Hartree equation: Difference between revisions
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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 [[ | 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> | ||
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 [Nak-p3] (see also [[ | * 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 [[GiOz1993]]) | ||
** For <math>n>1\,</math> and <math>n \ge 1\,</math> this is in [[ | ** 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> [[ | *** 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]]. | ||
** For <math>n < 1, n \ge 3\,,</math> and <math>1 - n/2 < s < 1\,</math> 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 [[ | *** Many earlier results in [[HaKakNm1998]], [[HaKaiNm1998]], [[HaNm2001]], [[HaNm1998b]] | ||
** In the Gevrey and real analytic categories there are some large data results in [[ | ** In the Gevrey and real analytic categories there are some large data results in [[GiVl2000]], [[GiVl2000b]], [[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 [[ | ** For small decaying data one has some invertibility of the wave operators [[HaNm1998]] | ||
A variant of the Hartree equations is the [[Schrodinger-Poisson system]]. | A variant of the Hartree equations is the [[Schrodinger-Poisson system]]. | ||
Revision as of 14:33, 10 August 2006
The Hartree equation is of the form
where
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \rightarrow n\,} (perhaps after suitable normalization of the kernel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |x|^{-n}\,} , which would otherwise blow up). The analysis divides into the short-range case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n > 1\,} , the long-range case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 < n < 1\,} , and the borderline (or critical) case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1\,.} Generally speaking, the smaller values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\,} are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresopnds to focusing.
The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} critical value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\,} is 4, in particular the equation is always subcritical in four or fewer dimensions. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n<4\,} one has global existence of energy solutions. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y\,,} twisted by a Fourier multiplier with symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/2 n < 1\,} but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ln (t)\,} must be replaced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t^{n-1}/(n-1)\,.}
The existence and mapping properties of these operators is only partly known:
- When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 2\,}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1\,,}
the wave operators map \hat{H^s} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H^s}\,}
for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1/2\,}
and are continuous and open [Nak-p3] (see also GiOz1993)
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n>1\,}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 1\,}
this is in NwOz1992
- In the defocusing case, all solutions in suitable spaces have asymptotic states in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} , and one has asymptotic completeness when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n > 4/3\,} HaTs1987.
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n < 1, n \ge 3\,,}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \le 1\,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 1.\,}
- For small decaying data one has some invertibility of the wave operators HaNm1998
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n>1\,}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \ge 1\,}
this is in NwOz1992
A variant of the Hartree equations is the Schrodinger-Poisson system.
