NLS with potential

(Thanks to Remi Carles for much help with this section. - Ed.)

One can ask what happens to the NLS when a potential is added, thus

$i\partial _{t}u+Du+|u|^{p-1}u=Vu\,$

where $V\,$ is real and time-independent. The behavior depends on whether $V\,$ is positive or negative, and how $V\,$ grows as $|x|\rightarrow \infty \,$. In the following results we suppose that $V\,$ grows like some sort of power of $x\,$ (this can be made precise with estimates on the derivatives of $V\,$, etc.) A particularly important case is that of the quadratic potential $V=\pm |x|^{2};\,$ this can be used to model a confining magnetic trap for Bose-Einstein condensation. Most of the mathematical research has gone into the isotropic quadatic potentials, but anisotropic ones (given by quadratic forms other than $|x|^{2}\,$) are also of physical interest.

• If $V\,$is linear, i.e. $V(x)=Ex\,,$ then the potential can in fact be eliminated by a change of variables [CarNky-p]
• If $V\,$is smooth, positive, and has bounded derivatives up to order 2 (i.e. is quadratic or subquadratic), then the theory is much the same as when there is no potential - one has decay estimates, Strichartz estimates, and the usual local and global well posedness theory (see Fuj1979, Fuj1980, Oh1989)
• When $V\,$is exactly a positive quadratic potential $V=w^{2}|x|^{2},\,$ then one has blowup for the focusing nonlinearity for negative energy in the $L^{2}\,$ supercritical or critical, $H^{1}\,$ subcritical case Zhj2000a, Car2002b, Zhj2005.
  • In the $L^{2}\,$ critical case one can in fact eliminate this potential by a change of variables Car2002c. One consequence of this change of variables is that one can convert the usual solitary wave solution for NLS into a solution that blows up in finite time (cf. how the pseudoconformal transform is used to achieve a similar effect without the potential).
• When $V\,$is exactly a negative quadratic potential, one can prevent blowup even in the focusing case if the potential is sufficiently strong [Car-p]. Indeed, one has a scattering theory in this case [Car-p]
• If $V\,$grows faster than quadratic, then there are significant problems due to the failure of smoothness of the fundamental solution; if $V\,$is also negative, then even the linear theory fails (for instance, the Hamiltonian need not be essentially self-adjoint on test functions). However for positive superquadratic potentials partial results are still possible YaZgg2001.

Much work has also been done on the semiclassical limit of these equations; see for instance BroJer2000, Ker2002, [CarMil-p], Car2003. For work on standing waves for NLS with quadratic potential, see Zhj2000a, Zhj2000b,Fuk2001, Fuk2003, FukOt2003, FukOt2003b. Furthermore, the work about sharp criteria of global existence for NLS with quadratic potential, see Zhj2005, ChgZhj2006a, ChgZhj2006b, ShjZhj2006, ChgZhj2007,.

One component of the theory of NLS with potential is that of Strichartz estimates with potential, which in turn may be derived from dispersive estimates with potential, although it is possible to obtain Strichartz estimates without a dispersive inequality. Both types of estimates can only be expected to hold after first projecting to the absolutely continuous part of the spectrum (although this is not necessary if the potential is positive).

The situation for dispersive estimates (which imply Strichartz), and related estimates such as local $L^{2}\,$ decay, and of $L^{p}\,$ boundedness of wave operators (which implies both the dispersive inequality and Strichartz) is as follows. Here we consider potentials which could oscillate; relying mostly on magnitude bounds on $V\,$rather than on symbol-type bounds.

