We show the scattering for a one dimensional nonlinear Schrödinger equation with a non-negative, repulsive potential V such that , and a mass-supercritical non-linearity. We follow the approach of concentration-compacity/rigidity first introduced by Kenig and Merle.
We consider the following one dimensional defocusing, non linear Schrödinger equation with a potential
For the mass-supercritical () homogeneous equation
We prove the scattering of solutions of (1.1) in dimension one for sufficiently regular, non-negative and repulsive potential V.
Let and be such that . We suppose moreover that V is non-negative and repulsive: and . Then, every solution of (1.1) with potential V scatters in .
We use the strategy of concentration-compacity/rigidity first introduced by Kenig and Merle in , and extented to the intercritical case by Holmer and Roudenko in , Duyckaerts, Holmer and Roudenko in . In the case of a potential, the main difficulty is the lack of translation invariance of the equation. Notice that Hong obtained in  the same result in the three dimensional case for the focusing equation. However, his approach cannot be extended to lower dimensions, as it requires endpoint Strichartz estimates which are not available. Banica and Visciglia treated in  the case of the non linear Schrödinger equation with a Dirac potential on the line, and we follow their approach. The Dirac potential is more singular, but it allows the use of explicit formulas that are not available in the present more general framework.
In dimension one or two, assume that V is smooth and compactly supported, and such that . Then the operator has a negative eigenvalue: as a consequence, the hypothesis of positivity of V cannot be relaxed as in dimension three, where  only supposes that the potential has a small negative part, and, in the same way, the hypothesis of repulsivity, which is needed for the rigidity, cannot be relaxed to having a small positive part.
The hypothesis are needed to show that the operator verifies the hypothesis of the abstract profile decomposition of , whereas the hypothesis and are needed in the rigidity part.
In the focusing, mass-supercritical case
We will denote by V a potential on the line satisfying the hypothesis of Theorem 1, α will be a real number such that . We set
From now on, we will fix the four following Strichartz exponents
Recall that we assume all along the paper that V is in and non negative. Goldberg and Schlag obtained in particular in  the dispersive estimate for the Schrödinger operator under these assumptions.
Indeed, they require the hypothesis of absence of resonances at zero energy. We claim that for this hypothesis is satisfied: by the definition of , if there is a resonance at zero, the solutions of
Proposition 1(Dispersive estimate ).
Let be such that . Then, for all , we have
Note that, interpolating the previous dispersive estimate (2.2) with the mass conservation law, we obtain immediately for all
Proposition 2(Strichartz estimates).
For all , all , all and all
The estimates (2.5)–(2.8) are exactly the same as (3.1)–(3.4) of , with the operator instead of . As the proof of  relies only on the admissible Strichartz estimates (2.4) that are given by Proposition 1, the same proof holds here. Finally, (2.9) enters on the frame of the non-admissible inhomogeneous Strichartz estimates of Theorem 1.4 of Foschi’s paper . □
We will need the three following classical perturbative results, which follow immediately from the previous Strichartz inequalities:
Let be a solution of (1.1). If , then u scatters in .
Proof of Propositions 3and 4.
For every there exists and such that the following occurs. Let be a solution of the following integral equation with source term
Theorem(Astract profile decomposition ).
Let be a self adjoint operator such that:
for some positive constants and for all ,
let . Then, as n goes to infinity
let , be sequences of real numbers, and . Then
for any fixed j,
orthogonality of the parameters:
decay of the reminder:
orthogonality of the Hilbert norm:
We will see that the self-adjoint operator verifies the hypothesis of the previous theorem.
Assumption (3.1). Because V is positive and by the Sobolev embedding ,
Assumption (3.2). We have
Assumption (3.3). It is an immediate consequence of the dispersive estimate and the translation invariance of the norms. Indeed, because , if , there exists a , compactly supported function such that
Assumption (3.4). We will show that
Note that is a solution of the following linear Schrödinger equation with zero initial data
Let us fix . and the functions of vanish at infinity, so, using the compacity in time, there exists such that
Assumption (3.5). We decompose
4.Non linear profiles
Propositions 7, 8 and 9 are the analogous of Propositions 3.4 and 3.6 of . The non linear Schrödinger equation with a Dirac potential is more singular, but it allows the use of explicit formulas that are not available in the present more general framework.
