Partial differential equations with matricial coefficients and generalized translation operators

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.