Unique Continuation for the Magnetic Schr\"odinger Equation

TL;DR

This paper proves the unique continuation property for the many-body magnetic Schrödinger equation, ensuring solutions that vanish on a positive measure set are identically zero, with implications for quantum chemistry.

Contribution

It establishes the unique continuation property for the magnetic Schrödinger equation with many-body potentials, including atomic and molecular Hamiltonians.

Findings

Unique continuation holds for solutions vanishing on positive measure sets.

Applicable to potentials composed of one-body and two-body functions.

Supports theoretical foundations in density-functional theories.

Abstract

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schr\"odinger equation. This property guarantees that if a solution of the Schr\"odinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.