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Partial differential equations with matricial coefficients and generalized translation operators


Author: N. H. Mahmoud
Journal: Trans. Amer. Math. Soc. 352 (2000), 3687-3706
MSC (2000): Primary 35A25, 35C15; Secondary 34B30
DOI: https://doi.org/10.1090/S0002-9947-00-02451-X
Published electronically: March 16, 2000
MathSciNet review: 1650030
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Abstract:

Let $\Delta_{\alpha }$ be the Bessel operator with matricial coefficients defined on $(0,\infty )$ by

\begin{equation*}\Delta_{\alpha }U(t)=U''(t)+\frac{2\alpha +I}{t}U'(t)\end{equation*}

where $\alpha$ is a diagonal matrix and let $q $ be an $n\times n$ matrix-valued function. In this work, we prove that there exists an isomorphism $X$ on the space of even ${\mathcal C}^{\infty}$, $\mathbb{C} ^n$-valued functions which transmutes $\Delta_{\alpha}$and $(\Delta_{\alpha}+q)$. This allows us to define generalized translation operators and to develop harmonic analysis associated with $(\Delta_{\alpha}+q)$. By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.


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Additional Information

N. H. Mahmoud
Affiliation: Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060 Tunis, Tunisie
Email: houda.mahmoud@insat.rnu.tn

DOI: https://doi.org/10.1090/S0002-9947-00-02451-X
Keywords: Singular differential operators, Bessel functions, transmutation operators, generalized translations, Riemann function, product formula
Received by editor(s): July 30, 1996
Received by editor(s) in revised form: January 30, 1998
Published electronically: March 16, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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