math>n>1\,</math> and <math>n \ge 1\,</math> this is in [[NwOz1992]]
 
*** 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> [[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> [[HaTs1987]].

Latest revision as of 17:30, 9 July 2007

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 (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 corresponds 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 to for and are continuous and open Na-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 Na-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.