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.
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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)
Amri, B.: Riesz transforms for Dunkl Hermite expansions. J. Math. Anal. Appl. 423, 646–659 (2015)
Amri, B.: The \(L^p\)-continuity of imaginary powers of the Dunkl harmonic oscillator. Indian J. Pure Appl. Math. 46, 239–249 (2015)
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)
de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113(1), 147–162 (1993)
Deleaval, L., Kriegler, C.: Dunkl spectral multipliers with values in UMD lattices. J. Funct. Anal. 272(5), 2132–2175 (2017)
Dunkl, C.F.: Differential–difference operators associated to reflextion groups. Trans. Am. Math. 311(1), 167–183 (1989)
Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)
Duong, X.T., Robinson, D.W.: Semigroup kernels, poisson bounds, and holomorphic functional calculus. J. Func. Anal. 142, 89–128 (1996)
Duong, X.T., McIntosh, A.: Singular integral operators with non smooth kernels on irregular domains. Rev. Mat. Iberoam. 15, 233–265 (1999)
Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)
Li, H.: Estimations \(L^p\) des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999)
Lin, C., Liu, H.: \(BMO_L(\mathbb{H}^n)\) spaces and Carleson measures for Schrödinger operators. Adv. Math. 228, 1631–1688 (2011)
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)
Nowak, A., Stempak, K.: Riesz transforms for the Dunkl harmonic oscillator. Math. Z. 262, 539–556 (2009)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)
Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192, 519–542 (1998)
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)
Schep, A.R.: Kernel operators. Indag. Math. Proc. 82, 39–53 (1979)
Simon, B.: Maximal and minimal Schrödinger forms. J. Oper. Theory 1, 37–47 (1979)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)
Thangavelu, S., Xu, Y.: Convolution operator and maximal function for Dunkl transform. J. Anal. Math. 97, 25–55 (2005)
Trimèche, K.: Paley–Wiener theorems for Dunkl transform and Dunkl translation operators. Integr. Transforms Spec. Funct. 13, 17–38 (2002)
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The authors would like to express their sincere thanks to the referee for his comments and suggestions.
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Communicated by Daniel Aron Alpay.
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Amri, B., Hammi, A. Dunkl–Schrödinger Operators. Complex Anal. Oper. Theory 13, 1033–1058 (2019). https://doi.org/10.1007/s11785-018-0834-1
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DOI: https://doi.org/10.1007/s11785-018-0834-1