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Dispersive estimates for matrix and scalar Schrödinger operators in dimension five. (English) Zbl 1373.35266

Summary: We investigate the boundedness of the evolution operators \(e^{itH}\) and \(e^{it\mathcal{H}}\) in the sense of \(L^{1}\to L^{\infty}\) for both the scalar Schrödinger operator \(H=-\Delta+V\) and the non-selfadjoint matrix Schrödinger operator \[ \mathcal H= \bigg[\begin{matrix} -\Delta+\mu-V_1 \quad -V_2\quad \\ \quad V_2 \qquad \Delta-\mu+V_1\end{matrix} \bigg] \] in dimension five. Here \(\mu>0\) and \(V_{1}, V_{2}\) are real-valued decaying potentials. The matrix operator arises when linearizing about a standing wave in certain nonlinear partial differential equations. We apply some natural spectral assumptions on \(\mathcal{H}\), including regularity of the edges of the spectrum \(\pm\mu\).

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables , National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 1964. · Zbl 0171.38503
[2] S. Agmon, Spectral properties of Schrödinger operators and scattering theory , Ann. Scuola Norm. Sup Pisa Cl. Sci. (4) 2 (1975), no. 2, 151-218. · Zbl 0315.47007
[3] M. Beceanu, A critical centre-stable manifold for the cubic focusing Schroedinger equation in three dimensions , Ph.D. thesis, Univ. Chicago, 2009, available at \arxivurl arXiv:
[4] M. Beceanu, New estimates for time-independent Schrödinger equation , Duke Math. J. 159 (2011), no. 3, 417-477. · Zbl 1229.35224 · doi:10.1215/00127094-1433394
[5] M. Beceanu and M. Golberg, Schrödinger dispersive estimates for a scaling-critical class of potentials , Comm. Math. Phys. 314 (2012), 471-481. · Zbl 1250.35047 · doi:10.1007/s00220-012-1435-x
[6] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires , C. R. Acad. Sci. Paris. 293 (1981), 489-492. · Zbl 0492.35010
[7] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state , Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313-345. · Zbl 0533.35029 · doi:10.1007/BF00250555
[8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions , Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347-375. · Zbl 0556.35046 · doi:10.1007/BF00250556
[9] 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), no. 6, 63-102; translation in St. Petersburg Math. J. 4 (1993), no. 6, 1111-1142. · Zbl 0853.35112
[10] V. S. Buslaev and G. S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations , Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 75-98. · Zbl 0841.35108
[11] F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates for the Schrödinger equation in dimensions four and five , Asymptot. Anal. 62 (2009), no. 3-4, 125-145. · Zbl 1163.35482 · doi:10.3233/ASY-2008-0916
[12] F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates for the Schrödinger equation with potentials of critical regularity , Cubo 11 (2009), no. 5, 57-70. · Zbl 1184.35084
[13] F. Cardoso, C. Cuevas and G. Vodev, High frequency dispersive estimates for the Schrödinger equation in high dimensions , Asymptot. Anal. 71 (2011), no. 4, 207-225. · Zbl 1225.35214
[14] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations , Comm. Math. Phys. 85 (1982), 549-561. · Zbl 0513.35007 · doi:10.1007/BF01403504
[15] C. V. Coffman, Uniqueness of positive solutions of \(\Delta u - u + u^3 = 0\) and a variational characterization of other solutions , Arch. Ration. Mech. Anal. 46 (1972), 81-95. · Zbl 0249.35029 · doi:10.1007/BF00250684
[16] A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy , Comm. Pure Appl. Math. 56 (2003), no. 11, 1565-1607. · Zbl 1072.35165 · doi:10.1002/cpa.10104
[17] S. Cuccagna, Stablization of solutions to nonlinear Schrö dinger equations, Comm. Pure Appl. Math. 54 (2001), no. 9, 1110-1145. · Zbl 1031.35129 · doi:10.1002/cpa.1018
[18] L. Demanet and W. Schlag, Numerical verification of a gap condition for linearized NLS , Nonlinearity 19 (2006), 829-852. · Zbl 1106.35044 · doi:10.1088/0951-7715/19/4/004
[19] M. B. Erdoğan, M. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions , Forum Math. 21 (2009), no. 4, 687-722. · Zbl 1181.35208 · doi:10.1515/FORUM.2009.035
[20] M. B. Erdoğan and W. R. Green, Dispersive estimates for the Schrödinger equation with \(C^{(n-3)/2}\) potentials in odd dimensions , Internat. Math. Res. Notices 13 (2010), 2532-2565. · Zbl 1200.35036
[21] M. B. Erdoğan and W. R. Green, Dispersive estimates for matrix Schrödinger operators in dimension two . Discrete Contin. Dyn. Syst. 33 (2013), no. 10, 4473-4495. · Zbl 1277.35294 · doi:10.3934/dcds.2013.33.4473
