Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unique Continuation for the Magnetic Schrödinger Equation

Andre Laestadius, Michael Benedicks, Markus Penz

Abstract

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger 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.

Unique Continuation for the Magnetic Schrödinger Equation

Abstract

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger 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.

Paper Structure

This paper contains 7 sections, 9 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Unique-continuation property and the Hohenberg--Kohn theorem
  3. Prerequisites
  4. Main Results
  5. Application to CDFT
  6. Proofs of the main results
  7. Conclusion

Key Result

Theorem 2

Let $N\geq 1$. Assume that $W_A\in L_\mathrm{loc}^{3N/2}(\mathbb R^{3N})$ and each component of $A$ is an element of $L_\mathrm{loc}^{3N}(\mathbb R^{3N})$. Then the Schrödinger equation has the UCP from sets of positive measure, i.e., if a solution $\psi\in H_\mathrm{loc}^{2N/(N+2)}(\mathbb R^{3N})$

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Theorem 2: Theorem 1.2 in Regbaoui Regbaoui
  • Remark 2
  • Definition 3
  • Remark 3
  • Remark 4
  • Theorem 4
  • Lemma 5: Corollary 1.1 in Kurata Kurata2
  • proof : Proof of Theorem \ref{['Thm:adaptedK']}
  • ...and 14 more