Unique continuation for many-body Schr\"odinger operators and the Hohenberg-Kohn theorem

TL;DR

This paper establishes the strong unique continuation property for many-body Schrödinger operators with certain potentials and proves the Hohenberg-Kohn theorem under these conditions, advancing the mathematical foundation of Density Functional Theory.

Contribution

It proves the unique continuation property and the Hohenberg-Kohn theorem for many-body Schrödinger operators with potentials in $L^p_{loc}$, independent of particle number.

Findings

Strong unique continuation property proven for specified Schrödinger operators.

Hohenberg-Kohn theorem established under new potential regularity conditions.

Results hold independently of the number of particles.

Abstract

We prove the strong unique continuation property for many-body Schr\"odinger operators with an external potential and an interaction potential both in $L^p_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$, independently of the number of particles. With the same assumptions, we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.