Schrödinger equations with rough Hamiltonians

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 366-384. doi: 10.1016/j.jfa.2006.12.019.
[2]	A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027.
[3]	A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., 4 (1997), 53-67. doi: 10.4310/MRL.1997.v4.n1.a6.
[4]	A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes. Comm. Partial Differential Equations, 22 (1997), 165-184. doi: 10.1080/03605309708821259.
[5]	F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II, Osaka J. Math., 47 (2010), 739-786.
[6]	E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers, J. Fourier Anal. Appl., 18 (2012), 661-684. doi: 10.1007/s00041-011-9214-1.
[7]	E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators, J. Math. Pures Appl., 99 (2013), 219-233. doi: 10.1016/j.matpur.2012.06.012.
[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), 081506 (17 pages).
[9]	E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591. doi: 10.1016/j.jmaa.2008.12.027.
[10]	E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534. doi: 10.1016/j.jfa.2007.09.015.
[11]	E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces, J. Fourier Anal. Appl., 16 (2010), 311-339. doi: 10.1007/s00041-009-9111-z.
[12]	E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal., 9 (2010), 1-21. doi: 10.3934/cpaa.2010.9.1.
[13]	E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators, Trans. Amer. Math. Soc., 361 (2009), 6049-6071. doi: 10.1090/S0002-9947-09-04848-X.
[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), 1550001 (33 pages). doi: 10.1142/S0129055X15500014.
[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), 271-290. doi: 10.1007/s00208-005-0672-0.
[16]	I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory, 34 (1988), 605-612. doi: 10.1109/18.9761.
[17]	H. G. Feichtinger, Modulation spaces on locally compact abelian groups, in Wavelets and their Applications (eds. M. Krishna, R. Radhaand and S. Thangavelu), Chennai, India, Allied Publishers, New Delhi, 2003, 99-140; Updated version of a technical report, University of Vienna, 1983.
[18]	K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0003-1.
[19]	K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam., 22 (2006), 703-724. doi: 10.4171/RMI/471.
[20]	K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier, 58 (2008), 2279-2314.
[21]	F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. för Mat., 39 (2001), 311-338. doi: 10.1007/BF02384559.
[22]	L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer-Verlag, 1985.
[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), 223-231. doi: 10.2748/tmj/1341249372.
[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), 175-183.
[25]	K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, (2012), 41-48.
[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), 733-753. doi: 10.1016/j.jfa.2013.08.017.
[27]	H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.
[28]	N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8510-1.
[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), 329-371.
[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), 3869-3883. doi: 10.1090/S0002-9939-09-09968-7.
[31]	F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743. doi: 10.1016/j.jfa.2014.05.007.
[32]	R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, in Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, 171-192.
[33]	M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progress in Mathematics, 301, Birkhäuser, 2012, 267-283. doi: 10.1007/978-3-0348-0454-7_14.
[34]	H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), 797-835.
[35]	G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.
[36]	J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192. doi: 10.4310/MRL.1994.v1.n2.a6.
[37]	J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, École Polytech., Palaiseau, 1995, 21pp.
[38]	D. Tataru, Phase space transforms and microlocal analysis, in Phase Space Analysis of Partial Differential Equations, Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, 505-524.
[39]	D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., 15 (2002), 419-442. doi: 10.1090/S0894-0347-01-00375-7.
[40]	B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052. doi: 10.1016/j.jfa.2013.08.009.
[41]	B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250. doi: 10.1016/j.jde.2007.04.009.
[42]	B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[43]	B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.
[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), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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), 366-384. doi: 10.1016/j.jfa.2006.12.019.
[2]	A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027.
[3]	A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., 4 (1997), 53-67. doi: 10.4310/MRL.1997.v4.n1.a6.
[4]	A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes. Comm. Partial Differential Equations, 22 (1997), 165-184. doi: 10.1080/03605309708821259.
[5]	F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II, Osaka J. Math., 47 (2010), 739-786.
[6]	E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers, J. Fourier Anal. Appl., 18 (2012), 661-684. doi: 10.1007/s00041-011-9214-1.
[7]	E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators, J. Math. Pures Appl., 99 (2013), 219-233. doi: 10.1016/j.matpur.2012.06.012.
[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), 081506 (17 pages).
[9]	E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591. doi: 10.1016/j.jmaa.2008.12.027.
[10]	E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534. doi: 10.1016/j.jfa.2007.09.015.
[11]	E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces, J. Fourier Anal. Appl., 16 (2010), 311-339. doi: 10.1007/s00041-009-9111-z.
[12]	E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal., 9 (2010), 1-21. doi: 10.3934/cpaa.2010.9.1.
[13]	E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators, Trans. Amer. Math. Soc., 361 (2009), 6049-6071. doi: 10.1090/S0002-9947-09-04848-X.
[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), 1550001 (33 pages). doi: 10.1142/S0129055X15500014.
[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), 271-290. doi: 10.1007/s00208-005-0672-0.
[16]	I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory, 34 (1988), 605-612. doi: 10.1109/18.9761.
[17]	H. G. Feichtinger, Modulation spaces on locally compact abelian groups, in Wavelets and their Applications (eds. M. Krishna, R. Radhaand and S. Thangavelu), Chennai, India, Allied Publishers, New Delhi, 2003, 99-140; Updated version of a technical report, University of Vienna, 1983.
[18]	K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0003-1.
[19]	K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam., 22 (2006), 703-724. doi: 10.4171/RMI/471.
[20]	K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier, 58 (2008), 2279-2314.
[21]	F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. för Mat., 39 (2001), 311-338. doi: 10.1007/BF02384559.
[22]	L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer-Verlag, 1985.
[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), 223-231. doi: 10.2748/tmj/1341249372.
[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), 175-183.
[25]	K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, (2012), 41-48.
[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), 733-753. doi: 10.1016/j.jfa.2013.08.017.
[27]	H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.
[28]	N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010. doi: 10.1007/978-3-7643-8510-1.
[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), 329-371.
[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), 3869-3883. doi: 10.1090/S0002-9939-09-09968-7.
[31]	F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743. doi: 10.1016/j.jfa.2014.05.007.
[32]	R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, in Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, 171-192.
[33]	M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progress in Mathematics, 301, Birkhäuser, 2012, 267-283. doi: 10.1007/978-3-0348-0454-7_14.
[34]	H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), 797-835.
[35]	G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.
[36]	J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192. doi: 10.4310/MRL.1994.v1.n2.a6.
[37]	J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, École Polytech., Palaiseau, 1995, 21pp.
[38]	D. Tataru, Phase space transforms and microlocal analysis, in Phase Space Analysis of Partial Differential Equations, Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, 505-524.
[39]	D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., 15 (2002), 419-442. doi: 10.1090/S0894-0347-01-00375-7.
[40]	B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052. doi: 10.1016/j.jfa.2013.08.009.
[41]	B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250. doi: 10.1016/j.jde.2007.04.009.
[42]	B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[43]	B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.
[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), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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