  The Floquet-Magnus expansion is a useful tool to calculate an effective
Hamiltonian for periodically driven systems. In this study, we investigate the
convergence of the expansion for a one-body nonlinear system in a continuous
space, a driven anharmonic oscillator. In this model, all eigenstates of the
time evolution operator are found to be localized in energy space, and the
expectation value of the energy is bounded from above. We first propose a
general procedure to estimate the radius of convergence of the Floquet-Magnus
expansion for periodically driven systems with an unbounded energy spectrum. By
applying it to the driven anharmonic oscillator, we numerically show that the
expansion diverges for all driving frequencies even if the anharmonicity is
arbitrarily small. This conclusion contradicts the widely accepted belief that
the divergence of the Floquet-Magnus expansion is a direct consequence of
quantum ergodicity, which implies that each eigenstate of the time evolution
operator is a linear combination of all available eigenstates of the
unperturbed Hamiltonian and the system heats up to infinite temperature after
long intervals.