Let , be such that . Then, up to a subsequence
Up to a subsequence, we can assume that or . Let us assume for example .
As a first step, we will show that
To obtain (4.1), we are now reduced to show that for fixed
Let , be such that , be the unique solution to (1.2) with initial data ψ, and . Then, up to a subsequence
We follow the same spirit of proof as for Proposition 7. We begin to show that
It remains to show that for fixed,
Let , , be such that and , U be a solution to (1.2) such that
Finally, we will need the following Proposition of non linear scattering:
Let . Then there exists , solution of (1.1) such that
The same proof as , Proposition 3.5, holds, as it involves only the analogous Strichartz estimates. □
5.Construction of a critical element
We have now all the tools to extract a critical element following the approach of . Let
If , then there exists , , such that the corresponding solution of (1.1) verifies that is relatively compact in .
(1) If . By the orthogonality condition, notice that this can happen only for one profile. Because , we have , so the solution of (1.1) with data scatters. If this case happens, let be this solution, otherwise, we set .
(2) If and . Let be the unique solution to (1.2) with initial data . We set .
(4) If and . Let be a solution to (1.2) such that
In this section, we will show that the critical solution constructed in the previous one assuming the fact that cannot exist.
We will need the following classical result concerning the compact families of
Suppose that is relatively compact in . Then, for any , there exists such that
Classic, see e.g. . □
Now, we can show the rigidity Proposition needed to end the proof:
Suppose that is a solution of (1.1) such that is relatively compact in . Then .
By a classical elementary computation, we get the following virial identities:
Let be a solution to (1.1) and χ be a compactly supported, regular function. Then
Now, we assume by contradiction that . Let be such that for and for , set and
We are now in position to end the proof of Theorem 1:
Proof of Theorem 1.
If , then the Proposition 11 allows us to extract a critical element , , such that the corresponding solution of (1.1) verifies that is relatively compact in . By Proposition 13, such a critical solution cannot exist, so and by Proposition 3, all the solutions of (1.1) scatter in . □
The author thanks N. Visciglia for having submitted him this problem, enlighting discussions and his warm welcome in Pisa, J. Zheng for his helpfull comments, his Ph.D. advisor F. Planchon for his disponibility and advices, and the referee for his careful reading and his constructive criticism.
V. Banica and N. Visciglia, Scattering for NLS with a delta potential, J. Differential Equations 260(5) (2016), 4410–4439. doi:10.1016/j.jde.2015.11.016.
J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 62(7) (2009), 920–968. doi:10.1002/cpa.20278.
T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett. 15(6) (2008), 1233–1250. doi:10.4310/MRL.2008.v15.n6.a13.
L. Fanelli and N. Visciglia, The lack of compactness in the Sobolev–Strichartz inequalities, J. Math. Pures Appl. (9) 99(3) (2013), 309–320. doi:10.1016/j.matpur.2012.06.015.
D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math. 54(10) (2011), 2037–2062. doi:10.1007/s11425-011-4283-9.
D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2(1) (2005), 1–24. doi:10.1142/S0219891605000361.
M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251(1) (2004), 157–178. doi:10.1007/s00220-004-1140-5.
J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282(2) (2008), 435–467. doi:10.1007/s00220-008-0529-y.
Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal. 15 (2016), 1571–1601, doi:10.3934/cpaa.2016003.
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120(5) (1998), 955–980, available at: http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi:10.1353/ajm.1998.0039.
C.E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166(3) (2006), 645–675. doi:10.1007/s00222-006-0011-4.
K. Nakanishi, Energy scattering for nonlinear Klein–Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169(1) (1999), 201–225. doi:10.1006/jfan.1999.3503.
F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42(2) (2009), 261–290.
N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett. 16(5) (2009), 919–926. doi:10.4310/MRL.2009.v16.n5.a14.