Schrodinger equations: Difference between revisions

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==Overview==


==Non-linear Schrodinger equations==
There are many '''nonlinear Schrodinger equations''' in the literature, all of which are perturbations of one sort or another of the [[free Schrodinger equation]]. One general class of such equations takes the form
 
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<center>'''Overview'''</center>
 
The free Schrodinger equation
 
<center><math>i \partial_t u + \Delta u = 0</math></center>
 
where u is a complex-valued function in <math>R^{d+1}</math>, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. This equation can be modified in many ways, notably by adding a potential or an obstacle, but we shall be interested in non-linear perturbations such as


<center><math>i \partial_t u + \Delta u = f (u, \overline{u}, Du, D \overline{u})</math></center>
<center><math>i \partial_t u + \Delta u = f (u, \overline{u}, Du, D \overline{u})</math></center>


where <math>D</math> denotes spatial differentiation. In such full generality, we refer to this equation as a ''derivative non-linear Schrodinger equation'' (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a [#nls semilinear Schrodinger equation] (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.
where <math>D</math> denotes spatial differentiation. In such full generality, we refer to this equation as a [[derivative non-linear Schrodinger equation]] (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a [[semilinear Schrodinger equation]] (NLS). These equations (particularly the [[cubic NLS]]) arise as model equations from several areas of physics.
 
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==Semilinear Schrodinger (NLS)==
 
[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section. However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]
 
The semilinear Schrodinger equation is
 
<center>i u<sub>t</sub> + <font face="Symbol">D</font> u + <font face="Symbol">l</font> |u|^{p-1} u = 0</center>
 
for p>1. (One can also add a potential term, which leads to many physically interesting problems, however the field of Schrodinger operators with potential is far too vast to even attempt to summarize here). In order to consider this problem in H<sup>s</sup> one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume
 
<center>p is an odd integer, or p > [s]+1.</center>
 
This is a Hamiltonian flow with the Hamiltonian
 
<center>H(u) = <font face="Symbol">ò</font> |<font face="Symbol">Ñ</font> u|<sup>2</sup>/2 - <font face="Symbol">l</font> |u|^{p+1}/(p+1) dx</center>
 
and symplectic form
 
<center>{u, v} = Im <font face="Symbol">ò</font> u <u>v</u> dx.</center>
 
From the phase invariance u -> exp(i <font face="Symbol">q</font>) u one also has conservation of the L<sup>2</sup> norm. The case <font face="Symbol">l</font> > 0 is focussing; <font face="Symbol">l</font> < 0 is defocussing.
 
The scaling regularity is s<sub>c</sub> = d/2 - 2/(p-1). The most interesting values of p are the ''L''<sup>2</sup>''-critical'' or ''pseudoconformal'' power p=1+4/d and the ''H''<sup>1</sup>''-critical'' power p=1+4/(d-2) for d>2 (for d=1,2 there is no H<sup>1</sup> conformal power). The power p = 1 + 2/d is also a key exponent for the scattering theory (as this is when the non-linearity |u|^{p-1} u has about equal strength with the decay t^{-d/2}). The cases p=3,5 are the most natural for physical applications since the non-linearity is then a polynomial. The cubic NLS in particular seems to appear naturally as a model equation for many different physical contexts, especially in dispersive, weakly non-linear perturbations of a plane wave. For instance, it arises as a model for dilute Bose-Einstein condensates. <br /><br />
 
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Dimension d
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Scattering power 1+2/d
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L<sup>2</sup>-critical power 1+4/d
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H<sup>1</sup>-critical power 1+4/(d-2)
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2
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3
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5/3
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3/2
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2
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9/5
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7/3
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6
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4/3
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5/3
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2
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The pseudoconformal transformation of the Hamiltonian gives that the time derivative of
 
<center><nowiki>|| (x + 2it </nowiki><font face="Symbol">Ñ</font>) u ||<sup>2</sup>_2 - 8 <font face="Symbol">l</font> t<sup>2</sup>/(p+1) || u ||_{p+1}^{p+1}</center>
 
is equal to
 
<center>4dt<font face="Symbol">l</font>(p-(1+4/d))/(p+1) ||u||_{p+1}^{p+1}.</center>
 
This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. xu(0) in L<sup>2</sup>), especially in the L<sup>2</sup>-critical case p=1+4/d (when the above derivative is zero). The L<sup>2</sup> norm of xu(0) is sometimes known as the ''pseudoconformal charge''.
 
