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Quantitative Hardy inequality for magnetic Hamiltonians

Luca Fanelli, Hynek Kovarik

Abstract

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.

Quantitative Hardy inequality for magnetic Hamiltonians

Abstract

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.
Paper Structure (6 sections, 13 theorems, 132 equations)

This paper contains 6 sections, 13 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. A Hardy inequality for the Pauli operator
  3. Radial magnetic fields with compact support
  4. Hardy inequality for magnetic Schrödinger operators
  5. Aharonov-Bohm magnetic field
  6. One-dimensional weighted Hardy inequalities

Key Result

Theorem 2.2

Let $B$ satisfy Assumption ass-B and suppose that $0<\alpha \not\in\mathbb{Z}$. Let $A\in L^\infty(\mathbb{R}^2;\mathbb{R}^2)$ be such that $\nabla \times A=B$ in the sense of distributions. Then holds for all $u\in H^1(\mathbb{R}^2)$ and all $\rho>0$ with $\beta_{+}(B;\rho)$ is defined in beta.

Theorems & Definitions (23)

  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Proposition 2.8
  • proof
  • ...and 13 more