# 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 - A, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473
##### 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, (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. Google Scholar [3] R. Asad and G. Simpson, Embedded eigenvalues and the nonlinear Schrödinger equation,, Journal of Mathematical Physics, 52 (2011). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [35] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations,, J. Funct. Anal., 49 (1982), 10. 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. 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. 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, (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. 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. doi: 10.1002/cpa.20066. Google Scholar [41] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two,, Comm. Math. Phys., 257 (2005), 87. 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. 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, 163 (2007), 255. Google Scholar [44] W. Schlag, Stable manifolds for an orbitally unstable NLS,, Ann. of Math. (2), 169 (2009), 139. 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. doi: 10.1007/BF02096557. Google Scholar [46] W. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. 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. 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. 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. Google Scholar

show all references

##### 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, (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. Google Scholar [3] R. Asad and G. Simpson, Embedded eigenvalues and the nonlinear Schrödinger equation,, Journal of Mathematical Physics, 52 (2011). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [35] M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations,, J. Funct. Anal., 49 (1982), 10. 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. 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. 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, (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. 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. doi: 10.1002/cpa.20066. Google Scholar [41] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two,, Comm. Math. Phys., 257 (2005), 87. 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. 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, 163 (2007), 255. Google Scholar [44] W. Schlag, Stable manifolds for an orbitally unstable NLS,, Ann. of Math. (2), 169 (2009), 139. 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. doi: 10.1007/BF02096557. Google Scholar [46] W. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. 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. 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. 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. Google Scholar
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