Energy-localized Egorov theorem for matrix-valued Schrödinger operator without Cordes's hyperbolicity condition

We establish a semi-classical Egorov type theorem for a matrix-valued Schr"{o}dinger operator, frequently arising in important applications, which does not satisfy typically imposed hypotheses, e.g., Cordes's hyperbolicity condition. The proof combines a number of new ideas and the techniques involve $h$-pseudodifferential operator calculus, Riesz projection formula, energy localization, and detailed study of the Heisenberg equations on symbol level using the time-dependent Schr"{o}dinger equation to justify an auxiliary localized initial condition which allows for a detailed analysis of the Hamiltonian flow and the derivation of key estimates for the solutions to new auxiliary Heisenberg equations.