Provided the initial state, the Runge-Gross theorem establishes that the time-dependent (TD) external potential of a system of non-relativistic electrons determines uniquely their TD electronic density, and vice versa (up to a constant in the potential). This theorem requires the TD external potential and density to be Taylor-expandable around the initial time of the propagation. This paper presents an extension without this restriction. Given the initial state of the system and evolution of the density due to some TD scalar potential, we show that a perturbative (not necessarily weak) TD potential that induces a non-zero divergence of the external force-density, inside a small spatial subset and immediately after the initial propagation time, will cause a change in the density within that subset, implying that the TD potential uniquely determines the TD density. In this proof, we assume unitary evolution of wavefunctions and first-order differentiability (which does not imply analyticity) in time of the internal and external force-densities, electronic density, current density, and their spatial derivatives over the small spatial subset and short time interval.

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16.

If the internal and external force densities satisfy this condition, then the density and current density meet this condition as well.

17.

In this work, the limit t0 is always taken along the right-hand side (positive axis).

18.

H.c.: Hermitian conjugate.

19.

For convenience, we use the mean-value theorem in the form f(ξ)(ba)=abf(s)ds, where a < ξ < b. Although it is common to take the intermediate value (ξ) such that aξb, it is easy to show that a < ξ < b also holds: Consider baf(s)ds=F(b)F(a), where F is the antiderative. Using the mean-value theorem for derivatives, we have F(b)F(a)=abf(s)ds=F(ξ)(ba), where F(ξ)=f(ξ) and a < ξ < b.

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