The equation is invariant under Gallilean transformations
 
<center>u(x,t) -> exp(i (v.x/2 - |v|<sup>2</sup> t)) u(x-vt, t).</center>
 
This can be used to show ill-posedness below L<sup>2</sup> in the focusing case [KnPoVe-p], and also in the defocusing case [CtCoTa-p2]. (However if the non-linearity is replaced by a non-invariant expression such as <u>u</u><sup>2</sup>, then [#Quadratic_NLS one can go below L<sup>2</sup>]).
 
From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form
 
<center><font face="Symbol">ò</font> <font face="Symbol">ò</font> |u|^{p+1}/|x| dx dt</center>
 
in the defocussing case in terms of the H^{1/2} norm. These are useful for limiting the number of times the solution can concentrate at the origin; this is especially handy in the radially symmetric case.
 
In the other direction, one has LWP for s <font face="Symbol">³</font> 0, s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]]; see also [[Bibliography#Ts1987|Ts1987]]; for the case s=1, see [[Bibliography#GiVl1979|GiVl1979]]. In the L<sup>2</sup>-subcritical cases one has GWP for all s<font face="Symbol">³</font>0 by L<sup>2</sup> conservation; in all other cases one has GWP and scattering for small data in H<sup>s</sup>, s <font face="Symbol">³</font> s<sub>c</sub>. These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements [[Bibliography#Pl2000|Pl2000]], [Pl-p4]. This can then be used to obtain self-similar solutions, see [[Bibliography#CaWe1998|CaWe1998]], [[Bibliography#CaWe1998b|CaWe1998b]], [[Bibliography#RiYou1998|RiYou1998]], [MiaZg-p1], [MiaZgZgx-p], [MiaZgZgx-p2], [[Bibliography#Fur2001|Fur2001]].
 
Now suppose we remove the regularity assumption that p is either an odd integer or larger than [s]+1. Then some of the above results are still known to hold:
 
* ? In the H^1 subcritical case one has GWP in H^1, assuming the nonlinearity is smooth near the origin [[Bibliography#Ka1986|Ka1986]]
** In R^6 one also has Lipschitz well-posedness [BuGdTz-p5]
 
<br /> In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for s > 0, s<sub>c</sub> if p > 1, with the endpoint s=0 being attained when 1 <= p <= 4 [[Bibliography#Bo1993|Bo1993]]. In particular one has GWP in L^2 when p < 4, or when p=4 and the data is small norm.For 6 > p <font face="Symbol">³</font> 4 one also has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[Bibliography#Bo1995c|Bo1995c]]. (For p=6 one needs to impose a smallness condition on the L<sup>2</sup> norm or assume defocusing; for p>6 one needs to assume defocusing). <br />
 
* For the defocussing case, one has GWP in the H<sup>1</sup>-subcritical case if the data is in H<sup>1</sup>. To improve GWP to scattering, it seems that needs p to be L<sup>2</sup> super-critical (i.e. p > 1 + 4/d). In this case one can obtain scattering if the data is in L<sup>2</sup>(|x|<sup>2</sup> dx) (since one can then use the pseudo-conformal conservation law).
** In the d <font face="Symbol">³</font> 3 cases one can remove the L<sup>2</sup>(|x|<sup>2</sup> dx) assumption [[Bibliography#GiVl1985|GiVl1985]] (see also [[Bibliography#Bo1998b|Bo1998b]]) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of t^{-d/2} is integrable). In this case one can even relax the H<sup>1</sup> norm to H<sup>s</sup> for some s<1 [[references:CoKeStTkTa-p7 CoKeStTkTa-p7]].
** For d=1,2 one can also remove the L<sup>2</sup>(|x|<sup>2</sup> dx) assumption [[Bibliography#Na1999c|Na1999c]] by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.
 
<br /> In the L^2-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
 
<center>d<sup>2</sup><sub>t</sub> <font face="Symbol">ò</font> x<sup>2</sup> |u|<sup>2</sup> dx ~ H(u);</center>
 
see e.g. [[Bibliography#OgTs1991|OgTs1991]]. By scaling this implies that we have instantaneous blowup in H^s for s < s_c in the focusing case. In the defocusing case blowup <br /> is not known, but the H^s norm can still get arbitrarily large arbitrarily quickly for s < s_c [CtCoTa-p2]
 
Suppose we are in the L^2 subcritical case p < 1 + 2/d, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy E. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is H<sup>1</sup>-stable [[Bibliography#Ws1985|Ws1985]], [[Bibliography#Ws1986|Ws1986]]. Below the H^1 norm, this is not known, but polynomial upper bounds on the instability are in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]].Multisolitons are also asymptotically stable under smooth decaying perturbations [[Bibliography#Ya1980|Ya1980]], [[Bibliography#Grf1990|Grf1990]], [[Bibliography#Zi1997|Zi1997]], [RoScgSf-p], [RoScgSf-p2], provided that p is betweeen 1+2/d and 1+4/d.
 
