Simplifying the Schrieffer-Wolff transformation I: Universal solution to SWT

The Schrieffer-Wolff transformation (SWT) is a crucial tool in quantum mechanics for simplifying Hamiltonians by decoupling energy subspaces. Current implementations, however, often lack generality or require heuristic truncations for infinite-dimensional systems. We present a systematic framework that overcomes these limitations, providing a general closed-form solution for the SWT generator without ad hoc assumptions or Hilbert space truncations. This approach extends naturally to time-dependent systems with periodic perturbations, covering all frequency regimes. By keeping intact the operator form of perturbed Hamiltonians, our results open the doors to new analytical insights into infinite-dimensional systems. Furthermore, these results are also applicable to systems with periodic driving, thus expanding the SWT's utility in analyzing quantum systems beyond static conditions.