# American Institute of Mathematical Sciences

October  2013, 33(10): 4473-4495. doi: 10.3934/dcds.2013.33.4473

## Dispersive estimates for matrix Schrödinger operators in dimension two

 1 Department of Mathematics, University of Illinois, Urbana, IL 61801, United States 2 Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, United States

Received  November 2012 Revised  February 2013 Published  April 2013

We consider the non-selfadjoint operator $H = \left[\begin{array}{cc} -\Delta + \mu-V_1 & -V_2\\ V_2 & \Delta - \mu + V_1 \end{array} \right]$ where $\mu>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\mathbb{R}^2)\times L^1(\mathbb{R}^2)\to L^\infty(\mathbb{R}^2)\times L^\infty(\mathbb{R}^2)$ dispersive decay estimates for the evolution $e^{it H}P_{ac}$. We also obtain the following weighted estimate $$\|w^{-1} e^{it\mathcal H}P_{ac}f\|_{L^\infty(\mathbb R^2)\times L^\infty(\mathbb R^2)} ≲ \frac{1}{|t|\log^2(|t|)} \|w f\|_{L^1(\mathbb R^2)\times L^1(\mathbb R^2)},\,\,\,\,\,\,\,\, |t| >2,$$with $w(x)=\log^2(2+|x|)$.
Citation: M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473
##### References:
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Green, Dispersive estimates for the Schrodinger equation for $C^\frac{n-3}{2}$ potentials in odd dimensions, Int. Math. Res. Notices, 13 (2010), 2532-2565. doi: 10.1093/imrn/rnp221.  Google Scholar [16] M. B. Erdoǧan and W. R. Green, Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy, To appear in Trans. Amer. Math. Soc., (2012). Google Scholar [17] M. B. Erdoǧan and W. R. Green, A weighted dispersive estimate for Schrödinger operators in dimension two, Comm. Math. Phys., 319 (2013), 791-811 doi: 10.1007/s00220-012-1640-7.  Google Scholar [18] M. B. Erdoǧan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I, Dynamics of PDE, 1 (2004), 359-379. doi: 10.1007/BF02789446.  Google Scholar [19] M. B. Erdoǧan and W. 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Simpson, Spectral analysis for matrix Hamiltonian operators, Nonlinearity, 24 (2011), 389-429. doi: 10.1088/0951-7715/24/2/003.  Google Scholar [34] T. Mizumachi, Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 47 (2007), 599-620.  Google Scholar [35] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56. doi: 10.1016/0022-1236(82)90084-2.  Google Scholar [36] C.-A. Pillet and E. C. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differ. Eqs., 141 (1997), 310-326. doi: 10.1006/jdeq.1997.3345.  Google Scholar [37] J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys., 61 (1978), 149-168. doi: 10.1007/BF01609491.  Google Scholar [38] M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis, IV: Analysis of Operators," Academic Press, New York, NY, 1972.  Google Scholar [39] 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 [40] 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 [41] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two, Comm. Math. Phys., 257 (2005), 87-117. doi: 10.1007/s00220-004-1262-9.  Google Scholar [42] W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar [43] W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007).  Google Scholar [44] W. Schlag, Stable manifolds for an orbitally unstable NLS, Ann. of Math. (2), 169 (2009), 139-227. doi: 10.4007/annals.2009.169.139.  Google Scholar [45] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146. doi: 10.1007/BF02096557.  Google Scholar [46] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar [47] R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Commun. Math. Phys., 215 (2000), 343-356. doi: 10.1007/s002200000298.  Google Scholar [48] K. Yajima, $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys., 208 (1999), 125-152. doi: 10.1007/s002200050751.  Google Scholar [49] K. Yajima, The $L^p$ Boundedness of wave operators for Schrödinger operators with threshold singularities I. The odd dimensional case, J. Math. Sci. Univ. Tokyo, 13 (2006), 43-94.  Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.  Google Scholar [2] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151-218.  Google Scholar [3] R. Asad and G. Simpson, Embedded eigenvalues and the nonlinear Schrödinger equation, Journal of Mathematical Physics, 52 (2011), 033511. doi: 10.1063/1.3567152.  Google Scholar [4] M. Beceanu, A critical center-stable manifold for Schrödinger's equation in three dimensions, Comm. Pure Appl. Math., 65 (2012), 431-507. doi: 10.1002/cpa.21387.  Google Scholar [5] 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.  Google Scholar [6] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [7] D. Bollé, F. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 175-204.  Google Scholar [8] V. S. Buslaev and G. S. Perelman, Scattering for the nonlinear Schrödinger equation: States that are close to a soliton, (Russian) Algebra i Analiz, 4 (1992), 63-102; translation in St. Petersburg Math. J., 4 (1993), 1111-1142.  Google Scholar [9] F. Cardosa, C. Cuevas and G. Vodev, Dispersive estimates for the Schrödinger equation in dimensions four and five, Asymptot. Anal., 62 (2009), 125-145. doi: 10.3233/ASY-2009-0916.  Google Scholar [10] O. Costin, M. Huang and W. Schlag, On the spectral properties of $L_{\pm}$ in three dimensions, Nonlinearity, 25 (2012), 125-164. doi: 10.1088/0951-7715/25/1/125.  Google Scholar [11] S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145. doi: 10.1002/cpa.1018.  Google Scholar [12] S. Cuccagna and T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284 (2008), 51-77. doi: 10.1007/s00220-008-0605-3.  