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## SCATTERING FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH A GENERAL ONE-DIMENSIONAL CONFINEMENT

### Citations

460 | Methods of modern mathematical physics. II. Fourier analysis, selfadjointness - Reed, Simon - 1975 |

407 |
Semilinear Schrödinger Equations
- Cazenave
- 2003
(Show Context)
Citation Context ...ation A0(t) = A0 = Id, A1(t) = A1 = M 1/2 x , A2(t) = A2 = ∇y, A3(t) = y + it∇y = itei|y|2/(2t)∇y ( · e−i|y|2/(2t) ) = e−itHyeitH . The operator A3 is the standard Galilean operator on Rd−1, see e.g. =-=[4]-=-, so the last identity stems from the fact that e−itMx commutes with both ei t 2∆y and y. We readily have: Lemma 2.6. The operators Aj satisfy the following properties: • Commutation: for j ∈ {0, . . ... |

113 | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3
- Colliander, Keel, et al.
(Show Context)
Citation Context ...nsion cases d = 1, 2, by introducing more intricate Morawetz estimates. The most recent approach to prove asymptotic completeness in H1 relies on the introduction of interaction Morawetz estimates in =-=[6]-=-, an approach which has been revisited since, in particular in [16] and [10]. In the anisotropic case, interaction Morawetz have been used in [1] and [19] with two different angles: in both cases, it ... |

93 |
On a class of nonlinear Schrödinger equations
- Ginibre, Velo
- 1978
(Show Context)
Citation Context ...TIC COMPLETENESS In this section, we prove Theorem 1.5. Three approaches are available to prove asymptotic completeness for nonlinear Schrödinger equations (without potential). The initial approach (=-=[8]-=-) consists in working with a Σ regularity. This makes it possible to use the operator x+ it∇, whose main properties are essentially those stated in Lemma 2.6, and to which an important evolution law (... |

66 |
Remarks on the energy scattering for nonlinear Klein-Gordon and Schrodinger equations.
- Nakanishi
- 2001
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Citation Context ...on assumption on the data, and allows to work in H1(Rd), provided that σ > 2/d. It is based on Morawetz inequalities: asymptotic completeness is then established in [13, 9] for the case d > 3, and in =-=[15]-=- for the low dimension cases d = 1, 2, by introducing more intricate Morawetz estimates. The most recent approach to prove asymptotic completeness in H1 relies on the introduction of interaction Moraw... |

57 |
Decay and scattering of solutions of a nonlinear Schrödinger equation
- Lin, Strauss
- 1978
(Show Context)
Citation Context ...cal approach relaxes the localization assumption on the data, and allows to work in H1(Rd), provided that σ > 2/d. It is based on Morawetz inequalities: asymptotic completeness is then established in =-=[13, 9]-=- for the case d > 3, and in [15] for the low dimension cases d = 1, 2, by introducing more intricate Morawetz estimates. The most recent approach to prove asymptotic completeness in H1 relies on the i... |

43 |
Rapidly decaying solutions of the nonlinear Schrödinger equation
- Cazenave, Weissler
- 1992
(Show Context)
Citation Context ...e long range scattering in one dimension). A technical difference with [19] is that for the Cauchy problem, we do not make use of inhomogeneous Strichartz for non-admissible pairs like established in =-=[5, 7, 20]-=-, and for scattering theory, such estimates are not needed when d 6 4. We emphasize that here, the potential V can have essentially any behavior, provided that it remains bounded from below. It can be... |

38 | Inhomogeneous Strichartz estimates
- Foschi
(Show Context)
Citation Context ...e long range scattering in one dimension). A technical difference with [19] is that for the Cauchy problem, we do not make use of inhomogeneous Strichartz for non-admissible pairs like established in =-=[5, 7, 20]-=-, and for scattering theory, such estimates are not needed when d 6 4. We emphasize that here, the potential V can have essentially any behavior, provided that it remains bounded from below. It can be... |

32 |
Bilinear virial identities and applications
- Planchon, Vega
(Show Context)
Citation Context ...ates. The most recent approach to prove asymptotic completeness in H1 relies on the introduction of interaction Morawetz estimates in [6], an approach which has been revisited since, in particular in =-=[16]-=- and [10]. In the anisotropic case, interaction Morawetz have been used in [1] and [19] with two different angles: in both cases, it starts with the choice of an anisotropic weight in the virial compu... |

25 | Nonlinear Schrödinger equations with repulsive harmonic potential and applications
- Carles
(Show Context)
Citation Context ...e of σ below 2/d; see e.g. [4]. Unfortunately, this conservation law seems to be bound to isotropic frameworks: an analogous identity is available in the presence on an isotropic quadratic potential (=-=[3]-=-), but in our present framework, anisotropy seems to rule out a similar algebraic miracle. The second historical approach relaxes the localization assumption on the data, and allows to work in H1(Rd),... |

20 | Inhomogeneous Strichartz estimates for the Schrödinger equation, preprint - Vilela |

19 | Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity - ABDALLAH, CASTELLA, et al. |