[22] M. B. Erdoğan 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 , Dyn. Partial Differ. Equ. 1 (2004), 359-379. · Zbl 1080.35102 · doi:10.4310/DPDE.2004.v1.n4.a1
[23] M. B. Erdoğan 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. · Zbl 1146.35324 · doi:10.1007/BF02789446
[24] 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), no. 3, 277-346. · Zbl 1142.35060
[25] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I , J. Funct. Anal. 74 (1987), no. 1, 160-197. · Zbl 0656.35122 · doi:10.1016/0022-1236(87)90044-9
[26] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II , J. Funct. Anal. 94 (1990), 308-348. · Zbl 0711.58013 · doi:10.1016/0022-1236(90)90016-E
[27] M. Goldberg, A Dispersive bound for three-dimensional Schrödinger operators with zero energy eigenvalues , Comm. Partial Differential Equations 35 (2010), 1610-1634. · Zbl 1223.35265 · doi:10.1080/03605302.2010.493967
[28] M. Goldberg, Dispersive bound for the three-dimensional Schrödinger equation with almost critical potentials , Geom. Funct. Anal. 16 (2006), no. 3, 517-536. · Zbl 1158.35408 · doi:10.1007/s00039-006-0568-5
[29] M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials , Amer. J. Math. 128 (2006) 731-750. · Zbl 1096.35027 · doi:10.1353/ajm.2006.0025
[30] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three , Comm. Math. Phys. 251 (2004), no. 1, 157-178. · Zbl 1086.81077 · doi:10.1007/s00220-004-1140-5
[31] M. Goldberg and M. Visan, A counterexample to dispersive estimates , Comm. Math. Phys. 266 (2006), no. 1, 211-238. · Zbl 1110.35073 · doi:10.1007/s00220-006-0013-5
[32] P. D. Hislop and I. M. Sigal, Introduction to spectral theory. With applications to Schrödinger operators , Applied Mathematical Sciences, vol. 113, Springer, New York, 1996. · Zbl 0855.47002 · doi:10.1007/978-1-4612-0741-2
[33] A. Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions results in \(L^2(\R^m)\), \(m\geq5\) , Duke Math. J. 47 (1980), no. 1, 57-80. · Zbl 0437.47009 · doi:10.1215/S0012-7094-80-04706-7
[34] A. Jensen and T. Kato, Spectral properties of Schrödinger and time-decay of wave functions , Duke Math. J. 46 (1979), no. 3, 583-611. · Zbl 0448.35080 · doi:10.1215/S0012-7094-79-04631-3
[35] J.-L. Journé, A. Soffer and C.D. Sogge, Decay estimates for Schrödinger operators , Comm. Pure Appl. Math. 44 (1991), 573-611. · Zbl 0743.35008 · doi:10.1002/cpa.3160440504
[36] M. K. Kwong, Uniqueness of positive solutions of \(\Delta u + u^p = 0\) in \(\R^n\) , Arch. Ration. Mech. Anal. 65 (1989), 243-266. · Zbl 0676.35032 · doi:10.1007/BF00251502
[37] J. Marzuola, Dispersive estimates using scattering theory for matrix Hamiltonian equations , Discrete Contin. Dyn. Syst. 30 (2011), no. 4, 995-1036. · Zbl 1254.35021 · doi:10.3934/dcds.2011.30.995
[38] J. Marzuola and G. Simpson, Spectral analysis for matrix Hamiltonian operators , Nonlinearity 24 (2011), 389-429. · Zbl 1213.35371 · doi:10.1088/0951-7715/24/2/003
[39] K. McLeod and J. Serrin, Nonlinear Schrödinger equation. Uniqueness of positive solutions of \(\Delta u + f (u) = 0\) in \(\R^n\) , Arch. Ration. Mech. Anal. 99 (1987), 115-145. · Zbl 0667.35023 · doi:10.1007/BF00275874
[40] S. Moulin and G. Vodev, Low-frequency dispersive estimates for the Schrödinger group in higher dimensions , Asymptot. Anal. 55 (2007), no. 1-2, 49-71. · Zbl 1219.35053
[41] M. Murata, Asymptotic expansions in time for solutions of Schrö dinger-type equations, J. Funct. Anal. 49 (1982), no. 1, 10-56. · Zbl 0499.35019 · doi:10.1016/0022-1236(82)90084-2
[42] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation , Ann. Henri Poincaré 2 (2001), no. 4, 605-673. · Zbl 1007.35087 · doi:10.1007/PL00001048
[43] G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations , Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051-1095. · Zbl 1067.35113 · doi:10.1081/PDE-200033754
[44] J. Rauch, Local decay of scattering solutions to Schrö dinger’s equation, Comm. Math. Phys. 61 (1978), no. 2, 149-168. · Zbl 0381.35023 · doi:10.1007/BF01609491
[45] M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, IV: Analysis of operators , Academic Press, New York, NY, 1972. · Zbl 0242.46001
[46] I. Rodnianski and W. Schlag, Time decay for solutions of the Schrödinger equations with rough and time-dependent potentials , Invent. Math. 155 (2004), no. 3, 451-513. · Zbl 1063.35035 · doi:10.1007/s00222-003-0325-4
[47] I. Rodnianski, W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS , preprint, 2004.
