Sobolev type theorems for an operator with singularity
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- by Shuji Watanabe PDF
- Proc. Amer. Math. Soc. 125 (1997), 129-136 Request permission
Abstract:
Spaces of Sobolev type are discussed, which are defined by the operator with singularity: $\mathcal {D} = d/dx - (c/x)R$, where $Ru(x) = u(-x)$ and $c > 1$. This operator appears in a one-dimensional harmonic oscillator governed by Wigner’s commutation relations. Smoothness of $u$ and continuity of $u / x^{\beta }$ ($\beta > 0$) are studied where $u$ is in each space of Sobolev type, and results similar to Sobolev’s lemma are obtained. The proofs are carried out based on a generalization of the Fourier transform. The results are applied to the Schrödinger equation.References
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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
- Received by editor(s): January 30, 1995
- Received by editor(s) in revised form: May 31, 1995
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 129-136
- MSC (1991): Primary 46E35, 47B25, 81Q10
- DOI: https://doi.org/10.1090/S0002-9939-97-03523-5
- MathSciNet review: 1343728