Schrodinger:Maxwell-Schrodinger system

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Maxwell-Schrodinger system in

This system is a partially non-relativistic analogue of the [wave:mkg Maxwell-Klein-Gordon system]., coupling a U(1) connection Aa with a complex scalar field u. The Lagrangian density is

\int Fab Fab + 2 Im u D u - Dj u Dj u

giving rise to the system of PDE

i ut = Dj u Dj u/2 + A u
da Fab = Jb

where the current density Jb is given by

J = |u|^2; Jj = - Im u Dj u

As with the MKG system, there is a gauge invariance for the connection; one can place A in the Lorentz, Coulomb, or Temporal gauges (other choices are of course possible).

Let us place u in H^s, and A in H^sigma x H^{sigma-1}. The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be s = sigma = 1/2.

  • In the Lorentz and Temporal gauges, one has LWP for s >= 5/3 and s-1 <= sigma <= s+1, (5s-2)/3 [NkrWad-p]
    • For smooth data (s=sigma > 5/2) in the Lorentz gauge this is in NkTs1986 (this result works in all dimensions)
  • Global weak solutions are known in the energy class (s=sigma=1) in the Lorentz and Coulomb gauges GuoNkSr1996. GWP is still open however.
  • Modified wave operators have been constructed in the Coulomb gauge in the case of vanishing magnetic field in [GiVl-p3], [GiVl-p5]. No smallness condition is needed on the data at infinity.
    • A similar result for small data is in Ts1993
  • In one dimension, GWP in the energy class is known Ts1995
  • In two dimensions, GWP for smooth solutions is known TsNk1985