Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 004, 16 pages      arXiv:1301.4196      https://doi.org/10.3842/SIGMA.2014.004

Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

Jesús A. Álvarez López a and Manuel Calaza b
a) Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
b) Laboratorio de Investigación 10, Servicio de Reumatología, Instituto de Investigaciółn Sanitaria, Hospital Clínico Universitario, 15706 Santiago de Compostela, Spain

Received September 09, 2013, in final form January 06, 2014; Published online January 10, 2014

Abstract
Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line.

Key words: Dunkl harmonic oscillator; Sobolev embedding; generalized Hermite functions; Schwartz space.

pdf (428 kb)   tex (24 kb)

References

  1. Álvarez López J., Calaza M., Application of the method of Bonan-Clark to the generalized Hermite polynomials, arXiv:1101.5022.
  2. Álvarez López J., Calaza M., Witten's perturbation on strata, arXiv:1205.0348.
  3. Askey R., Wainger S., Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708.
  4. Bonan S.S., Clark D.S., Estimates of the Hermite and the Freud polynomials, J. Approx. Theory 63 (1990), 210-224.
  5. Brasselet J.P., Hector G., Saralegi M., L2-cohomologie des espaces stratifiés, Manuscripta Math. 76 (1992), 21-32.
  6. Brüning J., Lesch M., Hilbert complexes, J. Funct. Anal. 108 (1992), 88-132.
  7. Cheeger J., On the Hodge theory of Riemannian pseudomanifolds, in Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., Vol. 36, Amer. Math. Soc., Providence, R.I., 1980, 91-146.
  8. Cheeger J., Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657.
  9. Chihara T.S., Generalized Hermite polynomials, Ph.D. Thesis, Purdue University, 1955.
  10. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York, 1978.
  11. Dickinson D., Warsi S.A., On a generalized Hermite polynomial and a problem of Carlitz, Boll. Un. Mat. Ital. 18 (1963), 256-259.
  12. Dunkl C.F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
  13. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  14. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
  15. Dunkl C.F., Symmetric functions and BN-invariant spherical harmonics, J. Phys. A: Math. Gen. 35 (2002), 10391-10408, math.CA/0207122.
  16. Dutta M., Chatterjea S.K., More K.L., On a class of generalized Hermite polynomials, Bull. Inst. Math. Acad. Sinica 3 (1975), 377-381.
  17. Erdélyi A., Asymptotic forms for Laguerre polynomials, J. Indian Math. Soc. 24 (1960), 235-250.
  18. Goresky M., MacPherson R., Morse theory and intersection homology theory, Astérisque 101 (1983), 135-192.
  19. Hörmander L., The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften, Vol. 256, 2nd ed., Springer-Verlag, Berlin, 1990.
  20. Mather J., Notes on topological stability, Hardvard University, 1970, available at http://www.math.princeton.edu/facultypapers/mather/notes_on_topological_stability.pdf.
  21. Muckenhoupt B., Asymptotic forms for Laguerre polynomials, Proc. Amer. Math. Soc. 24 (1970), 288-292.
  22. Muckenhoupt B., Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc. 147 (1970), 433-460.
  23. Nagase M., L2-cohomology and intersection homology of stratified spaces, Duke Math. J. 50 (1983), 329-368.
  24. Nagase M., Sheaf theoretic L2-cohomology, Adv. Stud. Pure Math. 8 (1986), 273-279.
  25. Nowak A., Stempak K., Imaginary powers of the Dunkl harmonic oscillator, SIGMA 5 (2009), 016, 12 pages, arXiv:0902.1958.
  26. Nowak A., Stempak K., Riesz transforms for the Dunkl harmonic oscillator, Math. Z. 262 (2009), 539-556, arXiv:0802.0474.
  27. Plyushchay M., Hidden nonlinear supersymmetries in pure parabosonic systems, Internat. J. Modern Phys. A 15 (2000), 3679-3698, hep-th/9903130.
  28. Roe J., Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series, Vol. 395, 2nd ed., Longman, Harlow, 1998.
  29. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396, math.CA/9307224.
  30. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  31. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
  32. Szegö G., Orthogonal polynomials, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  33. Thom R., Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240-284.
  34. Verona A., Stratified mappings - structure and triangulability, Lecture Notes in Math., Vol. 1102, Springer-Verlag, Berlin, 1984.
  35. Watanabe S., An embedding theorem of Sobolev type for an operator with singularity, Proc. Amer. Math. Soc. 125 (1997), 839-848.
  36. Watanabe S., A generalized Fourier transform and its applications, Integral Transforms Spec. Funct. 13 (2002), 321-344.
  37. Watanabe S., An embedding theorem of Sobolev type, Integral Transforms Spec. Funct. 15 (2004), 369-374.
  38. Witten E., Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.
  39. Yang L.M., A note on the quantum rule of the harmonic oscillator, Phys. Rev. 84 (1951), 788-790.

Previous article  Next article   Contents of Volume 10 (2014)