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An embedding theorem of Sobolev type
for an operator with singularity


Author: Shuji Watanabe
Journal: Proc. Amer. Math. Soc. 125 (1997), 839-848
MSC (1991): Primary 35G10, 46E35, 47B25
DOI: https://doi.org/10.1090/S0002-9939-97-03642-3
MathSciNet review: 1353406
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Abstract: We discuss spaces of Sobolev type which are defined by the operator with singularity: ${\cal D} = d/dx - (c/x)R$, where $Ru(x) = u(-x)$ and $c > 1$. This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of $u$ and continuity of $u / x^{\beta }$ ($\beta > 0$) where $u$ is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.


References [Enhancements On Off] (What's this?)

  • 1. J. A. Goldstein, Semigroups of linear operators and applications, Oxford University Press, New York, 1985 / Clarendon Press, Oxford, 1985. MR 87c:47056
  • 2. Y. Ohnuki and S. Kamefuchi, Quantum field theory and parastatistics, University of Tokyo Press, Tokyo, 1982 / Springer-Verlag, Berlin, Heidelberg and New York, 1982. MR 85b:81001
  • 3. Y. Ohnuki and S. Watanabe, Self-adjointness of the operators in Wigner's commutation relations, J. Math. Phys. 33 (1992), 3653-3665. MR 93h:81065
  • 4. Y. Ohnuki and S. Watanabe, Properties of the operators in Wigner's commutation relations (in preparation).
  • 5. N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan 34 (1982), 677-701. MR 84i:47021
  • 6. M. Reed and B. Simon, Methods of modern mathematical physics, vol. II, Fourier analysis, self-adjointness, Academic Press, New York, 1975. MR 58:12429b
  • 7. H. Sohr, Über die Selbstadjungiertheit von Schrödinger- Operatoren, Math. Z. 160 (1978), 255-261. MR 80d:35040
  • 8. M. Watanabe and S. Watanabe, Self-adjointness of the momentum operator with a singular term, Proc. Amer. Math. Soc. 107 (1989), 999-1004. MR 90g:81035
  • 9. S. Watanabe, Sobolev type theorems for an operator with singularity, Proc. Amer. Math. Soc. (to appear). CMP 95:16
  • 10. E. P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations ?, Phys. Rev. 77 (1950), 711-712. MR 11:706e
  • 11. L. M. Yang, A note on the quantum rule of the harmonic oscillator, Phys. Rev. 84 (1951), 788-790. MR 13:804e

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Additional Information

Shuji Watanabe
Affiliation: Department of Mathematics, Toyota National College of Technology, Eisei-cho 2-1, Toyota-shi 471, Japan

DOI: https://doi.org/10.1090/S0002-9939-97-03642-3
Keywords: Embedding theorem of Sobolev type, operator with singularity, partial differential equations with singular coefficients
Received by editor(s): September 22, 1995
Additional Notes: Research partially supported by Grant-in-Aid for Scientific Research (No. 07740175), Ministry of Education, Science, Sports and Culture
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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