Ground-state inverse potentials in the analytic setting

Establish the existence of an external potential reproducing prescribed ground-state densities in the same analytic regularity setting as the time-dependent inverse problem, thereby extending the paper’s inverse-potential theory to ground states.

Background

The paper proves a local-in-time inverse result for time-dependent densities on the torus, assuming analytic spatial regularity and strict positivity of the density. The authors identify the corresponding existence theory for ground states in the same analytic framework as an important unresolved direction.

This problem concerns the static analogue of the time-dependent density-to-potential map and is distinct from the already established time-dependent theorem.

References

Important open problems are the existence of $V$ for ground states in the same (analytic) setting, or the extension to point nuclei.

— The inverse problem of time-dependent density functional theory on the torus  (2609.08984 - Lauritsen et al., 8 Sep 2026) in Section 1, Introduction

The case of higher dimensions is largely open.

— The inverse problem of time-dependent density functional theory on the torus  (2609.08984 - Lauritsen et al., 8 Sep 2026) in Remark 2.14, “Non-interacting and interacting $v$-representability”