On Matrix Schr\"odinger Unitary Groups in Particular Representations of Finite Dimensional Quantum Dynamical Systems

In this paper we study some particular types of matrix Schr\"odinger semigroups of the form $\exp(-it\mathbb{H})$ where $\mathbb{H}\in M_N(\mathbf{C})$ is the Hamiltonian of a given quantum dynamical system modeled in the finite dimensional Hilbert space $\mathcal{H}$. Once we have defined a particular matrix Schr\"odinger unitary group we perform some estimates for its approximation and its corresponding implementation in the numerical solution of the finite dimensional Schr\"odinger evolution equation to that it is related.