American Institute of Mathematical Sciences

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October  2015, 35(10): 4805-4821. doi: 10.3934/dcds.2015.35.4805

Schrödinger equations with rough Hamiltonians

Elena Cordero 1, , Fabio Nicola 2, and  Luigi Rodino 1,

1. 

Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino
2. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  July 2014 Revised  January 2015 Published  April 2015

We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.
    Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.
Keywords: pseudodifferential operators, Schrödinger equation, semilinear equations., Sjöstrand class, modulation spaces.
Mathematics Subject Classification: Primary: 35S10; Secondary: 35Q41, 42B3.
Citation: Elena Cordero, Fabio Nicola, Luigi Rodino. Schrödinger equations with rough Hamiltonians. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4805-4821. doi: 10.3934/dcds.2015.35.4805
References:
[1]

A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces,, J. Funct. Anal., 246 (2007), 366. doi: 10.1016/j.jfa.2006.12.019. Google Scholar

[2]

A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces,, Bull. Lond. Math. Soc., 41 (2009), 549. doi: 10.1112/blms/bdp027. Google Scholar

[3]

A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators,, Math. Res. Lett., 4 (1997), 53. doi: 10.4310/MRL.1997.v4.n1.a6. Google Scholar

[4]

A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes., Comm. Partial Differential Equations, 22 (1997), 165. doi: 10.1080/03605309708821259. Google Scholar

[5]

F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II,, Osaka J. Math., 47 (2010), 739. Google Scholar

[6]

E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers,, J. Fourier Anal. Appl., 18 (2012), 661. doi: 10.1007/s00041-011-9214-1. Google Scholar

[7]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators,, J. Math. Pures Appl., 99 (2013), 219. doi: 10.1016/j.matpur.2012.06.012. Google Scholar

[8]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class,, J. Math. Phys., 55 (2014). Google Scholar

[9]

E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation,, J. Math. Anal. Appl., 353 (2009), 583. doi: 10.1016/j.jmaa.2008.12.027. Google Scholar

[10]

E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation,, J. Funct. Anal., 254 (2008), 506. doi: 10.1016/j.jfa.2007.09.015. Google Scholar

[11]

E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces,, J. Fourier Anal. Appl., 16 (2010), 311. doi: 10.1007/s00041-009-9111-z. Google Scholar

[12]

E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators,, Commun. Pure Appl. Anal., 9 (2010), 1. doi: 10.3934/cpaa.2010.9.1. Google Scholar

[13]

E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators,, Trans. Amer. Math. Soc., 361 (2009), 6049. doi: 10.1090/S0002-9947-09-04848-X. Google Scholar

[14]

E. Cordero, F. Nicola and L. Rodino, Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potential,, Rev. Math. Phys., 27 (2015). doi: 10.1142/S0129055X15500014. Google Scholar

[15]

P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schroedinger equation with a potential in $L^r_t L^s_x$,, Math. Ann., 333 (2005), 271. doi: 10.1007/s00208-005-0672-0. Google Scholar

[16]

I. Daubechies, Time-frequency localization operators: A geometric phase space approach,, IEEE Trans. Inf. Theory, 34 (1988), 605. doi: 10.1109/18.9761. Google Scholar

[17]

H. G. Feichtinger, Modulation spaces on locally compact abelian groups,, in Wavelets and their Applications (eds. M. Krishna, (2003), 99. Google Scholar

[18]

K. Gröchenig, Foundations of Time-Frequency Analysis,, Applied and Numerical Harmonic Analysis, (2001). doi: 10.1007/978-1-4612-0003-1. Google Scholar

[19]

K. Gröchenig, Time-frequency analysis of Sjöstrand's class,, Rev. Mat. Iberoam., 22 (2006), 703. doi: 10.4171/RMI/471. Google Scholar

[20]

K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization,, Ann. Inst. Fourier, 58 (2008), 2279. Google Scholar

[21]

F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra,, Ark. för Mat., 39 (2001), 311. doi: 10.1007/BF02384559. Google Scholar

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III,, Springer-Verlag, (1985). Google Scholar

[23]

K. Kato, M. Kobayashi and S. Ito, Representation of Schrödinger operator of a free particle via short time Fourier transform and its applications,, Tohoku Math. J., 64 (2012), 223. doi: 10.2748/tmj/1341249372. Google Scholar

[24]

K. Kato, M. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator,, SUT J. Math., 47 (2011), 175. Google Scholar

[25]

K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation,, RIMS Kôkyûroku Bessatsu, (2012), 41. Google Scholar

[26]

K. Kato, M. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials,, J. Funct. Anal., 266 (2014), 733. doi: 10.1016/j.jfa.2013.08.017. Google Scholar

[27]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators,, Comm. Pure Appl. Math., 58 (2005), 217. doi: 10.1002/cpa.20067. Google Scholar