One can go beyond scattering and ask for asymptotic completeness and existence of the wave operators. When p <font face="Symbol">£</font> 1 + 2/d this is not possible due to the poor decay in time in the non-linear term [[Bibliography#Bb1984|Bb1984]], [[Bibliography#Gs1977b|Gs1977b]], [[Bibliography#Sr1989|Sr1989]], however at p = 1+2/d one can obtain modified wave operators for data with suitable regularity, decay, and moment conditions [[Bibliography#Oz1991|Oz1991]], [[Bibliography#GiOz1993|GiOz1993]], [[Bibliography#HaNm1998|HaNm1998]], [[Bibliography#ShiTon2004|ShiTon2004]], [[Bibliography#HaNmShiTon2004|HaNmShiTon2004]]. In the regime between the L<sup>2</sup> and H<sup>1</sup> critical powers the wave operators are well-defined in the energy space [[Bibliography#LnSr1978|LnSr1978]], [[Bibliography#GiVl1985|GiVl1985]], [[Bibliography#Na1999c|Na1999c]]. At the L<sup>2</sup> critical exponent 1 + 4/d one can define wave operators assuming that we impose an L<sup>p</sup><sub>x,t</sub> integrability condition on the solution or some smallness or localization condition on the data [[Bibliography#GiVl1979|GiVl1979]], [[Bibliography#GiVl1985|GiVl1985]], [[Bibliography#Bo1998|Bo1998]] (see also [[Bibliography#Ts1985|Ts1985]] for the case of finite pseudoconformal charge). Below the L<sup>2</sup> critical power one can construct wave operators on certain spaces related to the pseudo-conformal charge [[Bibliography#CaWe1992|CaWe1992]], [[Bibliography#GiOz1993|GiOz1993]], [[Bibliography#GiOzVl1994|GiOzVl1994]], [[Bibliography#Oz1991|Oz1991]]; see also [[Bibliography#GiVl1979|GiVl1979]], [[Bibliography#Ts1985|Ts1985]]. For H<sup>s</sup> wave operators were also constructed in [[Bibliography#Na2001|Na2001]]. However in order to construct wave operators in spaces such as L<sup>2</sup>(|x|<sup>2</sup> dx) (the space of functions with finite pseudoconformal charge) it is necessary that p is larger than or equal to the rather unusual power
 
<center>1 + 8 / (sqrt(d<sup>2</sup> + 12d + 4) + d - 2);</center>
 
see [[Bibliography#NaOz2002|NaOz2002]] for further discussion.
 
Many of the global results for H<sup>s</sup> also hold true for L<sup>2</sup>(|x|^{2s} dx). Heuristically this follows from the pseudo-conformal transformation, although making this rigorous is sometimes difficult. Sample results are in [[Bibliography#CaWe1992|CaWe1992]], [[Bibliography#GiOzVl1994|GiOzVl1994]], [[Bibliography#Ka1995|Ka1995]], [[Bibliography#NkrOz1997|NkrOz1997]], [NkrOz-p]. See [[Bibliography#NaOz2002|NaOz2002]] for further discussion.
 
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==NLS on manifolds and obstacles==
 
The NLS has also been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in H<sup>1</sup> [BuGdTz-p3], while for smooth three-dimensional compact surfaces and p=3 one has LWP in H<sup>s</sup> for s>1, together with weak solutions in H<sup>1</sup> [BuGdTz-p3]. In the special case of a sphere one has LWP in H^{d/2 + 1/2} for d<font face="Symbol">³</font>3 and p < 5 [BuGdTz-p3].
 
<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>For the cubic equation on two-dimensional surfaces one has LWP in H^s for s > ½ [BuGdTz-p3]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s >= 1 one has GWP [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]] and regularity [[Bibliography#BrzGa1980|BrzGa1980]]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s < 0 uniform ill-posedness can be obtained by adapting the argument in [[Bibliography#BuGdTz2002|BuGdTz2002]] or [CtCoTa-p]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For the [#Cubic_NLS_on_RxT sphere], [#Cubic_NLS_on_RxT cylinder], or [#Cubic_NLS_on_T^2 torus] more precise results are known
 
A key tool here is the development of Strichartz estimates on curved space. For general manifolds one has all the L^q_t L^r_x Strichartz estimates (locally in time), but with a loss of 1/q derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of 2/q derivatives). When the manifold is flat outside of a compact set and obeys a non-trapping condition, the optimal Strichartz estimates (locally in time) were obtained in [StTt-p]. <br /> When instead the manifold is decaying outside of a compact set and obeys a non-trapping condition, the Strichartz estimates (locally in time) with an epsilon loss were obtained by Burq [Bu-p3]; in the special case of L^4 estimates on R^3, and for non-trapping asymptotically conic manifolds, the epsilon was removed in [HslTaWun-p]
 
Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows.
 
