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March  2017, 37(3): 1201-1226. doi: 10.3934/dcds.2017050

Strichartz estimates for charge transfer models

Gong Chen

Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615, USA

Received  October 2015 Revised  November 2016 Published  December 2016

In this note, we prove Strichartz estimates for scattering states of scalar charge transfer models in $\mathbb{R}^{3}$. Following the idea of Strichartz estimates based on [3,10], we also show that the energy of the whole evolution is bounded independently of time without using the phase space method, as for example, in [5]. One can easily generalize our arguments to $\mathbb{R}^{n}$ for $n≥q3$. We also discuss the extension of these results to matrix charge transfer models in $\mathbb{R}^{3}$.

Keywords: Strichartz estimates, charge transfer model, energy boundedness.
Mathematics Subject Classification: Primary:35Q35, 37K40.
Citation: Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050
References:
[1]

K. Cai, Fine properties of charge transfer models, preprint, arXiv: math/0311048. Google Scholar

[2]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.  doi: 10.1002/cpa.1018.  Google Scholar

[3]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-textendashtrapping potential, J. Differential Equations, 256 (2014), 1395-1466.  doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[4]

M. B. Erdogan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅱ, J. Anal. Math., 99 (2006), 199-248.  doi: 10.1007/BF02789446.  Google Scholar

[5]

J. M. Graf, Phase space analysis of the charge transfer model, Helv. Physica Acta, 63 (1990), 107-138.   Google Scholar

[6]

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

[7]

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

[8]

L. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations Universitext. Springer, New York, 2009. doi: 978-0-387-84898-3.  Google Scholar

[9]

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

[10]

I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math., 58 (2005), 149-216.  doi: 10.1002/cpa.20066.  Google Scholar

[11]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, in Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. , (eds. J. Bourgain, C. Kenig and S. Klainerman), Princeton Univ. Press, 163 (2007), 255-285.  Google Scholar

[12]

K. Yajima, The Wk, p-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.  Google Scholar

[13]

K. Yajima, A multichannel scattering theory for some time dependent Hamiltonians, charge transfer problem, Comm. Math. Phys., 75 (1980), 153-178.  doi: 10.1007/BF01222515.  Google Scholar

show all references

References:
[1]

K. Cai, Fine properties of charge transfer models, preprint, arXiv: math/0311048. Google Scholar

[2]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.  doi: 10.1002/cpa.1018.  Google Scholar

[3]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-textendashtrapping potential, J. Differential Equations, 256 (2014), 1395-1466.  doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[4]

M. B. Erdogan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: Ⅱ, J. Anal. Math., 99 (2006), 199-248.  doi: 10.1007/BF02789446.  Google Scholar

[5]

J. M. Graf, Phase space analysis of the charge transfer model, Helv. Physica Acta, 63 (1990), 107-138.   Google Scholar

[6]

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

[7]

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

[8]

L. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations Universitext. Springer, New York, 2009. doi: 978-0-387-84898-3.  Google Scholar

[9]

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

[10]

I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math., 58 (2005), 149-216.  doi: 10.1002/cpa.20066.  Google Scholar

[11]

W. Schlag, Dispersive estimates for Schrödinger operators: A survey, in Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. , (eds. J. Bourgain, C. Kenig and S. Klainerman), Princeton Univ. Press, 163 (2007), 255-285.  Google Scholar

[12]

K. Yajima, The Wk, p-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.  Google Scholar

[13]

K. Yajima, A multichannel scattering theory for some time dependent Hamiltonians, charge transfer problem, Comm. Math. Phys., 75 (1980), 153-178.  doi: 10.1007/BF01222515.  Google Scholar

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