[28]

N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators,, Pseudo-Differential Operators. Theory and Applications, (2010). doi: 10.1007/978-3-7643-8510-1. Google Scholar

[29]

N. Lerner and Y. Morimoto, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators,, Publications of the Research Institute for Mathematical Sciences (Kyoto University), 43 (2007), 329. Google Scholar

[30]

A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces,, Proc. Amer. Math. Soc., 137 (2009), 3869. doi: 10.1090/S0002-9939-09-09968-7. Google Scholar

[31]

F. Nicola, Phase space analysis of semilinear parabolic equations,, J. Funct. Anal., 267 (2014), 727. doi: 10.1016/j.jfa.2014.05.007. Google Scholar

[32]

R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases,, in Gabor Analysis and Algorithms, (1998), 171. Google Scholar

[33]

M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations,, in Evolution Equations of Hyperbolic and Schrödinger Type, (2012), 267. doi: 10.1007/978-3-0348-0454-7_14. Google Scholar

[34]

H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients,, Ann. Inst. Fourier (Grenoble), 48 (1998), 797. Google Scholar

[35]

G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients,, Comm. Partial Differential Equations, 27 (2002), 1337. doi: 10.1081/PDE-120005841. Google Scholar

[36]

J. Sjöstrand, An algebra of pseudodifferential operators,, Math. Res. Lett., 1 (1994), 185. doi: 10.4310/MRL.1994.v1.n2.a6. Google Scholar

[37]

J. Sjöstrand, Wiener type algebras of pseudodifferential operators,, in Séminaire sur les Équations aux Dérivées Partielles, (1995), 1994. Google Scholar

[38]

D. Tataru, Phase space transforms and microlocal analysis,, in Phase Space Analysis of Partial Differential Equations, (2004), 505. Google Scholar

[39]

D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III,, J. Amer. Math. Soc., 15 (2002), 419. doi: 10.1090/S0894-0347-01-00375-7. Google Scholar

[40]

B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data,, J. Funct. Anal., 265 (2013), 3009. doi: 10.1016/j.jfa.2013.08.009. Google Scholar

[41]

B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations,, J. Differential Equations, 239 (2007), 213. doi: 10.1016/j.jde.2007.04.009. Google Scholar

[42]

B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data,, J. Differential Equations, 232 (2007), 36. doi: 10.1016/j.jde.2006.09.004. Google Scholar

[43]

B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I,, World Scientific Publishing Co. Pte. Ltd., (2011). doi: 10.1142/9789814360746. Google Scholar

[44]

B. Wang, Z. Lifeng and G. Boling, Isometric decomposition operators, function spaces $E^\lambda_p,q$ and applications to nonlinear evolution equations,, J. Funct. Anal., 233 (2006), 1. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

show all references

References:
[1]

A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces,, J. Funct. Anal., 246 (2007), 366. doi: 10.1016/j.jfa.2006.12.019. Google Scholar

[2]

A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces,, Bull. Lond. Math. Soc., 41 (2009), 549. doi: 10.1112/blms/bdp027. Google Scholar

[3]

A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators,, Math. Res. Lett., 4 (1997), 53. doi: 10.4310/MRL.1997.v4.n1.a6. Google Scholar

[4]

A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes., Comm. Partial Differential Equations, 22 (1997), 165. doi: 10.1080/03605309708821259. Google Scholar

[5]

F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II,, Osaka J. Math., 47 (2010), 739. Google Scholar

[6]

E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers,, J. Fourier Anal. Appl., 18 (2012), 661. doi: 10.1007/s00041-011-9214-1. Google Scholar

[7]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators,, J. Math. Pures Appl., 99 (2013), 219. doi: 10.1016/j.matpur.2012.06.012. Google Scholar

[8]

E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class,, J. Math. Phys., 55 (2014). Google Scholar

[9]

E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation,, J. Math. Anal. Appl., 353 (2009), 583. doi: 10.1016/j.jmaa.2008.12.027. Google Scholar

[10]

E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation,, J. Funct. Anal., 254 (2008), 506. doi: 10.1016/j.jfa.2007.09.015. Google Scholar

[11]

E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces,, J. Fourier Anal. Appl., 16 (2010), 311. doi: 10.1007/s00041-009-9111-z. Google Scholar

[12]

E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators,, Commun. Pure Appl. Anal., 9 (2010), 1. doi: 10.3934/cpaa.2010.9.1. Google Scholar

[13]

E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators,, Trans. Amer. Math. Soc., 361 (2009), 6049. doi: 10.1090/S0002-9947-09-04848-X. Google Scholar

[14]

E. Cordero, F. Nicola and L. Rodino, Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potential,, Rev. Math. Phys., 27 (2015). doi: 10.1142/S0129055X15500014. Google Scholar

