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Sobolev type theorems
for an operator with singularity

Author: Shuji Watanabe
Journal: Proc. Amer. Math. Soc. 125 (1997), 129-136
MSC (1991): Primary 46E35, 47B25, 81Q10
MathSciNet review: 1343728
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Abstract: Spaces of Sobolev type are discussed, 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. 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.

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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

Keywords: Sobolev type theorem, operator with singularity, Schr\"odinger equation, self-adjointness
Received by editor(s): January 30, 1995
Received by editor(s) in revised form: May 31, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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