Monatshefte für Mathematik

, Volume 168, Issue 2, pp 279–303 | Cite as

Analyticity of the Schrödinger propagator on the Heisenberg group

  • S. ParuiEmail author
  • P. K. Ratnakumar
  • S. Thangavelu


We discuss the analytic extension property of the Schrödinger propagator for the Heisenberg sublaplacian and some related operators. The result for the sublaplacian is proved by interpreting the sublaplacian as a direct integral of an one parameter family of dilated special Hermite operators.


Schrödinger equation Oscillatory group Special Hermite expansion Heisenberg group Sublaplacian 

Mathematics Subject Classification (1991)

Primary 22E30 Secondary 35G10 47A63 


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  1. 1.
    Bargmann V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cholewinski F.M.: Generalised Fock spaces and associated operators. SIAM J. Math. 15, 177–202 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Erdelyi A. et al.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)Google Scholar
  4. 4.
    Folland G.B.: Harmonic Analysis in Phase Space. Annals of Math Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  5. 5.
    Folland G.B.: Real Analysis, Modern Techniques and Their Applications. Wiley-Interscience, New York (1984)zbMATHGoogle Scholar
  6. 6.
    Furioli G., Veneruso A.: Strichartz inequalities for the Schrödinger equation with the full Laplacian on the Heisenberg group. Studia Math. 160(2), 157–178 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Garg R., Thangavelu S.: On the Hermite expansions of functions from the Hardy class. Studia Math. 198(2), 177195 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goldberg M., Schlag W.: Dispersive estimates for Schrödinger operator in dimension one and three. Commun. Math. Phys. 251(1), 157–178 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hayashi N., Saitoh S.: Analyticity and smoothing effect of Schrödinger equation. Ann. Inst. Henri Poincare 52(2), 163–173 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jenson A.: Commutator methods and a smoothing property of the Schrödinger evolution group. Mathe. Z. 191, 53–59 (1986)CrossRefGoogle Scholar
  11. 11.
    Journé J.L., Soffer A., Sogge C.D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. XLIV, 573–604 (1991)CrossRefGoogle Scholar
  12. 12.
    Karp, D.: Square summability with geometric weights for classical orthogonal expansions. In: Begehr, H.G.W., et al. (eds.) Advances in Analysis, pp. 407–422. World Scientific, Singapore (2005)Google Scholar
  13. 13.
    Keel M., Tao T.: End point Strichartz estimates. Am. J. Math. 120, 955–980 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Krötz B., Thangavelu S., Xu Y.: The heat kernel transform for the Heisenberg group. J. Funct. Anal. 225, 301–336 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lebedev N.N.: Special Functions and Their Applications. Dover, New York (1992)Google Scholar
  16. 16.
    Nandakumaran A.K., Ratnakumar P.K.: Schrödinger equation and the regularity of the oscillatory semigroup for the Hermite operator. J. Funct. Anal 224, 371–385 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nandakumaran A.K., Ratnakumar P.K.: Corrigendum Schrödinger equation and the regularity of the oscillatory semigroup for the Hermite operator. J. Funct. Anal. 224, 719–720 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ratnakumar P.K.: On Schrödinger propogator for the special Hermite operator. J. Fourier Anal. Appl. 14(2), 286–300 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Rudin W.: Principle of Mathematical Analysis. McGraw-Hill International Editions, New York (1976)Google Scholar
  20. 20.
    Shohat J., Tamarkin J.: The Problem of Moments. American Math. Soc., Mathematical Surveys, vol. II. AMS, New York (1943)Google Scholar
  21. 21.
    Strichartz R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Strichartz R.S.: Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87, 51–148 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Stempak K.: On connections between Hankel, Laguerre and Jacobi transplantations. Tohoku Math. J. (2) 54(4), 471–493 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Stempak K., Trebels W.: On weighted transplantation and multipliers for Laguerre expansions. Math. Ann. 300, 203–219 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Thangavelu S.: Lectures on Hermite and Laguerre Expansions, Mathematical Notes, vol. 42. Princeton University Press, Princeton (1993)Google Scholar
  26. 26.
    Thangavelu S.: Harmonic Analysis on the Heisenberg Group. Progress in Math., vol. 154. Birkhäuser, Boston (1998)CrossRefGoogle Scholar
  27. 27.
    Thangavelu S.: Hermite and Laguerre semigroups: some recent developments. Séminaires et Congrès 25, 251–284 (2012)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of MathematicsNational Institute of Science Education and ResearchBhubaneswarIndia
  2. 2.Harish-Chandra Research InstituteJhunsi, AllahabadIndia
  3. 3.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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