Exact solutions to operator differential equations

This paper considers the Heisenberg equations of motion \. {}q=-iq, H, \. {}p=-ip, 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=p^{2}2+q^{4}4.