Exactly solved Schr\"odinger equations with time-dependent Hamiltonians

We present the analytical, exact, explicit, and assumption free formulas for the evolution operators corresponding to four instances of time-dependent Hamiltonians relevant to quantum spin batteries including two stochastic cases.

We demonstrate how to recover and go beyond existing expansions and approximations directly from the exact solutions giving, for example, an explicit exact formula for Floquet Hamiltonians at all orders.

The exact solutions are obtained through a completely novel combination of three mathematical techniques, the $\star$-algebra, path-sums and Omega calculus, which we briefly overview.

These are widely applicable to other non-autonomous differential systems.