* If (p-1)(d-2) < 2 then one has GWP in H^1 assuming a coercivity condition (e.g. if the nonlinearity is defocusing) [BuGdTz-p4].
** Note there is a loss compared with the non-obstacle theory, where one expects the condition to be (p-1)(d-2) < 4.
** The same is true for the endpoint d=3, p=3 if the energy is sufficiently small [BuGdTz-p4].
** If d <= 4 then the flow map is Lipschitz [BuGdTz-p4]
** For d=2, p <= 3 this is in [[Bibliography#BrzGa1980|BrzGa1980]], [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]]
* If p < 1 + 2/d then one has GWP in L^2 [BuGdTz-p4]
** For d=3 GWP for smooth data is in [[Bibliography#Jor1961|Jor1961]]
** Again, in the non-obstacle theory one would expect p < 1 + 4/d
** if p < 1 + 1/d then one also has strong uniqueness in the class L^2 [BuGdTz-p4]
 
On a domain in R^d, with Dirichlet boundary conditions, the results are as follows.
 
* Local well-posedness in H^s for s > d/2 can be obtained by energy methods.
* In two dimensions when p <=3, global well-posedness in the energy class (assuming energy less than the ground state, in the p=3 focusing case) is in [[Bibliography#BrzGa1980|BrzGa1980]], [[Bibliography#Vd1984|Vd1984]], [[Bibliography#OgOz1991|OgOz1991]], [[references.html Ca1989]].More precise asymptotics of a minimal energy blowup solution in the focusing p=3 case are in [BuGdTz-p], [Ban-p3]
* When p > 1 + 4/d blowup can occur in the focusing case [[Bibliography#Kav1987|Kav1987]]
 
GWP and scattering for defocusing NLS on Schwarzchild manifolds for radial data is in [[Bibliography#LabSf1999|LabSf1999]]
 
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==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
 
<center>i u<sub>t</sub> + <font face="Symbol">D</font> u + <font face="Symbol">l</font><nowiki>|u|^{p-1} u = V u</nowiki></center>
 
where V is real and time-independent. The behavior depends on whether V is positive or negative, and how V grows as |x| -> infinity. 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 = +- |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) = E.x, 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 [[Bibliography#Fuj1979|Fuj1979]], [[Bibliography#Fuj1980|Fuj1980]], [[Bibliography#Oh1989|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 [[Bibliography#Car2002b|Car2002b]].
** In the L^2 critical case one can in fact eliminate this potential by a change of variables [[Bibliography#Car2002c|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 [[Bibliography#YaZgg2001|YaZgg2001]].
 
Much work has also been done on the semiclassical limit of these equations; see for instance [[Bibliography#BroJer2000|BroJer2000]], [[Bibliography#Ker2002|Ker2002]], [CarMil-p], [[Bibliography#Car2003|Car2003]]. For work on standing waves for NLS with quadratic potential, see [[Bibliography#Fuk2001|Fuk2001]], [[Bibliography#Fuk2003|Fuk2003]], [[Bibliography#FukOt2003|FukOt2003]], [[Bibliography#FukOt2003b|FukOt2003b]].
 
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 [[Bibliography#Wed2000|Wed2000]]; the stronger L^p boundedness of wave operators in this case was established in [[Bibliography#Wed1999|Wed1999]], [[Bibliography#ArYa2000|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 [[Bibliography#Ya1999|Ya1999]], [[Bibliography#JeYa2002|JeYa2002]]. The method does not quite extend to p=1,infinity and thus does not directly imply the dispersive estimate although it does give Strichartz estimates for 1 < p < infinity.
** Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in [[Bibliography#JeNc2001|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 [[Bibliography#JouSfSo1991|JouSfSo1991]]
** A related local L^2 decay estimate was obtained for exponentially decaying potentials in [[Bibliography#Ra1978|Ra1978]]; this was refined to polynomially decaying potentials (e.g. <x>^{-3-}) (with additional resolvent estimates at low frequencies) in [[Bibliography#JeKa1980|JeKa1980]].
** L^p boundedness of wave operators was established in [[Bibliography#Ya1995|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 [[Bibliography#JeKa1980|JeKa1980]]).
** Dispersive estimates have also been proven for potentials whose Rollnik and global Kato norms are both smaller than the critical value of 4pi [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]
**# This has been extended to potentials decaying exactly like |x|^2 and d >= 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 >= 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 [[Bibliography#JouSfSo1991|JouSfSo1991]]
** Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in [[Bibliography#Je1980|Je1980]], [[Bibliography#Je1984|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 [[Bibliography#NieSf2003|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 >= d/2, as well as inverse square potentials 1/|x|^2, and linear combinations of these, have local smoothing [[Bibliography#RuVe1994|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 [[Bibliography#Ya1980|Ya1980]], [[Bibliography#Grf1990|Grf1990]], [[Bibliography#Zi1997|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 <= p <= 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.
 
