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Time-Frequency Analysis of Schrödinger Propagators

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Evolution Equations of Hyperbolic and Schrödinger Type

Part of the book series: Progress in Mathematics ((PM,volume 301))

Abstract

We present a survey on recent results concerning applications of Time-Frequency Analysis to the study of Fourier Integral Operators (FIOs). In particular, we focus on Schrödinger-type FIOs, showing that Gabor frames provide optimally sparse representations of such operators. Using Maple software, new numerical examples for the Harmonic Oscillator are provided.

Mathematics Subject Classification. 35S30,47G30,42C15.

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References

  1. Asada K, Fujiwara D (1978) On some oscillatory transformation in 𝐿2(ℝ𝑛). Japan J Math 4:299–361

    MathSciNet  Google Scholar 

  2. A. B´enyi, K. Gr¨ochenig, C. Heil, and K. Okoudjou. Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Operator Theory, 54:389–401, 2005.

    Google Scholar 

  3. A. B´enyi, K. Gr¨ochenig, K.A. Okoudjou and L.G. Rogers. Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal., 246(2):366–384, 2007.

    Google Scholar 

  4. Berra M (2011) Time-frequency analysis of Fourier Integral Operators and Applications. Master Thesis, July

    Google Scholar 

  5. A. B´enyi, L. Grafakos, K. Gr¨ochenig and K. Okoudjou. A class of Fourier multipliers for modulation spaces, Applied and Computational Harmonic Analysis, 19:131–139, 2005.

    Google Scholar 

  6. F.A. Berezin and M.A. Shubin. The Schr¨odinger equation. Mathematics and its Applications (Soviet Series), 66, Kluwer Academic Publishers Group, 1991.

    Google Scholar 

  7. P. Boggiatto, E. Cordero, and K. Gr¨ochenig. Generalized Anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations and Operator Theory, 48:427–442, 2004.

    Google Scholar 

  8. Boulkhemair A (1997) Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math Res Lett 4:53–67

    MathSciNet  MATH  Google Scholar 

  9. E.J. Cand´es and L. Demanet. Curvelets and Fourier Integral Operators, C. R. Math. Acad. Sci. Paris, 336(5):395–398, 2003.

    Google Scholar 

  10. E.J. Cand´es and L. Demanet. The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58:1472–1528, 2005.

    Google Scholar 

  11. E.J. Cand´es, L. Demanet and L. Ying. Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29(6):2464–2493, 2007.

    Google Scholar 

  12. C. Carath´eodory. Variationsrechnung und partielle Differentialglichungen erster Ordnung. Teubner, Berlin, (1935) Leipzig 1956. Holden-Day, San Francisco, English transl., p 1965

    Google Scholar 

  13. Concetti F, Toft J (2009) Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols I. Ark Mat 47(2):295–312

    Article  MathSciNet  MATH  Google Scholar 

  14. Concetti F, Garello G, Toft J (2010) Schatten–von Neumann properties for Fourier integral operators with non-smooth symbols II. Osaka J Math 47(3):739–786

    MathSciNet  MATH  Google Scholar 

  15. F. Concetti, G. Garello, J. Toft. Trace ideals for Fourier integral operators with non-smooth symbols III. Preprint, available at arXiv:0802.2352.

    Google Scholar 

  16. Cordero E, Nicola F (2008) Metaplectic representation on Wiener amalgam spaces and applications to the Schr¨odinger equation. J Funct Anal 254:506–534

    Article  MathSciNet  MATH  Google Scholar 

  17. Cordero E, Nicola F (2010) Boundedness of Schr¨odinger Type Propagators on Modulation Spaces. J Fourier Anal Appl 16:311–339

    Article  MathSciNet  MATH  Google Scholar 

  18. Cordero E, Nicola F, Rodino L (2010) Time-frequency Analysis of Fourier Integral Operators. Comm Pure Appl Anal 9(1):1–21

    Article  MathSciNet  MATH  Google Scholar 

  19. Cordero E, Nicola F, Rodino L (2009) Sparsity of Gabor representation of Schr¨odinger propagators. Appl Comput Harmon Anal 26(3):357–370

    Article  MathSciNet  MATH  Google Scholar 

  20. Cordero E, Nicola F, Rodino L (2009) Boundedness of Fourier Integral Operators on ℱ𝐿𝑝 spaces. Trans Amer Math Soc 361(11):6049–6071

    Article  MathSciNet  MATH  Google Scholar 

  21. A. C´ordoba and C. Fefferman. Wave packets and Fourier integral operators. Comm. Partial Differential Equations, 3(11):979–1005, 1978.

