The quantum harmonic oscillator has a fixed energy ladder: the Sobolev norms of its eigenstates are determined by the quantum number. Add a time-dependent perturbation, and these norms can grow — the system absorbs energy from the perturbation indefinitely.
The paper on generalized reducibility and Sobolev norm growth (arXiv: 2603.20834) shows that the growth rate can be prescribed. For many sub-exponential functions f(t) — whether monotone or oscillatory — there exist time-decaying perturbations of the harmonic oscillator whose Sobolev norms grow like f(t).
The method is constructive: the perturbation is tailored to produce the desired growth profile. “Generalized reducibility” is the framework — a weakening of the classical reducibility program (which seeks to conjugate the time-dependent system to a time-independent one). In the generalized version, the conjugation is only approximate, and the approximation error drives the norm growth.
The through-claim: energy growth is programmable through perturbation design. The quantum harmonic oscillator is stable; its Sobolev norms are constant. But perturbations — even time-decaying ones — can program any sub-exponential growth rate into the system. The perturbation is a script; the norm growth is the execution. The connection between the perturbation's structure and the norm's behavior is explicit and constructive.
2603.20834. Mathematical physics / quantum harmonic oscillator / Sobolev norms / energy growth / perturbation theory.