Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Spectra of Higher Spin Operators on the Sphere

  • Doojin Hong (Department of Mathematics, University of North Dakota)
  • Received : 2022.01.07
  • Accepted : 2022.09.29
  • Published : 2023.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

We present explicit formulas for the spectra of higher spin operators on the subbundle of the bundle of spinor-valued trace free symmetric tensors that are annihilated by Clifford multiplication over the standard sphere in odd dimension. In the even dimensional case, we give the spectra of the square of such operators. The Dirac and Rarita-Schwinger operators are zero-form and one-form cases, respectively. We also give eigenvalue formulas for the conformally invariant differential operators of all odd orders on the subbundle of the bundle of spinor-valued forms that are annihilated by Clifford multiplication in both even and odd dimensions on the sphere.

Keywords

References

  1. T. P. Branson, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal, 106(2)(1992), 314-328. https://doi.org/10.1016/0022-1236(92)90050-S
  2. T. Branson, Nonlinear phenomena in the spectral theory of geometric linear differential operators, Proc. Symp. Pure Math., 59(1996), 27-65.
  3. T. Branson, Stein-Weiss operators and ellipticity, J. Funct. Anal., 151(2)(1997), 334-383. https://doi.org/10.1006/jfan.1997.3162
  4. T. Branson, Spectra of self-gradients on shperes, J. Lie Theory, 9(2)(1999), 491-506.
  5. T. Branson, Automated Symbolic Computation in Spin Geometry, Clifford Analysis and its Applications, 25(2001), 27-38. https://doi.org/10.1007/978-94-010-0862-4_3
  6. T. Branson and O. Hijazi, Bochner-Weitzenbock formulas associated with the Rarita-Schwinger operator, Internat. J. Math., 13(2)(2002), 137-182. https://doi.org/10.1142/S0129167X02001174
  7. T. Branson, G. Olafsson and B. Orsted, Spectrum generating operators, and intertwining operators for representations induced from a maximal parabolic subgroups, J. Funct. Anal., 135(1)(1996), 163-205. https://doi.org/10.1006/jfan.1996.0008
  8. T. Branson and B. Orsted, Spontaneous generation of eigenvalues, J. Geom. Phys., 56(11)(2006), 2261-2278. https://doi.org/10.1016/j.geomphys.2005.12.001
  9. J. Bures, F. Sommen, V. Soucek and P. Van Lancker, Symmetric Analogues of Rarita-Schwinger equations, Ann. Global Anal. Geom., 21(3)(2002), 215-240. https://doi.org/10.1023/A:1014923601006
  10. A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on Sn and Pn(ℂ), Osaka J. Math., 15(1978), 515-546.
  11. Y. Kosmann, Derivees de Lie des spineurs, Ann. Mat. Pura Appl., 91(1972), 317-395. https://doi.org/10.1007/BF02428822
  12. D. Hong, Intertwinors on Functions over the Product of Spheres, SIGMA Symmetry Integrability Geom. Methods Appl., 7(2011), 7pp.
  13. E. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math., 90(1968), 163-196.  https://doi.org/10.2307/2373431