Exact solutions to operator differential equations

This paper considers the Heisenberg equations of motion $\dot{q}=-i[q, H]$, $\dot{p}=-i[p, H]$, for the quantum-mechanical Hamiltonian $H(p, q)$ having one degree of freedom. It is a commonly held belief that such operator differential equations are intractable. However, a technique is presented here that allows one to obtain exact, closed-form solutions for huge classes of Hamiltonians. This technique, which is a generalization of the classical action-angle-variable methods, allows us to solve, albeit formally and implicitly, the operator differential equations of the anharmonic oscillator whose Hamiltonian is $H=\frac{p^2}{2}+\frac{q^4}{4}$.