Partial differential equations with matricial coefficients and generalized translation operators

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

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.