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Resolvent expansions of 3D magnetic Schroedinger operators and Pauli operators

Arne Jensen, Hynek Kovarik

Abstract

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in $L^2(\mathbb{R}^3)$ and $L^2(\mathbb{R}^3;\mathbb{C}^2)$, respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g.~finite rank perturbations, are discussed as well.

Resolvent expansions of 3D magnetic Schroedinger operators and Pauli operators

Abstract

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in and , respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g.~finite rank perturbations, are discussed as well.
Paper Structure (10 sections, 22 theorems, 148 equations)

This paper contains 10 sections, 22 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. The set-up
  3. Properties of the free resolvent
  4. The factored resolvent equation
  5. Main results
  6. The case $V\geq 0.$
  7. The Pauli operator
  8. The case $V=0$.
  9. General perturbations
  10. Computation of $(m_{P,0}+S_{P,1})^{-1}$

Key Result

Lemma 2.2

There exists a vector potential $A\colon\mathbb{R}^3\to\mathbb{R}^3$ with $\mathop{\mathrm{curl}}\nolimits A=B$ such that

Theorems & Definitions (43)

  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 3.1: JK
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.3
  • Lemma 4.4: jn
  • Lemma 4.5: JK
  • ...and 33 more