A Class of Matrix Schr\"odinger Bispectral Operators

We prove the bispectrality of some class of matrix Schr\"odinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the Laurent series and after that obtaining local solutions using estimations in the Frobenius norm. Furthermore, the characterization of the algebra of polynomial eigenvalues in the spectral variable is given using some family of functions $\left\{ P_{k}\right\}_{k\in \mathbb{N}}$ with the remarkable property of satisfying a general version of the Leibniz rule.