In the proof for Schrödinger solutions with time-dependent potentials it was already noted, that $H^2$ regularity is guaranteed, i.e. initial states in $D(H)=H^2$ stay in this set during evolution. A natural question is now for which potential classes $H^{2m}$ regularity is fulfilled. Indeed this can answered with the same strategy: Instead of $H(t)^{-1}H(t)$ identities $H(t)^{-m}H(t)^m$ are introduced and switched with the evolution operators. The $H(t)^{m}H(s)^{-m} = K_m(t,s) + \mathrm{id}$ can be estimated again, but the potential norm is now in the $\mathrm{Lip}([0,T],K^m)$ Sobolev–Kato–Lipschitz space.

Why is this important? In TDDFT related proofs the so-called “divergence of force density equation” is of great utility: \[ -\nabla\cdot (n \nabla v) = q - \partial_t^2 n. \] The term $q$ not written out here includes fourth order weak derivatives of $\psi(t)$ and thus $H^4$ stability has to be guaranteed by the external potential $v$. The discussed potential class is thus part of a rigorous framework in which further results from TDDFT can be discussed.

Another important application is for functional derivatives of $\psi[v]$ with respect to the external potential $v$ that enter in linear response theory. Let $U[v]$ be the evolution system yielding solutions to Schrödinger's equation with external potential $v$, then \[ \delta \psi[v;w] = \lim_{\lambda \rightarrow 0} \frac{1}{\lambda} \big(U[v+\lambda w]-U[v]\big) \psi_0 \] is the Gâteaux derivative of $\psi[v]$ in direction $w$. The Gâteaux derivative corresponds to the directional derivative in multivariable calculus, whereas the Fréchet derivative is the generalization of the total derivative. Such a difference of evolutions has already appeared in the proofs before, so a similar technique can be employed that leads again to Sobolev–Kato–Lipschitz spaces as a condition for functional differentiability.