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Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian

Louis Garrigue

Abstract

We prove the strong unique continuation property for many-body Pauli operators with external potentials, interaction potentials and magnetic fields in $L^p\loc(\R^d)$, and with magnetic potentials in ${L^{q}\loc(\R^d)}$, where ${p > \max(2d/3,2)}$ and ${q > 2d}$. For this purpose, we prove a singular Carleman estimate involving fractional Laplacian operators.

Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian

Abstract

We prove the strong unique continuation property for many-body Pauli operators with external potentials, interaction potentials and magnetic fields in , and with magnetic potentials in , where and . For this purpose, we prove a singular Carleman estimate involving fractional Laplacian operators.

Paper Structure

This paper contains 11 sections, 10 theorems, 94 equations.

Table of Contents

  1. Main results
  2. Carleman estimates for singular weights
  3. Unique continuation properties
  4. Hohenberg-Kohn theorems in presence of magnetic fields
  5. Fixed magnetic fields
  6. Ill-posedness of the Hohenberg-Kohn theorem for Spin-Current DFT
  7. Hohenberg-Kohn for the Maxwell-Schrödinger model
  8. Proofs of Carleman inequalities
  9. Proof of Theorem \ref{['carl']}
  10. Proof of Corollary \ref{['fraccarl']}
  11. Proof of the strong unique continuation property

Key Result

Theorem 1.1

Let $0 < \alpha \leqslant 1/2$, and let us define $\phi(x) \vcentcolon= -\ln \left|x\right| + (-\ln \left|x\right|)^{-\alpha}$ for $\left|x\right| \leqslant 1/2$. In dimension $n$, there exist constants $c_{n}$ and $\tau_n \geqslant 1$ such that for any $\tau \geqslant \tau_n$ and any ${u \in C^{\in

Theorems & Definitions (15)

  • Theorem 1.1: Carleman inequality
  • Corollary 1.2: Fractional Carleman inequality
  • Theorem 1.3: Strong UCP for systems with gradients
  • Corollary 1.4: Strong UCP for the many-body Pauli operator
  • Theorem 1.5: Hohenberg-Kohn with a fixed magnetic field
  • Remark 1.6
  • Theorem 1.7: Hohenberg-Kohn for Maxwell DFT
  • proof
  • Proposition 3.1: Figueiredo-Gossez with magnetic term
  • proof
  • ...and 5 more