[48] I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models , Comm. Pure Appl. Math. 58 (2005), no. 2, 149-216. · Zbl 1130.81053 · doi:10.1002/cpa.20066
[49] W. Schlag, Dispersive estimates for Schrödinger operators: A survey , Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255-285. · Zbl 1143.35001
[50] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two , Comm. Math. Phys. 257 (2005), no. 1, 87-117. · Zbl 1134.35321 · doi:10.1007/s00220-004-1262-9
[51] W. Schlag, Spectral theory and nonlinear partial differential equations: A survey , Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 703-723. · Zbl 1121.35121 · doi:10.3934/dcds.2006.15.703
[52] W. Schlag, Stable manifolds for an orbitally unstable nonlinear Schrödinger equation , Ann. of Math. (2) 169 (2009), no. 1, 139-227. · Zbl 1180.35490 · doi:10.4007/annals.2009.169.139
[53] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations , Comm. Math. Phys. 91 (1983), no. 3, 313-327. · Zbl 0539.35067 · doi:10.1007/BF01208779
[54] J. Shatah and W. Strauss, Instability of nonlinear bound states , Comm. Math. Phys. 100 (1985), no. 2, 173-190. · Zbl 0603.35007 · doi:10.1007/BF01212446
[55] A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations , Comm. Math. Phys. 133 (1990), 119-146. · Zbl 0721.35082 · doi:10.1007/BF02096557
[56] A. Soffer and M. Weinstein, Multichannel nonlinear scattering. II. The case of anisotropic potentials and data , J. Differential Equations 98 (1992), 376-390. · Zbl 0795.35073 · doi:10.1016/0022-0396(92)90098-8
[57] W. Strauss, Existence of solitary waves in higher dimensions , Comm. Math. Phys. 55 (1977), 149-162. · Zbl 0356.35028 · doi:10.1007/BF01626517
[58] W. A. Strauss, Nonlinear wave equations , CBMS Regional Conference Series in Mathematics, vol. 73, Amer. Math. Soc., Providence, RI, 1989. · Zbl 0714.35003
[59] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse , Applied Mathematical Sciences, vol. 139, Springer, New York, 1999. · Zbl 0928.35157
[60] G. Vodev, Dispersive estimates of solutions to the Schrö dinger equation, Ann. Henri Poincaré 6 (2005), no. 6, 1179-1196. · Zbl 1084.81031 · doi:10.1007/s00023-005-0237-5
[61] G. Vodev, Dispersive estimates of solutions to the Schrö dinger equation in dimensions \(n\geq4\), Asymptot. Anal. 49 (2006), no. 1-2, 61-86. · Zbl 1111.35006
[62] R. Weder, \(L^p - L^{p'}\) estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrö dinger equation with a potential, J. Funct. Anal. 170 (2000), no. 1, 37-68. · Zbl 0943.34070 · doi:10.1006/jfan.1999.3507
[63] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations , SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. · Zbl 0583.35028 · doi:10.1137/0516034
[64] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations , Comm. Pure Appl. Math. 39 (1986), no. 1, 51-67. · Zbl 0594.35005 · doi:10.1002/cpa.3160390103
[65] 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. · Zbl 1115.35094
[66] K. Yajima, \(L^p\)-boundedness of wave operators for two-dimensional Schrödinger operators , Comm. Math. Phys. 208 (1999), no. 1, 125-152. · Zbl 0961.47004 · doi:10.1007/s002200050751
[67] K. Yajima, Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue , Comm. Math. Phys. 259 (2005), no. 2, 475-509. · Zbl 1079.81021 · doi:10.1007/s00220-005-1375-9
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