Propagation of analytic singularities for the Schrödinger Equation

André Martinez[1]; Shu Nakamura[2]; Vania Sordoni[1]

  • [1] Università di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy
  • [2] Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan 153-8914

Séminaire Équations aux dérivées partielles (2007-2008)

  • Volume: 2007-2008, page 1-14

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Martinez, André, Nakamura, Shu, and Sordoni, Vania. "Propagation of analytic singularities for the Schrödinger Equation." Séminaire Équations aux dérivées partielles 2007-2008 (2007-2008): 1-14. <http://eudml.org/doc/11174>.

@article{Martinez2007-2008,
affiliation = {Università di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy; Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan 153-8914; Università di Bologna, Dipartimento di Matematica, Piazza di Porta San Donato 5, 40127 Bologna, Italy},
author = {Martinez, André, Nakamura, Shu, Sordoni, Vania},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {infinite speed of propagation; short-range type perturbations},
language = {eng},
pages = {1-14},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Propagation of analytic singularities for the Schrödinger Equation},
url = {http://eudml.org/doc/11174},
volume = {2007-2008},
year = {2007-2008},
}

TY - JOUR
AU - Martinez, André
AU - Nakamura, Shu
AU - Sordoni, Vania
TI - Propagation of analytic singularities for the Schrödinger Equation
JO - Séminaire Équations aux dérivées partielles
PY - 2007-2008
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2007-2008
SP - 1
EP - 14
LA - eng
KW - infinite speed of propagation; short-range type perturbations
UR - http://eudml.org/doc/11174
ER -