• When $d=1\,$ one has dispersive estimates whenever $<x>V\,$is $L^{1}\,$ [GbScg-p].
  • For potentials such that $<x>^{3/2+}V\,$is in $L^{1}\,$, this is essentially in Wed2000; the stronger $L^{p}\,$ boundedness of wave operators in this case was established in Wed1999, ArYa2000.
• When $d=2\,,$ relatively little is known.
  • $L^{p}\,$ boundedness of wave operators for potentials decaying like $<x>^{-6-}\,,$ assuming 0 is not a resonance nor eigenvalue, is in Ya1999, JeYa2002. The method does not quite extend to $p=1,\infty \,$ and thus does not directly imply the dispersive estimate although it does give Strichartz estimates for $1<p<\infty \,.$
  • Local $L^{2}\,$ decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in JeNc2001
• When $d=3\,$ one has dispersive estimates whenever $V\,$decays like $<x>^{-3-}\,$ and 0 is neither a eigenvalue nor resonance [GbScg-p]
  • For potentials which decay like $<x>^{-7-}\,$ and whose Fourier transform is in $L^{1}\,,$ a version of this estimate is in JouSfSo1991
  • A related local $L^{2}\,$ decay estimate was obtained for exponentially decaying potentials in Ra1978; this was refined to polynomially decaying potentials (e.g. $<x>^{-3-}\,$) (with additional resolvent estimates at low frequencies) in JeKa1980.
  • $L^{p}\,$ boundedness of wave operators was established in Ya1995 for potentials decaying like $<x>^{-5-}\,$ and for which 0 is neither an eigenvalue nor a resonance.
  • If 0 is a resonance one cannot expect to obtain the optimal decay estimate; at best one can hope for $t^{-1/2}\,$ (see JeKa1980).
  • Dispersive estimates have also been proven for potentials whose Rollnik and global Kato norms are both smaller than the critical value of $4\pi \,$ [RoScg-p]. Indeed their arguments partly extend to certain time-dependent potentials (e.g. quasiperiodic potentials), once one also imposes a smallness condition on the $L^{3/2}\,$ norm of $V\,$
  • If the potential is in $L^{2}\,$ and has finite global Kato norm, then one has dispersive estimates for high frequencies at least [RoScg-p].
  • Strichartz estimates have been obtained for potentials decaying like $<x>^{-2-}\,$ if 0 is neither a zero nor a resonance [RoScg-p]
    1. This has been extended to potentials decaying exactly like $|x|^{2}\,$ and $d\geq 3\,$ assuming some radial regularity and if the potential is not too negative [BuPlStaTv-p], [BuPlStaTv-p2]; in particular one has Strichartz estimates for potentials $V=a/|x|^{2},d\geq 3,\,$ and $a>-(n-2)^{2}/4\,$ (this latter condition is necessary to avoid bound states).
• For $d>3\,,$ most of the $d=3\,$ results should extend. For instance, the following is known.
  • For potentials which decay like $<x>^{-d-4-}\,$ and whose Fourier transform is in $L^{1}\,,$ dispersive estimates are in JouSfSo1991
  • Local $L^{2}\,$ decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in Je1980, Je1984.

For finite rank perturbations of the Laplacian, where each rank one perturbation is generated by a bump function and the bump functions are sufficiently far apart in physical space, decay and Schrodinger estimates were obtained in NieSf2003.The bounds obtained grow polynomially in the number of rank one perturbations.

Local smoothing estimates seem to be more robust than dispersive estimates, holding in a wider range of situations.For instance, in $R^{d}\,,$ any potential in $L^{p}\,$ for $p\geq d/2\,,$ as well as inverse square potentials $1/|x|^{2}\,,$ and linear combinations of these, have local smoothing RuVe1994.Note one does not need to project away the bound states for such estimates (as the bound states tend to already be rather smooth).However, for $p<d/2\,,$ one can have breakdown of local smoothing (and also dispersive and Strichartz estimates) [Duy-p]

For time-dependent potentials, very little is known.If the potential is small and quasiperiodic in time (or more generally, has a highly concentrated Fourier transform in time) then dispersive and Strichartz estimates were obtained in [RoScg-p]; the smallness is used to rule out bound states, among other things.In the important case of the charge transfer model (the time-dependent potential that arises in the stability analysis of multisolitons) see Ya1980, Grf1990, Zi1997, [RoScgSf-p], [RoScgSf-p2], where energy, dispersive, and Strichartz estimates are obtained, with application to the asymptotic stability of multisolitons.

The nonlinear interactions between the bound states of a Schrodinger operator with potential (which have no dispersion or decay properties in time) and the absolutely continuous portion of the spectrum (where one expects dispersion and Strichartz estimates) is not well understood.A preliminary result in this direction is in [GusNaTsa-p], which shows in $R^{3}$ that if there is only one bound state, and Strichartz estimates hold in the remaining portion of the spectrum, and the non-linearity does not have too high or too low a power (say $4/3\leq p\leq 4\,,$ or a Hartree-type nonlinearity) then every small $H^{1}\,$ solution will asymptotically decouple into a dispersive part evolving like the linear flow (with potential), plus a nonlinear bound state, with the energy and phase of this bound state eventually stabilizing at a constant.In [SfWs-p] the interaction of a ground state and an excited state is studied, with the generic behavior consisting of collapse to the ground state plus radiation.