Some properties of the potential-to-ground state map in quantum mechanics

TL;DR

This paper investigates the mathematical properties of the potential-to-ground state map in quantum mechanics, revealing its continuity, compactness, and the ill-posedness of related inverse problems, which impacts computational approaches.

Contribution

It proves the path-connectedness of the space of binding potentials and analyzes the local continuity and differential properties of the map, highlighting issues in inverse problem stability.

Findings

The space of binding potentials is path-connected.

The potential-to-ground state map is locally weak-strong continuous.

The differential of the map is compact, leading to ill-posed inverse problems.

Abstract

We analyze the map from potentials to the ground state in static many-body quantum mechanics. We first prove that the space of binding potentials is path-connected. Then we show that the map is locally weak-strong continuous and that its differential is compact. In particular, this implies the ill-posedness of the Kohn-Sham inverse problem.