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Separation of the Grushin differential operator in weighted Hilbert spaces

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Abstract

In this paper we investigate the separation property of the Grushin differential operator of the form

$Gu = - \frac{1} {2}\left( {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{x^4 }} {4}\frac{{\partial ^2 u}} {{\partial y^2 }}} \right) + Q(x,y)u(x,y), \forall (x,y) \in \Omega \subset R^2 $

in the Hilbert space L 2(Ω, H), as well as in the weighted Hilbert space L 2,k (Ω, H), with the operator potential Q(x, y) ∈ C 1(Ω, L(H)), where L(H) is the space of all bounded linear operators on the arbitrary Hilbert space H and Ω be an open subset of R 2.

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Author notes

  1. H. A. Atia

    Present address: Mathematics Department, Rabigh College of Science and Art, King Abdulaziz University, P. O. Box 344, Rabigh, 21911, Saudi Arabia

Authors and Affiliations

  1. Faculty of Science, Mathematics Department, Zagazig University, Zagazig, Egypt

    H. A. Atia

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  1. H. A. Atia
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Submitted by A. Lapin

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Atia, H.A. Separation of the Grushin differential operator in weighted Hilbert spaces. Lobachevskii J Math 32, 180–188 (2011). https://doi.org/10.1134/S1995080211030036

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  • DOI: https://doi.org/10.1134/S1995080211030036

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