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==Unique continuation==
 
A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if u is a solution to a nonlinear Schrodinger equation, and u(t_0) and u(t_1) is specified on a domain D at two different times t_0, t_1, does this uniquely specify the solution everywhere at all other intermediate times?
 
* For the 1D cubic NLS, with D equal to a half-line, and u assumed to vanish on D, this is in [[Bibliography#Zg1997|Zg1997]].
* For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in [[Bibliography#Bo1997b|Bo1997b]].
* For D the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in [[Bibliography#KnPoVe2003|KnPoVe2003]]
* For D in a half-plane, and allowing potentials in various Lebesgue spaces, this is in [IonKn-p]
* A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in [[reference:Isk1993 Isk1993]]
 
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==Derivative non-linear Schrodinger==
 
By derivative non-linear Schrodinger (D-NLS) we refer to equations of the form
 
<center>u<sub>t</sub> - i <font face="Symbol">D</font> u = f(u, <u>u</u>, Du, <u>Du</u>)</center>
 
where f is an analytic function of u, its spatial derivatives Du, and their complex conjugates which vanishes to at least second order at the origin. We often assume the natural gauge invariance condition
 
<center>f(exp(i <font face="Symbol">q</font>) <u>u</u>, exp(-i <font face="Symbol">q</font>) <u>u</u>, exp(i <font face="Symbol">q</font>) Du, exp(-i <font face="Symbol">q</font>) <u>Du</u>) = exp(i <font face="Symbol">q</font>) f(u, <u>u</u>, Du, <u>Du</u>).</center>
 
The main new difficulty here is the loss of regularity of one derivative in the non-linearity, which causes standard techniques such as the energy method to fail (since the energy estimate does not recover any regularity in the case of the Schrodinger equation). However, there are other estimates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain well-posedness results for sufficiently smooth and regular data. In the analytic category some existence results can be found in [[Bibliography#SnTl1985|SnTl1985]], [[Bibliography#Ha1990|Ha1990]].
 
An alternative strategy is to apply a suitable transformation in order to place the non-linearity in a good form. For instance, a term such as <u>u</u> <u>Du</u> is preferable to u Du (the Fourier transform is less likely to stay near the upper paraboloid, and these terms are more likely to disappear in energy estimates). One can often "gauge transform" the equation (in a manner dependent on the solution u) so that all bad terms are eliminated. In one dimension this can be done by fairly elementary methods (e.g. the method of integrating factors), but in higher dimensions one must use some pseudo-differential calculus.
 
In order to quantify the decay properties needed, we define H^{s,m} denote the space of all functions u for which <x>^m D<sup>s</sup> u is in L<sup>2</sup><nowiki>; thus s measures regularity and m measures decay. It is a well-known fact that the Schrodinger equation trades decay for regularity; for instance, data in H^{m,m'} instantly evolves to a solution locally in H^{m+m'} for the free Schrodinger equation and m, m' </nowiki><font face="Symbol">³</font> 0.
 
* If m <font face="Symbol">³</font> [d/2] + 4 is an integer then one has LWP in H^m \cap H^{m-2,2} [[Bibliography#Ci1999|Ci1999]]; see also [[Bibliography#Ci1996|Ci1996]], [[Bibliography#Ci1995|Ci1995]], [[Bibliography#Ci1994|Ci1994]].
** If f is cubic or better then one can improve this to LWP in H^m [[Bibliography#Ci1999|Ci1999]]. Furthermore, if one also has gauge invariance then data in H^{m,m'} evolves to a solution in H^{m+m'} for all non-zero times and all positive integers m' [[Bibliography#Ci1999|Ci1999]].
** If d=1 and f is cubic or better then one has LWP in H<sup>3</sup> [[Bibliography#HaOz1994b|HaOz1994b]].
*** For special types of cubic non-linearity one can in fact get GWP for small data in H^{0,4} \cap H^{4,0} [[Bibliography#Oz1996|Oz1996]].
** LWP in H<sup>s</sup> \cap H^{0,m} for small data for sufficiently large s, m was shown in [[Bibliography#KnPoVe1993c|KnPoVe1993c]]. The solution was also shown to have s+1/2 derivatives in L<sup>2</sup>_{t,x,loc}.
*** If f is cubic or better one can take m=0[[Bibliography#KnPoVe1993c|KnPoVe1993c]].
*** If f is quartic or better then one has GWP for small data in H<sup>s</sup>. [[Bibliography#KnPoVe1995|KnPoVe1995]]
*** For large data one still has LWP for sufficiently large s, m [[Bibliography#Ci1995|Ci1995]], [[Bibliography#Ci1994|Ci1994]].
 
