Does the standard tool for driven quantum systems actually work?

Most driven quantum systems — like atoms in laser fields or oscillators under periodic forcing — have Hamiltonians that are mathematically unbounded, yet the standard Floquet-Magnus expansion was only proven to work for bounded ones. This work rigorously extends the method to unbounded operators and, crucially, proves it gives accurate approximations even when the expansion itself doesn't formally converge. The fix matters: nearly every experimentally relevant quantum system falls outside the original theory's fine print.