2017, 37(10): 5355-5365. doi: 10.3934/dcds.2017233

Strichartz estimates for $N$-body Schrödinger operators with small potential interactions

Center for Mathematical Analysis & Computation (CMAC), Yonsei University, Seoul 03722, Korea

* Corresponding author: Younghun Hong

Received  January 2016 Revised  May 2017 Published  June 2017

Fund Project: The author is supported by NRF grant 2015R1A5A1009350

In this paper, we prove Strichartz estimates for $N$-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the $V_S^p$-norm of Koch and Tataru [19]. As an application, we prove scattering for $N$-body Schrödinger operators.

Citation: Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
References:
[1]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481. doi: 10.1007/s00220-012-1435-x.

[2]

J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp.

[3]

N. BurqF. PlanchonJ. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.

[4]

N. BurqF. PlanchonJ. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schr${\rm{\ddot d}}$inger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.

[5]

T. ChenY. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123. doi: 10.1007/s00205-016-1068-x.

[6]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.

[7]

X. Chen and J. Holmer, On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200. doi: 10.4171/JEMS/610.

[8]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600. doi: 10.1215/S0012-7094-80-04734-1.

[9]

M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750. doi: 10.1353/ajm.2006.0025.

[10]

M. Goldberg, Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal., 16 (2006), 517-536. doi: 10.1007/s00039-006-0568-5.

[11]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5.

[12]

G. Graf, Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101. doi: 10.1007/BF02278000.

[13]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.

[14]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349. doi: 10.1215/00127094-1415889.

[15]

J.-L. JourneA. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504.

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[17]

K. KirkpatrickB. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.

[18]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.

[19]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp.

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014).

[21]

I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨odinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4.

[22]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. , Princeton Univ. Press, 163 (2007), 255–285.

[23]

I. M. Sigal and A. Soffer, The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math., 126 (1987), 35-108. doi: 10.2307/1971345.

show all references

References:
[1]

M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481. doi: 10.1007/s00220-012-1435-x.

[2]

J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp.

[3]

N. BurqF. PlanchonJ. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.

[4]

N. BurqF. PlanchonJ. Stalker and S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schr${\rm{\ddot d}}$inger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.

[5]

T. ChenY. Hong and N. Pavlović, Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal., 224 (2017), 91-123. doi: 10.1007/s00205-016-1068-x.

[6]

T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.

[7]

X. Chen and J. Holmer, On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016), 1161-1200. doi: 10.4171/JEMS/610.

[8]

D. Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600. doi: 10.1215/S0012-7094-80-04734-1.

[9]

M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006), 731-750. doi: 10.1353/ajm.2006.0025.

[10]

M. Goldberg, Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal., 16 (2006), 517-536. doi: 10.1007/s00039-006-0568-5.

[11]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5.

[12]

G. Graf, Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990), 73-101. doi: 10.1007/BF02278000.

[13]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.

[14]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011), 329-349. doi: 10.1215/00127094-1415889.

[15]

J.-L. JourneA. Soffer and C. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. doi: 10.1002/cpa.3160440504.

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[17]

K. KirkpatrickB. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.

[18]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.

[19]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp.

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014).

[21]

I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨odinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/s00222-003-0325-4.

[22]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. , Princeton Univ. Press, 163 (2007), 255–285.

[23]

I. M. Sigal and A. Soffer, The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math., 126 (1987), 35-108. doi: 10.2307/1971345.

[1]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

[2]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

[3]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

[4]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[5]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094

[6]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[7]

Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177

[8]

Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941

[9]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803

[10]

Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127

[11]

Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034

[12]

Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109

[13]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771

[14]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481

[15]

Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183

[16]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

[17]

Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723

[18]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143

[19]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071

[20]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (2)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]