<br /> If the non-linearity consists mostly of the conjugate wave <u>u</u>, then one can do much better. For instance [Gr-p], when f = (D<u>u</u>)^k one can obtain LWP when s > s<sub>c</sub> = d/2 + 1 - 1/(k-1), s<font face="Symbol">³</font>1, and k <font face="Symbol">³</font> 2 is an integer; similarly when f = D(<u>u</u>^k) one has LWP when s > s<sub>c</sub> = d/2 - 1/(k-1), s <font face="Symbol">³</font>0, and k <font face="Symbol">³</font> 2 is an integer. In particular one has GWP in L<sup>2</sup> when d=1 and f = i(<u>u</u><sup>2</sup>)<sub>x</sub> and GWP in H<sup>1</sup> when d=1 and f = i({<u>u</u>}<sub>x</sub>)<sup>2</sup>. These results apply in both the periodic and non-periodic setting.
 
Non-linearities such as t^{-\alpha} |u<sub>x</sub><nowiki>|</nowiki><sup>2</sup> in one dimension have been studied in [[Bibliography#HaNm2001b|HaNm2001b]], with some asymptotic behaviour obtained.
 
In d=2 one has GWP for small data when the nonlinearities are of the form <u>u</u> <u>Du</u> + u Du [[Bibliography#De2002|De2002]].
 
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[[Category:Equations]]
 
 
 
==Quasilinear NLS (QNLS)==
 
These are general equations of the form
 
<center><math>u_t = i a(x,t,u,Du) D^2 u + b_1(x,t,u,Du) Du + b_2(x,t,u,Du) D<u>u</u> + first order terms</math>,</center>
 
where a, b_1, b_2, and the lower order terms are smooth functions of all variables.These general systems arise in certain physical models (see e.g. [[Bibliography#BdHaSau1997|BdHaSau1997]]).Also under certain conditions they can be derived from fully nonlinear Schrodinger equations by differentiating the equation.
 
In order to qualify as a quasilinear NLS, we require that the quadratic form a is real and elliptic.It is also natural to assume that the metric structure induced by a obeys a non-trapping condition (all geodesics eventually reach spatial infinity), as this is what is necessary for the optimal local smoothing estimate to occur.For a similar reason it is useful to assume that the magnetic field b_1 (or more precisely, the imaginary part of this field) is uniformly integrable along lines in space in the time independent case (for the time dependent case the criterion involves the bicharacteristic flow and is more complicated, see [[Bibliography#Ic1984|Ic1984]]); without this condition even the linear equation can be ill-posed.
 
A model example of QNLS is the equation
 
<center><math>\partial_t u = i (\Delta - V(x))u - 2iu h' (|u|^2 ) \Delta h(|u|^2) + i u g(|u|^2)</math></center>
 
 
for smooth functions <math>h,g</math>.
 
When V=0 local existence for small data is known in <math>H^6(R^n)</math> for <math>n=1,2,3</math> [[Bibliography#BdHaSau1997|BdHaSau1997]]
 
Under certain conditions on the initial data the LWP can be extended to GWP for n=2,3 [[Bibliography#BdHaSau1997|BdHaSau1997]].
 
For large data, LWP is known in <math>H^s(R^n)</math> for any n and any sufficiently large <math>s > s(n) </math>[[Bibliography#Col2002|Col2002]]
 
For suitable choices of V LWP is also known for <math>H^\infty(R^n)</math> for any n [[Bibliography#Pop2001|Pop2001]]; this uses the Nash-Moser iteration method.


In one dimension, the fully nonlinear Schrodinger equation has LWP in <math>H^\infty(R^n)</math> assuming a cubic nonlinearity [[Bibliography#Pop2001b|Pop2001b]].Other LWP results for the one-dimensional QNLS have been obtained by [LimPo-p] using gauge transform arguments.
One can generalize both the linear and nonlinear perturbations to these equations and consider
the class of [[quasilinear Schrodinger equations]] or even [[fully nonlinear Schrodinger equations]]. Needless to say, these equations are significantly more difficult to analyse than the simpler model cases discussed above.


