Partial differential equations with matricial coefficients and generalized translation operators
2000; American Mathematical Society; Volume: 352; Issue: 8 Linguagem: Inglês
10.1090/s0002-9947-00-02451-x
ISSN1088-6850
Autores Tópico(s)Differential Equations and Numerical Methods
ResumoLet Δ α \Delta _{\alpha } be the Bessel operator with matricial coefficients defined on ( 0 , ∞ ) (0,\infty ) by Δ α U ( t ) = U ( t ) + 2 α + I t U ′ ( t ) \begin{equation*}\Delta _{\alpha }U(t)=U(t)+\frac {2\alpha +I}{t}U’(t)\end{equation*} where α \alpha is a diagonal matrix and let q q be an n × n n\times n matrix-valued function. In this work, we prove that there exists an isomorphism X X on the space of even C ∞ {\mathcal C}^{\infty } , C n \mathbb {C}^n -valued functions which transmutes Δ α \Delta _{\alpha } and ( Δ α + q ) (\Delta _{\alpha }+q) . This allows us to define generalized translation operators and to develop harmonic analysis associated with ( Δ α + q ) (\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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