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 and Continuous Dynamical Systems, 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, 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.

[2]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151-218.

[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.

[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.

[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.

[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.

[7]

D. Bollé, F. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 175-204.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

W. R. Green, Dispersive estimates for matrix and scalar Schrödinger operators in dimension five, To appear in the Illinois J. Math., (2010).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

T. Mizumachi, Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 47 (2007), 599-620.

[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.

[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.

[37]

J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys., 61 (1978), 149-168. doi: 10.1007/BF01609491.

[38]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis, IV: Analysis of Operators," Academic Press, New York, NY, 1972.

[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.

[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.

[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.

[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.

[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).

[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.

[45]

A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146. doi: 10.1007/BF02096557.

[46]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[47]

R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Commun. Math. Phys., 215 (2000), 343-356. doi: 10.1007/s002200000298.

[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.

[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.

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, 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.

[2]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151-218.

[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.

[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.

[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.

[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.

[7]

D. Bollé, F. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincaré Phys. Théor., 48 (1988), 175-204.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

W. R. Green, Dispersive estimates for matrix and scalar Schrödinger operators in dimension five, To appear in the Illinois J. Math., (2010).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

T. Mizumachi, Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., 47 (2007), 599-620.

[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.

[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.

[37]

J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys., 61 (1978), 149-168. doi: 10.1007/BF01609491.

[38]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis, IV: Analysis of Operators," Academic Press, New York, NY, 1972.

[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.

[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.

[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.

[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.

[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).

[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.

[45]

A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys., 133 (1990), 119-146. doi: 10.1007/BF02096557.

[46]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[47]

R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Commun. Math. Phys., 215 (2000), 343-356. doi: 10.1007/s002200000298.

[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.

[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.

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