Schrodinger:Hartree equation

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 u_t + 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 references:HaTs1987 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 references#GiOz1993 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 references:GiOz1993 GiOz1993)
  • For n>1 and n > 1 this is in references:NwOz1992 NwOz1992
    • In the defocusing case, all solutions in suitable spaces have asymptotic states in L², and one has asymptotic completeness when n > 4/3 references:HaTs1987 HaTs1987.
  • For n < 1, n ³3, and 1 - n/2 < s < 1 this is in [Nak-p4]
    • Many earlier results in references:HaKakNm1998 HaKakNm1998, references:HaKaiNm1998 HaKaiNm1998, references:HaNm2001 HaNm2001, references: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 n< 1 and n > 1.
  • For small decaying data one has some invertibility of the wave operators references:HaNm1998 HaNm1998

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