Existence criteria for stabilization from the scaling behaviour of ionization probabilities

We provide a systematic derivation of the scaling behaviour of various quantities and, in particular, establish the scale invariance of the ionization probability. We discuss the gauge invariance of the scaling properties and the manner in which they can be exploited as a consistency check in explicit analytical expressions, in perturbation theory, in the Kramers–Henneberger and Floquet approximations, in upper and lower bound estimates and fully numerical solutions of the time dependent Schrödinger equation. The scaling invariance leads to a differential equation which has to be satisfied by the ionization probability and which yields an alternative criterion for the existence of atomic bound state stabilization.