[15]

P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schroedinger equation with a potential in $L^r_t L^s_x$,, Math. Ann., 333 (2005), 271. doi: 10.1007/s00208-005-0672-0. Google Scholar

[16]

I. Daubechies, Time-frequency localization operators: A geometric phase space approach,, IEEE Trans. Inf. Theory, 34 (1988), 605. doi: 10.1109/18.9761. Google Scholar

[17]

H. G. Feichtinger, Modulation spaces on locally compact abelian groups,, in Wavelets and their Applications (eds. M. Krishna, (2003), 99. Google Scholar

[18]

K. Gröchenig, Foundations of Time-Frequency Analysis,, Applied and Numerical Harmonic Analysis, (2001). doi: 10.1007/978-1-4612-0003-1. Google Scholar

[19]

K. Gröchenig, Time-frequency analysis of Sjöstrand's class,, Rev. Mat. Iberoam., 22 (2006), 703. doi: 10.4171/RMI/471. Google Scholar

[20]

K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization,, Ann. Inst. Fourier, 58 (2008), 2279. Google Scholar

[21]

F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra,, Ark. för Mat., 39 (2001), 311. doi: 10.1007/BF02384559. Google Scholar

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III,, Springer-Verlag, (1985). Google Scholar

[23]

K. Kato, M. Kobayashi and S. Ito, Representation of Schrödinger operator of a free particle via short time Fourier transform and its applications,, Tohoku Math. J., 64 (2012), 223. doi: 10.2748/tmj/1341249372. Google Scholar

[24]

K. Kato, M. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator,, SUT J. Math., 47 (2011), 175. Google Scholar

[25]

K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation,, RIMS Kôkyûroku Bessatsu, (2012), 41. Google Scholar

[26]

K. Kato, M. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials,, J. Funct. Anal., 266 (2014), 733. doi: 10.1016/j.jfa.2013.08.017. Google Scholar

[27]

H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators,, Comm. Pure Appl. Math., 58 (2005), 217. doi: 10.1002/cpa.20067. Google Scholar

[28]

N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators,, Pseudo-Differential Operators. Theory and Applications, (2010). doi: 10.1007/978-3-7643-8510-1. Google Scholar

[29]

N. Lerner and Y. Morimoto, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators,, Publications of the Research Institute for Mathematical Sciences (Kyoto University), 43 (2007), 329. Google Scholar

[30]

A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces,, Proc. Amer. Math. Soc., 137 (2009), 3869. doi: 10.1090/S0002-9939-09-09968-7. Google Scholar

[31]

F. Nicola, Phase space analysis of semilinear parabolic equations,, J. Funct. Anal., 267 (2014), 727. doi: 10.1016/j.jfa.2014.05.007. Google Scholar

[32]

R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases,, in Gabor Analysis and Algorithms, (1998), 171. Google Scholar

[33]

M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations,, in Evolution Equations of Hyperbolic and Schrödinger Type, (2012), 267. doi: 10.1007/978-3-0348-0454-7_14. Google Scholar

[34]

H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients,, Ann. Inst. Fourier (Grenoble), 48 (1998), 797. Google Scholar

[35]

G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients,, Comm. Partial Differential Equations, 27 (2002), 1337. doi: 10.1081/PDE-120005841. Google Scholar

[36]

J. Sjöstrand, An algebra of pseudodifferential operators,, Math. Res. Lett., 1 (1994), 185. doi: 10.4310/MRL.1994.v1.n2.a6. Google Scholar

[37]

J. Sjöstrand, Wiener type algebras of pseudodifferential operators,, in Séminaire sur les Équations aux Dérivées Partielles, (1995), 1994. Google Scholar

[38]

D. Tataru, Phase space transforms and microlocal analysis,, in Phase Space Analysis of Partial Differential Equations, (2004), 505. Google Scholar

[39]

D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III,, J. Amer. Math. Soc., 15 (2002), 419. doi: 10.1090/S0894-0347-01-00375-7. Google Scholar

[40]

B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data,, J. Funct. Anal., 265 (2013), 3009. doi: 10.1016/j.jfa.2013.08.009. Google Scholar

[41]

B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations,, J. Differential Equations, 239 (2007), 213. doi: 10.1016/j.jde.2007.04.009. Google Scholar

[42]

B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data,, J. Differential Equations, 232 (2007), 36. doi: 10.1016/j.jde.2006.09.004. Google Scholar

[43]

B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I,, World Scientific Publishing Co. Pte. Ltd., (2011). doi: 10.1142/9789814360746. Google Scholar

[44]

B. Wang, Z. Lifeng and G. Boling, Isometric decomposition operators, function spaces $E^\lambda_p,q$ and applications to nonlinear evolution equations,, J. Funct. Anal., 233 (2006), 1. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

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