    Google Scholar 

  22. Duistemaat JJ, Guillemin VW (1975) The spectrum of positive elliptic operators and periodic bicharacteristics. Invent Math 29:39–79

    Article  MathSciNet  Google Scholar 

  23. J.J. Duistemaat and L. H¨ormander. Fourier integral operators II. Acta Math., 128:183–269, 1972.

    Google Scholar 

  24. H.G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983, and also in Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors, Allied Publishers, 99–140, 2003.

    Google Scholar 

  25. Feichtinger HG (1981) On a new Segal algebra. Monatsh Math 92(4):269–289

    Article  MathSciNet  MATH  Google Scholar 

  26. H.G. Feichtinger and K. Gr¨ochenig. Gabor frames and time-frequency analysis of distributions. J. Funct. Anal., 146(2):464–495, 1997.

    Google Scholar 

  27. H.G. Feichtinger and K. Gr¨ochenig. Banach spaces related to integrable group representations and their atomic decompositions I. J. Funct. Anal., 86(2) 307–340, 1989.

    Google Scholar 

  28. H.G. Feichtinger and K.H. Gr¨ochenig. Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. f. Math., 108:129–148, 1989.

    Google Scholar 

  29. Folland GB (1989) Harmonic Analysis in Phase Space. Princeton Univ, Press, Princeton, NJ

    Google Scholar 

  30. Frazier M, Jawerth B (1990) A discrete transform and decomposition of distribution spaces. J Funct Anal 93:34–170

    Article  MathSciNet  MATH  Google Scholar 

  31. Galperin YV, Samarah S (2004) Time-frequency analysis on modulation spaces 𝑀𝑝, 𝑞 𝑚, 0 < 𝑝, 𝑞 ≤∞. Appl Comp Harm Anal 16:1–18

    Article  MathSciNet  MATH  Google Scholar 

  32. K. Gr¨ochenig. Foundations of Time-Frequency Analysis. Birkh¨auser, Boston, 2001.

    Google Scholar 

  33. K. Gr¨ochenig. An uncertainty principle related to the Poisson summation formula. Studia Math., 121(1):87–104, 1996.

    Google Scholar 

  34. K. Gr¨ochenig. Time-Frequency Analysis of Sj¨ostrand’s Class Rev. Mat. Iberoamericana, 22(2):703–724, 2006.

    Google Scholar 

  35. K. Gr¨ochenig and C. Heil. Modulation spaces and pseudodifferential operators. Integral Equations Operator Theory, 34(4):439–457, 1999.

    Google Scholar 

  36. K. Gr¨ochenig and M. Leinert. Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc., 17:1–18, 2004.

    Google Scholar 

  37. K. Gr¨ochenig, Z. Rzeszotnik, and T. Strohmer. Quantitative estimates for the finite section method. Preprint, 2008.

    Google Scholar 

  38. K. Gr¨ochenig and S. Samarah. Nonlinear Approximation with Local Fourier Bases. Constr. Approx., 16:317–331, 2000.

    Google Scholar 

  39. Guo K, Labate D (2007) Sparse shearlet representation of Fourier integral operators Electron. Res Announc Math Sci 14:7–19

    MathSciNet  Google Scholar 

  40. B. Helffer. Th´eorie Spectrale pour des Op´erateurs Globalement Elliptiques. Ast´erisque, Soci´et´e Math´ematique de France, 1984.