14 |
Smoothing property for Schrodinger equations with potential superquadratic at infinity
- Yajima, Zhang
- 2001
(Show Context)
Citation Context ... behavior, provided that it remains bounded from below. It can be bounded (in which case the term “confinement” is inadequate), or grow arbitrarily fast as x → ±∞. This is in sharp contrast with e.g. =-=[14, 22, 23]-=-, where Strichartz estimates (with loss) are established in the presence of superquadratic potentials, or with [2], where a functional calculus adapted to confining potentials is developed: in all the... |

8 | data scattering for the nonlinear Schrödinger equation on product spaces, Comm. P.D.E. 37 (2012) 125–135
- Tzvetkov, Visciglia, et al.
(Show Context)
Citation Context ...ns, for a total space dimension d, it is expected that the “scattering dimension” is d − n. This was proven systematically in the case of a harmonic confinement in [1], complemented by [12]; see also =-=[11, 18]-=-. Therefore, to prove asymptotic completeness thanks to Morawetz estimates, it is natural to assume σ > 2d−n (essentially because it is not known how to take advantage of these estimates otherwise, ex... |

5 | Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping. preprint. Archived at http://arxiv.org/abs/1408.6213
- HANI, THOMANN
(Show Context)
Citation Context ...sider in the present paper (see Remark 1.6): essentially, if the nonlinearity is short range on Rd−n, then it remains short range on Rd with n confined directions. Long range effects are described in =-=[12]-=-, in the case n = d− 1 and σ = 1 (cubic nonlinearity, which is exactly the threshold to have long range scattering in one dimension). A technical difference with [19] is that for the Cauchy problem, w... |

5 | On the decay of solutions to a class of defocusing NLS
- VISCIGLIA
(Show Context)
Citation Context ...sition 5.1, which corresponds to the one considered in [19], is more efficient then, and allows to prove Theorem 1.5 for all d > 2. SCATTERING FOR NLS WITH 1D CONFINEMENT 15 The first step stems from =-=[21]-=-: Theorem 1.3 and Proposition 5.1 imply that ‖u(t)‖Lrxy −→t→+∞ 0, ∀2 < r < 2d (d− 2)+ . The end of the proof is presented in [19], and is readily adapted to our framework: it consists in choosing suit... |

5 |
smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity
- Local
(Show Context)
Citation Context ... behavior, provided that it remains bounded from below. It can be bounded (in which case the term “confinement” is inadequate), or grow arbitrarily fast as x → ±∞. This is in sharp contrast with e.g. =-=[14, 22, 23]-=-, where Strichartz estimates (with loss) are established in the presence of superquadratic potentials, or with [2], where a functional calculus adapted to confining potentials is developed: in all the... |

4 |
Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials
- MIZUTANI
(Show Context)
Citation Context ... behavior, provided that it remains bounded from below. It can be bounded (in which case the term “confinement” is inadequate), or grow arbitrarily fast as x → ±∞. This is in sharp contrast with e.g. =-=[14, 22, 23]-=-, where Strichartz estimates (with loss) are established in the presence of superquadratic potentials, or with [2], where a functional calculus adapted to confining potentials is developed: in all the... |

3 | Scattering for nonlinear Schrödinger equation under partial harmonic confinement
- ANTONELLI, CARLES, et al.
(Show Context)
Citation Context ...paper, we prove the analogous result in the case of (1.1), as well as the existence of wave operators (Cauchy problem with behavior prescribed at infinite time). This extends some of the results from =-=[1]-=- where the special case of an harmonic potential V is considered. The properties related to the harmonic potentials are exploited to prove the existence of wave operators in the case of a multidimensi... |

2 |
Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations, Quart
- Quadratic
(Show Context)
Citation Context ... most recent approach to prove asymptotic completeness in H1 relies on the introduction of interaction Morawetz estimates in [6], an approach which has been revisited since, in particular in [16] and =-=[10]-=-. In the anisotropic case, interaction Morawetz have been used in [1] and [19] with two different angles: in both cases, it starts with the choice of an anisotropic weight in the virial computation fr... |

1 |
Modified scatering for the cubic Schrödinger equation on product spaces and applications. preprint, archived at http://arxiv.org/abs/1311
- HANI, PAUSADER, et al.
(Show Context)
Citation Context ...ns, for a total space dimension d, it is expected that the “scattering dimension” is d − n. This was proven systematically in the case of a harmonic confinement in [1], complemented by [12]; see also =-=[11, 18]-=-. Therefore, to prove asymptotic completeness thanks to Morawetz estimates, it is natural to assume σ > 2d−n (essentially because it is not known how to take advantage of these estimates otherwise, ex... |

1 |
and scattering for NLS onRd×T in the energy space. preprint. Archived at http: //arxiv.org/abs/1409.3938
- Well-posedness
- 2014
(Show Context)
Citation Context ...omain ([17, Theorem X.32]) D(H) = {f ∈ L2(Rd), −1 2 ∆f + V f ∈ L2(Rd)}. The goal of this paper is to understand the large time dynamics in (1.1). This framework is to be compared with the analysis in =-=[19]-=-, where there is no external potential (V = 0), but where the x variable belongs to the torus T (which is the only one-dimensional compact manifold without boundary). It is proven there that if a shor... |