Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1033–1058 | Cite as

Dunkl–Schrödinger Operators

  • Béchir AmriEmail author
  • Amel Hammi
Article

Abstract

In this paper, we consider the Schrödinger operators \(L_k=-\Delta _k+V\), where \(\Delta _k\) is the Dunkl–Laplace operator and V is a non-negative potential on \(\mathbb {R}^d\). We establish that \(L_k \) is essentially self-adjoint on \(C_0^\infty (\mathbb {R}^d)\). In particular, we develop a bounded \(H^\infty \)-calculus on \(L^p\) spaces for the Dunkl harmonic oscillator operator.

Keywords

Self-adjoint operator Schrödinger operator Dunkl operators 

Mathematics Subject Classification

Primary 47B25 35J10 Secondary 43A32 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the referee for his comments and suggestions.

References

  1. 1.
    Albrecht, D., Duong, X.T., McIntosh, A.: Operator theory and harmonic analysis. In: Proceedings of the Centre for Mathematics and Its Applications, vol. 34, pp. 77–136. CMA, ANU, Canberra (1996)Google Scholar
  2. 2.
    Amri, B.: Riesz transforms for Dunkl Hermite expansions. J. Math. Anal. Appl. 423, 646–659 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amri, B.: The \(L^p\)-continuity of imaginary powers of the Dunkl harmonic oscillator. Indian J. Pure Appl. Math. 46, 239–249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Badr, N., Ben Ali, B.: \(L^p\)-boundedness of Riesz transform related to Schrödinger operators on a manifold. Ann. Scuola Norm. Sup. di Pisa Cl. Sci. 5, 725–765 (2009)zbMATHGoogle Scholar
  5. 5.
    de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113(1), 147–162 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deleaval, L., Kriegler, C.: Dunkl spectral multipliers with values in UMD lattices. J. Funct. Anal. 272(5), 2132–2175 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dunkl, C.F.: Differential–difference operators associated to reflextion groups. Trans. Am. Math. 311(1), 167–183 (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duong, X.T., Robinson, D.W.: Semigroup kernels, poisson bounds, and holomorphic functional calculus. J. Func. Anal. 142, 89–128 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Duong, X.T., McIntosh, A.: Singular integral operators with non smooth kernels on irregular domains. Rev. Mat. Iberoam. 15, 233–265 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Li, H.: Estimations \(L^p\) des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lin, C., Liu, H.: \(BMO_L(\mathbb{H}^n)\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–1688 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McIntosh, A.: Operators which have an \(H ^\infty \) functional calculus. Mini conference on operator theory and partial differential equations. Proc. Centre Math. Anal. ANU 14, 210–231 (1986)Google Scholar
  15. 15.
    Nowak, A., Stempak, K.: Riesz transforms for the Dunkl harmonic oscillator. Math. Z. 262, 539–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)zbMATHGoogle Scholar
  17. 17.
    Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192, 519–542 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rösler, M.: Dunkl operators: theory and applications. In: Koelink, E., Van Assche, W. (eds.) Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes on Mathematics, vol. 1817, pp. 93–135. Springer, Berlin (2003)CrossRefGoogle Scholar
  19. 19.
    Schep, A.R.: Kernel operators. Indag. Math. Proc. 82, 39–53 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Simon, B.: Maximal and minimal Schrödinger forms. J. Oper. Theory 1, 37–47 (1979)zbMATHGoogle Scholar
  21. 21.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)zbMATHGoogle Scholar
  22. 22.
    Thangavelu, S., Xu, Y.: Convolution operator and maximal function for Dunkl transform. J. Anal. Math. 97, 25–55 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Trimèche, K.: Paley–Wiener theorems for Dunkl transform and Dunkl translation operators. Integr. Transforms Spec. Funct. 13, 17–38 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesTaibah UniversityAl-Madinah Al-MunawarahSaudi Arabia
  2. 2.Faculté des sciences de Tunis, Laboratoire d’Analyse Mathématique et ApplicationsUniversité Tunis El ManarTunisTunisie

Personalised recommendations