References

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  1. Bouclet, J.M., Tzvetkov, N., Strichartz estimates for long range perturbations, Amer. J. Math. vol 129, no. 6 (2007) Zbl1154.35077MR2369889
  2. Burq, N., Gérard, P., Tzvetkov, N. Strichartz inequalities and the non-linear Schrödinger equation on compact manifolds, Am. J. Math. 126 (2004), 569–605. Zbl1067.58027MR2058384
  3. Craig, W., Kappeler, T., Strauss, W., Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pures Appl. Math. 48 (1996), 769–860. Zbl0856.35106MR1361016
  4. Doi, S., Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318 (2000), 355–389. Zbl0969.35029MR1795567
  5. Doi, S., Singularities of solutions of Schrödinger equations for perturbed harmonic oscillators, Hyperbolic problems and related topics, Grad. Ser. Anal. 185–199, Int. Press, Somerville, MA, 2003. Zbl1047.35007MR2056850
  6. Ginibre, J., Velo, G., Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), 163–188. Zbl0762.35008MR1151250
  7. Hassel, A., Wunsch, J., The Schrödinger propagator for scattering metrics, Annals of Mathematics, 162 (2005) Zbl1126.58016MR2178967
  8. Hayashi, N., Kato, K., Analyticity and smoothing effect for the Schrödinger equation, Ann. Inst. H. Poincaré, Phys. Théo. 52 (1990), 163–173. Zbl0731.35047MR1051235
  9. Hayashi, N., Kato, K., Analyticity in time and Smoothing effect of solutions to nonlinear Schrödinger equations, Comm. Math. Phys. 184 (1997), 273–300. Zbl0871.35087MR1462747
  10. Ito, K.: Propagation of Singularities for Schrödinger Equations on the Euclidean Space with a Scattering Metric, Comm. P. D. E., 31 (12), 1735–1777 (2006). Zbl1117.35005MR2273972
  11. Kajitani, K., Wakabayashi, S., Analytic smoothing effect for Schrödinger type equations with variable coefficients, in: Direct and Inverse Problems of Mathematical Physics (Newark, DE 1997), Int. Soc. Anal. Comput. 5, Kluwer Acad. Publ. (Dordracht, 2000), 185–219. Zbl1043.35055MR1766299
  12. Kapitanski, L., Rodnianski, I., Yajima, K., On the fundamental solution of a perturbed harmonic oscillator, Topol. Methods Nonlinear Anal. 9 (1997), 77–106. Zbl0892.35035MR1483643
  13. Kapitanski, L., Safarov, Y., Dispersive smoothing for Schrödinger equations, Math. Res. Letters 3 (1996), 77–91. Zbl0860.35016MR1393385
  14. Kato, K., Taniguchi, K., Gevrey regularizing effect for nonlinear Schrödinger equations, Osaka J. Math 33 (1996), 863–880. Zbl0876.35109MR1435458
  15. Kato, T., Yajima, K., Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496. Zbl0833.47005MR1061120
  16. Kenig, C., Ponce, G., Vega, L., Oscillatory integrals and regularity of dispersive equations, Ind. Univ. Math. J. 40 (1991), 33–69. Zbl0738.35022MR1101221
  17. Martinez, A., An Introduction to Semiclassical and Microlocal Analysis, UTX Series, Springer-Verlag New-York, 2002. Zbl0994.35003MR1872698
  18. Martinez, A., Nakamura, S., Sordoni, V., Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. 59 (2006), 1330–1351. Zbl1122.35027MR2237289
  19. Martinez, A., Nakamura, S., Sordoni, V., Analytic wave front set for solutions to Schrödinger equations, Preprint 2007. Zbl1180.35016
  20. Mechergui, C., Equivalence between the analytic quadratic scattering wave front set and analytic homogeneus wave front set, Preprint 2007. 
  21. Melrose, R. B., Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992), Dekker, New York, 1994, pp. 85TH130. Zbl0837.35107MR1291640
  22. Morimoto, Y., Robbiano, L., Zuily, C., Remark on the smoothing for the Schrödinger equation, Indiana University Mathematic Journal 49 (2000), 1563–1579. Zbl0987.35009MR1838302
  23. Nakamura, S., Propagation of the Homogeneous Wave Front Set for Schrödinger Equations, Duke Math. J. 126, 349-367 (2005). Zbl1130.35023MR2115261
  24. Nakamura, S., Wave front set for solutions to Schrödinger equations, Preprint 2004 
  25. Nakamura, S.: Semiclassical singularity propagation property for Schrödinger equations, Preprint 2006 (http://www.arxiv.org/abs/math.AP/0605742), to appear in J. Math. Soc. Japan. 
  26. Robert, S., Autour de l’Approximation Semi-Classique, Birkhäuser (1987) Zbl0621.35001
  27. Robbiano, L., Zuily, C., Microlocal analytic smoothing effect for Schrödinger equation, Duke Math. J. 100 (1999), 93–129. Zbl0941.35014MR1714756
  28. Robbiano, L., Zuily, C., Effet régularisant microlocal analytique pour l’équation de Schrödinger: le cas des données oscillantes, Comm. Partial Differential Equations 100 (2000) 1891–1906. Zbl0959.35007
  29. Robbiano, L., Zuily, C., Analytic theory for the quadratic scattering wave front set and application to the Schrödinger equation, Soc. Math. France, Astérisque 283 (2002), 1–128. Zbl1029.35001MR1958605
  30. Robbiano, L., Zuily, C., Strichartz estimates for Schrödinger equations with variable coefficients, Mem. Soc. Math. France (N.S.) 101-102 (2005) Zbl1097.35002MR2193021
  31. Sjöstrand, J., Singularités analytiques microlocales, Soc. Math. France, Astérisque 95 (1982), 1–166. Zbl0524.35007MR699623
  32. Staffilani, G., Tataru, D., Strichartz estimates for a Schrödinger operator with non smooth coefficients, Com. P. D. E. 27 (2002), 1337–1372 Zbl1010.35015MR1924470
  33. Wunsch, J., Propagation of singularities and growth for Schrödinger operators, Duke Math. J. 98 (1999), 137-186. Zbl0953.35121MR1687567
  34. Yajima, K., On smoothing property of Schrödinger propagators. Functional-Analytic Methods for Partial Differential Equations, Lecture Notes in Math., 1450 (1990), 20–35, Springer, Berlin. Zbl0725.35084MR1084599
  35. Yajima, K., Schrödinger evolution equations and associated smoothing effect, Rigorous Results in Quantum Dynamics, World Sci. Publishing, River Edge, NJ, 1991, 167–185. MR1243040
  36. Yamazaki, M., On the microlocal smoothing effect of dispersive partial differential equations, Algebraic Analysis Vol. II, 911–926, Academic Press, Boston, MA,1988. Zbl0683.35010MR992503
  37. Zelditch, S., Reconstruction of singularities for solutions of Schrödinger equation, Comm. Math. Phys. 90 (1983), 1–26. Zbl0554.35031MR714610

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