    Google Scholar 

  41. B. Helffer and D. Robert. Comportement Asymptotique Precise du Spectre d’Op´erateurs Globalement Elliptiques dans ℝ𝑑. Sem. Goulaouic-Meyer-Schwartz 1980–81, ´Ecole Polytechnique, Expos´e II, 1980.

    Google Scholar 

  42. L. H¨ormander. Fourier integral operators I. Acta Math., 127:79–183, 1971.

    Google Scholar 

  43. L. H¨ormander. The Analysis of Linear Partial Differential Operators, Vol. III, IV. Springer-Verlag, 1985.

    Google Scholar 

  44. Lax P (1957) Asymptotic solutions of oscillatory initial value problems. Duke Math J 24:627–646

    Article  MathSciNet  MATH  Google Scholar 

  45. Kobayashi M (2006) Modulation spaces ℳ𝑝, 𝑞, for 0 < 𝑝, 𝑞 ≤ ∞. J Func Spaces Appl 4(2):329–341

    Article  MATH  Google Scholar 

  46. S.G. Krantz and H.R. Parks. The implicit function theorem. Birkh¨auser Boston Inc., Boston, 2002.

    Google Scholar 

  47. Kutyniok G, Labate D (2009) Resolution of the Wavefront Set using Continuous Shearlets. Trans Amer Math Soc 361(5):2719–2754

    Article  MathSciNet  MATH  Google Scholar 

  48. Rauhut H (2007) Coorbit Space Theory for Quasi-Banach Spaces. Studia Mathematica 180(3):237–253

    Article  MathSciNet  MATH  Google Scholar 

  49. R. Rochberg and K. Tachizawa. Pseudodifferential operators, Gabor frames, and local trigonometric bases. In Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkh¨auser Boston, Boston, MA, 171–192, 1998.

    Google Scholar 

  50. Ruzhansky M, Sugimoto M (2006) Global 𝐿2-boundedness theorems for a class of Fourier integral operators. Comm Partial Differential Equations 31(4–6):547–569

    Article  MathSciNet  MATH  Google Scholar 

  51. M.A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin, second edition, 2001. Translated from the 1978 Russian original by Stig I. Andersson.

    Google Scholar 

  52. G. Staffilani and D. Tataru. Strichartz estimates for a Schr¨odinger operator with nonsmooth coefficients. Comm. Partial Differential Equations, 27:1337–1372, 2002.

    Google Scholar 

  53. Stein EM (1993) Harmonic Analysis. Princeton University Press, Princeton

    Google Scholar 

  54. Tataru D (2002) Strichartz estimates for second-order hyperbolic operators with nonsmooth coefficients III. J Amer Math Soc 15:419–442

    Article  MathSciNet  MATH  Google Scholar 

  55. Toft J (2004) Continuity properties for modulation spaces, with applications to pseudodifferential calculus. II Ann Global Anal Geom 26(1):73–106

    Article  MathSciNet  MATH  Google Scholar 

  56. J. Toft. Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In J. Toft, M.W. Wong, H. Zhu (eds.) Modern Trends in Pseudo- Differential Operators, Operator Theory: Advances and Applications, Birkh¨auser Verlag Basel, 2007, 173–206.

    Google Scholar 

  57. F. Treves. Introduction to Pseudodifferential Operators and Fourier Integral Operators, Vol. I, II., Plenum Publ. Corp., New York, 1980.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Torino, via Carlo Alberto 10, I-10123, Torino, Italy

    Elena Cordero & Luigi Rodino

  2. Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, I-10129, Torino, Italy

    Fabio Nicola

Authors
  1. Elena Cordero
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  2. Fabio Nicola
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  3. Luigi Rodino
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Correspondence to Elena Cordero .

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Editors and Affiliations

  1. , Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

  2. , Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Mitsuru Sugimoto

  3. , Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Jens Wirth

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Cordero, E., Nicola, F., Rodino, L. (2012). Time-Frequency Analysis of Schrödinger Propagators. In: Ruzhansky, M., Sugimoto, M., Wirth, J. (eds) Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol 301. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0454-7_4

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