Google Scholar [13] S. Cuccagna and M. Tarulli, On asymptotic stability in energy space of ground states of NLS in 2D, Ann. I. H. Poincare, 26 (2009), 1361-1386. doi: 10.1016/j.anihpc.2008.12.001.  Google Scholar [14] L. Demanet and W. Schlag, Numerical verification of a gap condition for linearized NLS, Nonlinearity, 19 (2006), 829-852. doi: 10.1088/0951-7715/19/4/004.  Google Scholar [15] M. B. Erdoǧan and W. R. Green, Dispersive estimates for the Schrodinger equation for $C^\frac{n-3}{2}$ potentials in odd dimensions, Int. Math. Res. Notices, 13 (2010), 2532-2565. doi: 10.1093/imrn/rnp221.  Google Scholar [16] M. B. Erdoǧan and W. R. Green, Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy, To appear in Trans. Amer. Math. Soc., (2012). Google Scholar [17] M. B. Erdoǧan and W. R. Green, A weighted dispersive estimate for Schrödinger operators in dimension two, Comm. Math. Phys., 319 (2013), 791-811 doi: 10.1007/s00220-012-1640-7.  Google Scholar [18] M. B. Erdoǧan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I, Dynamics of PDE, 1 (2004), 359-379. doi: 10.1007/BF02789446.  Google Scholar [19] M. B. Erdoǧan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or eigenvalue at zero energy in dimension three: II, J. Anal. Math., 99 (2006), 199-248. doi: 10.1007/BF02789446.  Google Scholar [20] D. Finco and K. Yajima, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case, J. Math. Sci. Univ. Tokyo, 13 (2006), 277-346.  Google Scholar [21] F. Gesztesy, C. K. R. T. Jones, Y. Latushkin and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J., 49 (2000), 221-243. doi: 10.1512/iumj.2000.49.1838.  Google Scholar [22] M. Goldberg, Transport in the one-dimensional Schrödinger equation, Proc. Amer. Math. Soc., 135 (2007), 3171-3179. doi: 10.1090/S0002-9939-07-08897-1.  Google Scholar [23] 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.  Google Scholar [24] M. Goldberg and M. Visan, A counterexample to dispersive estimates, Comm. Math. Phys., 266 (2006), 211-238. doi: 10.1007/s00220-006-0013-5.  Google Scholar [25] W. R. Green, Dispersive estimates for matrix and scalar Schrödinger operators in dimension five, To appear in the Illinois J. Math., (2010).  Google Scholar [26] P. D. Hislop and I. M. Sigal, "Introduction to Spectral Theory. With Applications to Schrödinger Operators," Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2.  Google Scholar [27] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611. doi: 10.1215/S0012-7094-79-04631-3.  Google Scholar [28] A. Jensen and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Mat. Phys., 13 (2001), 717-754. doi: 10.1142/S0129055X01000843.  Google Scholar [29] A. Jensen and K. Yajima, A remark on $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys., 225 (2002), 633-637. doi: 10.1007/s002200100603.  Google Scholar [30] J.-L. Journé, 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 [31] E. Kirr and A. Zarnescu, On the asymptotic stability of bound states in 2D cubic Schrödinger equation, Comm. Math. Phys., 272 (2007), 443-468. doi: 10.1007/s00220-007-0233-3.  Google Scholar [32] J. Marzuola, Dispersive estimates using scattering theory for matrix Hamiltonian equations, Discrete Cont. Dyn. Syst. - Series A, 30 (2011), 995-1036 doi: 10.3934/dcds.2011.30.995.  Google Scholar [33] J. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators, Nonlinearity, 24 (2011), 389-429. doi: 10.1088/0951-7715/24/2/003.  Google Scholar [34] T. Mizumachi, Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 47 (2007), 599-620.  Google Scholar [35] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56. doi: 10.1016/0022-1236(82)90084-2.  Google Scholar [36] C.-A. Pillet and E. C. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations, J. Differ. Eqs., 141 (1997), 310-326. doi: 10.1006/jdeq.1997.3345.  Google Scholar [37] J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys., 61 (1978), 149-168. doi: 10.1007/BF01609491.  Google Scholar [38] M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis, IV: Analysis of Operators," Academic Press, New York, NY, 1972.  Google Scholar [39] 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 [40] 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 [41] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two, Comm. Math. Phys., 257 (2005), 87-117. doi: 10.1007/s00220-004-1262-9.  Google Scholar [42] W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar [43] W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, 255-285, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007).  Google Scholar [44] W. Schlag, Stable manifolds for an orbitally unstable NLS, Ann. of Math. (2), 169 (2009), 139-227. doi: 10.4007/annals.2009.169.139.  Google Scholar [45] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146. doi: 10.1007/BF02096557.  Google Scholar [46] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar [47] R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Commun. Math. Phys., 215 (2000), 343-356. doi: 10.1007/s002200000298.  Google Scholar [48] K. Yajima, $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys., 208 (1999), 125-152. doi: 10.1007/s002200050751.  Google Scholar [49] K. Yajima, The $L^p$ Boundedness of wave operators for Schrödinger operators with threshold singularities I. The odd dimensional case, J. Math. Sci. Univ. Tokyo, 13 (2006), 43-94.  Google Scholar
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