Resolvent expansions of 3D magnetic Schroedinger operators and Pauli operators

TL;DR

This paper derives detailed asymptotic expansions of the resolvent operators at the spectrum threshold for 3D magnetic Schrödinger and Pauli operators, using novel factorization techniques for first-order differential perturbations.

Contribution

It introduces a new factorization method for first-order differential perturbations, enabling explicit asymptotic resolvent expansions for magnetic Schrödinger and Pauli operators.

Findings

Explicit asymptotic expansions of resolvents at zero energy.

Leading and sub-leading terms are calculated explicitly.

Factorization schemes are applicable to more general perturbations.

Abstract

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schr\"odinger 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.