  Can classical systems be described analytically at all orders in their
interaction strength? For periodic and approximately periodic systems, the
answer is yes, as we show in this work. Our analytical approach, which we call
the \textit{Quantum Spectral Method}, is based on a novel application of Bohr's
correspondence principle, obtaining non-perturbative classical dynamics as the
classical limit of \textit{quantum matrix elements}. A major application of our
method is the calculation of self-force as the classical limit of atomic
radiative transitions. We demonstrate this by calculating an adiabatic
electromagnetic inspiral, along with its associated radiation, at all orders in
the multipole expansion. Finally, we propose a future application of the QSM to
compute scalar and gravitational self-force in Schwarzschild, analytically.