In general dimension, LWP for data in <math>H^{s,2}</math> for sufficiently large s has been obtained in [KnPoVe-p] assuming non-trapping, and asymptotic flatness of the metric a and of the magnetic field <math>Im b_1</math> (both decaying like <math>1/|x|^2</math> or better up to derivatives of second order).
One can combine these nonlinear perturbations with a [[free Schrodinger equation|linear perturbation]], leading for instance to the [[NLS with potential]] and the [[NLS on manifolds and obstacles]].
 
[[Category:Equations]]
 
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The perturbative theory of nonlinear Schrodinger equations (and the [[NLS|semilinear Schrodinger equations]] in particular) rests on a number of [[Schrodinger estimates|linear and nonlinear estimates for the free Schrodinger equation]].




==Specific Schrodinger Equations==
==Specific Schrodinger Equations==


Monomial semilinear Schrodinger equations are indexed by the degree of the nonlinearity and the spatial domain. A taxonomy of these and other specific Schrodinger equations appears on the [[Schrodinger:specific equations|specific equations]] page.
Monomial [[semilinear Schrodinger equation]]s can indexed by the degree of the nonlinearity, as follows.


===Quadratic NLS===
===Quadratic NLS===


Equations of the form
[[NLS]] equations of the form


<math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>
<math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>


which <math>Q(u, \overline{u})</math> a quadratic function of its arguments are [[quadratic NLS|quadratic nonlinear Schrodinger equations]].
with <math>Q(u, \overline{u})</math> a quadratic function of its arguments are [[quadratic NLS|quadratic nonlinear Schrodinger equations]].  They are mass-critical in four dimensions.


===Cubic NLS===
===Cubic NLS===


The [[cubic NLS| cubic nonlinear Schrodinger equation]] is of the form
The [[cubic NLS|cubic nonlinear Schrodinger equation]] is of the form


<math> i \partial_t u + \Delta u = \pm |u|^2 u</math>
<math> i \partial_t u + \Delta u = \pm |u|^2 u</math>
They are [[completely integrable]] in one dimension, mass-critical in two-dimensions, and energy-critical in four dimensions.


===Quartic NLS===
===Quartic NLS===


 
A [[NLS|nonlinear Schrodinger equation]] with nonlinearity of degree 4 is a [[quartic NLS|quartic nonlinear Schrodinger equation]].
A nonlinear Schrodinger equation with nonlinearity of degree 4 is a [[quartic NLS|quartic nonlinear Schrodinger equation]].


===Quintic NLS===
===Quintic NLS===


Equations of the form
[[NLS]] equations of the form


<math> i \partial_t u + \Delta u = \pm |u|^4 u</math>
<math> i \partial_t u + \Delta u = \pm |u|^4 u</math>


are [[quintic NLS|quintic nonlinear Schrodinger equations]].
are [[quintic NLS|quintic nonlinear Schrodinger equations]].  They are mass-critical in one dimension and energy-critical in three dimensions.


===Septic NLS===
===Septic NLS===


Equations of the form  
[[NLS]] equations of the form  
 


<math> i \partial_t u + \Delta u = \pm |u|^6 u</math>
<math> i \partial_t u + \Delta u = \pm |u|^6 u</math>
Line 391: Line 57:
===<math>L^2</math>-critical NLS===
===<math>L^2</math>-critical NLS===


The nonlinear Schrodinger equation
The [[NLS|nonlinear Schrodinger equation]]
 


<math> i \partial_t u + \Delta u = \pm |u|^{\frac{4}{d}} u</math>
<math> i \partial_t u + \Delta u = \pm |u|^{\frac{4}{d}} u</math>
Line 400: Line 65:
===Higher order NLS===
===Higher order NLS===


 
One can study higher-order NLS equations in which the Laplacian is replaced by a higher power. One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable [[cubic NLS]] on <math>R</math>. Another is the [[nonlinear Schrodinger-Airy system]].  A third class arises from the elliptic case of the [[Zakharov-Schulman system]].
One can study higher-order NLS equations in which the Laplacian is replaced by a higher power.One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable [Schrodinger:cubic NLS| cubic NLS] on <math>R</math> .Another is the [kdv:Schrodinger_Airy nonlinear Schrodinger-Airy equation].


===Schrodinger maps===
===Schrodinger maps===


A geometric Schodinger equation that has been intensively studied is the [[Schrodinger maps|Schrodinger map equation]].
A geometric [[derivative non-linear Schrodinger equation]] that has been intensively studied is the [[Schrodinger maps|Schrodinger map equation]].  This is the Schrodinger counterpart of the [[wave maps equation]].


