Schrodinger estimates

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Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms or , or in X^{s,b} spaces defined by

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.

Schrodinger Linear estimates

[More references needed here!]

On :

  • If , then
    • (Energy estimate)
    • (Strichartz estimates) Sz1977.
      • More generally, f is in whenever , and
        • The endpoint is true for KeTa1998. When it fails even in the BMO case Mo1998, although it still is true for radial functions Ta2000b, Stv-p.In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable Ta2000b, although there is a limit as to low little regularity one can impose MacNkrNaOz-p.
        • In the radial case there are additional weighted smoothing estimates available Vi2001
        • When one also has
        • When one can refine the assumption on the data in rather technical ways on the Fourier side, see e.g. VaVe2001.
        • When the estimate has a maximizer Kz-p2. This maximizer is in fact given by Gaussian beams, with a constant of Fc-p4. Similarly when with the estimate, which is also given by Gaussian beams with a constant of
    • (Kato estimates) When , Sl1987, Ve1988, ConSau1988.
      • When we instead have
    • (Maximal function estimates) In all dimensions one has for all
      • When one also has
      • When one also has The can be raised to TaVa2000b, with the corresponding loss in the exponent dictated by scaling. Improvements are certainly possible.
    • Variants of some of these estimates exist for manifolds, see BuGdTz-p
  • Fixed time estimates for free solutions:
    • (Energy estimate) If , then is also .
    • (Decay estimate) If , then has an norm of
    • Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

  • embds into Bo1993 (see also HimMis2001).
  • embeds into Bo1993. One cannot remove the from the exponent, however it is conjectured in Bo1993 that one might be able to embed into

On :

  • When embeds into (this is essentially in Bo1993)
    • The endpoint is probably false in every dimension.

Strichartz estimates are also available on more general manifolds, and in the presence of a potential. Inhomogeneous estimates are also available off the line of duality; see Fc-p2 for a discussion.

Schrodinger Bilinear Estimates

  • On R2 we have the bilinear Strichartz estimate Bo1999:

and BkOgPo1998

Also, if u has frequency and v has frequency then we have (see e.g. CoKeStTkTa2003b)

and similarly for .

  • The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R2 estimates. The analogues on are also known KnPoVe1996b:

In two dimensions, the endpoint linear Strichartz estimate continues to fail in the bilinear setting Ta2006c.

Schrodinger Trilinear estimates

  • On R we have the following refinement to the Strichartz inequality Gr-p2:

Schrodinger Multilinear estimates

  • In R2 we have the variant

where each factor is allowed to be conjugated if desired. See St1997b, CoDeKnSt-p.