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

MathSciNet review:
1353406

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss spaces of Sobolev type which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of and continuity of () where 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.

**1.**Jerome A. Goldstein,*Semigroups of linear operators and applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR**790497****2.**Yoshio Ohnuki and Susumu Kamefuchi,*Quantum field theory and parastatistics*, University of Tokyo Press, Tokyo; Springer-Verlag, Berlin, 1982. MR**687830****3.**Yoshio Ohnuki and Shuji Watanabe,*Selfadjointness of the operators in Wigner’s commutation relations*, J. Math. Phys.**33**(1992), no. 11, 3653–3665. MR**1185839**, 10.1063/1.529860**4.**Y. Ohnuki and S. Watanabe,*Properties of the operators in Wigner's commutation relations*(in preparation).**5.**Noboru Okazawa,*On the perturbation of linear operators in Banach and Hilbert spaces*, J. Math. Soc. Japan**34**(1982), no. 4, 677–701. MR**669276**, 10.2969/jmsj/03440677**6.**Michael Reed and Barry Simon,*Methods of modern mathematical physics. II. Fourier analysis, self-adjointness*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0493420****7.**Hermann Sohr,*Über die Selbstadjungiertheit von Schrödinger-Operatoren*, Math. Z.**160**(1978), no. 3, 255–261 (German). MR**497952**, 10.1007/BF01237039**8.**Michiaki Watanabe and Shuji Watanabe,*Selfadjointness of the momentum operator with a singular term*, Proc. Amer. Math. Soc.**107**(1989), no. 4, 999–1004. MR**984821**, 10.1090/S0002-9939-1989-0984821-8**9.**S. Watanabe,*Sobolev type theorems for an operator with singularity*, Proc. Amer. Math. Soc. (to appear). CMP**95:16****10.**Eugene P. Wigner,*Do the equations of motion determine the quantum mechanical commutation relations?*, Physical Rev. (2)**77**(1950), 711–712. MR**0035214****11.**L. M. Yang,*A note on the quantum rule of the harmonic oscillator*, Physical Rev. (2)**84**(1951), 788–790. MR**0046918**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
35G10,
46E35,
47B25

Retrieve articles in all journals with MSC (1991): 35G10, 46E35, 47B25

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