===Cubic DNLS on <math>R</math>===
===Cubic DNLS on <math>R</math>===


The [[Schrodinger:cubic DNLS on R]] deriviative cubic nonlinear Schrodinger equation has nonlinearity of the form <math>i \partial_x (|u|^2 u).</math>
The [[cubic DNLS on R|deriviative cubic nonlinear Schrodinger equation]] has nonlinearity of the form <math>i \partial_x (|u|^2 u).</math>


===Hartree Equation===
===Hartree Equation===


The [[Schrodinger:Hartree equation|Hartree equation]] has a nonlocal nonlinearity given by convolution.
The [[Hartree equation]] has a nonlocal nonlinearity given by convolution, as does the very similar [[Schrodinger-Poisson system]], and certain cases of the [[Davey-Stewartson system]].


===Maxwell-Schrodinger system===
===Maxwell-Schrodinger system===


A Schrodinger-wave system closely related to the [wave:Maxwell-Klein-Gordon|Maxwell-Klein-Gordon equation]] is the [[Schrodinger:Maxwell-Schrodinger system]].
A Schrodinger-wave system closely related to the [[Maxwell-Klein-Gordon equation]] is the [[Maxwell-Schrodinger system]].


==Schrodinger estimates==


 
[[Category:Schrodinger]]
Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms <math>L^q_t L^r_x</math> or <math>L^r_x L^q_t</math>, or in <math>X^{s,b}</math> spaces, defined by
[[Category:Equations]]
::<math>|| u ||_{X^{s,b}} = || u ||_{s,b} := || \langle \xi\rangle^s  \langle \tau -|\xi|^2\rangle^b \hat{u} ||_{L^2_{\tau,\xi}}.</math>
 
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 X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.  These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities.  See also [Bo1993b], [KlMa1993].
 
[[Category:Estimates]]
 
 
See [[Schrodinger estimates]]
 
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Latest revision as of 03:44, 8 February 2011

Overview

There are many nonlinear Schrodinger equations in the literature, all of which are perturbations of one sort or another of the free Schrodinger equation. One general class of such equations takes the form

where denotes spatial differentiation. In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a semilinear Schrodinger equation (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

One can generalize both the linear and nonlinear perturbations to these equations and consider the class of quasilinear Schrodinger equations or even fully nonlinear Schrodinger equations. Needless to say, these equations are significantly more difficult to analyse than the simpler model cases discussed above.

One can combine these nonlinear perturbations with a linear perturbation, leading for instance to the NLS with potential and the NLS on manifolds and obstacles.

The perturbative theory of nonlinear Schrodinger equations (and the semilinear Schrodinger equations in particular) rests on a number of linear and nonlinear estimates for the free Schrodinger equation.


Specific Schrodinger Equations

Monomial semilinear Schrodinger equations can indexed by the degree of the nonlinearity, as follows.

Quadratic NLS

NLS equations of the form

with a quadratic function of its arguments are quadratic nonlinear Schrodinger equations. They are mass-critical in four dimensions.

Cubic NLS

The cubic nonlinear Schrodinger equation is of the form

They are completely integrable in one dimension, mass-critical in two-dimensions, and energy-critical in four dimensions.

Quartic NLS

A nonlinear Schrodinger equation with nonlinearity of degree 4 is a quartic nonlinear Schrodinger equation.

Quintic NLS

NLS equations of the form

are quintic nonlinear Schrodinger equations. They are mass-critical in one dimension and energy-critical in three dimensions.

Septic NLS

NLS equations of the form

are septic nonlinear Schrodinger equations.

-critical NLS

The nonlinear Schrodinger equation

posed for is scaling invariant in . This family of nonlinear Schrodinger equations is therefore called the mass critical nonlinear Schrodinger equation.

Higher order NLS

One can study higher-order NLS equations in which the Laplacian is replaced by a higher power. One class of such examples comes from the infinite hierarchy of commuting flows arising from the completely integrable cubic NLS on . Another is the nonlinear Schrodinger-Airy system. A third class arises from the elliptic case of the Zakharov-Schulman system.

Schrodinger maps

A geometric derivative non-linear Schrodinger equation that has been intensively studied is the Schrodinger map equation. This is the Schrodinger counterpart of the wave maps equation.

Cubic DNLS on

The deriviative cubic nonlinear Schrodinger equation has nonlinearity of the form

Hartree Equation

The Hartree equation has a nonlocal nonlinearity given by convolution, as does the very similar Schrodinger-Poisson system, and certain cases of the Davey-Stewartson system.

Maxwell-Schrodinger system

A Schrodinger-wave system closely related to the Maxwell-Klein-Gordon equation is the Maxwell-Schrodinger system.