# Integrating Factors for Dirac-Schrodinger Operators: Improving Eigenvalue Estimates and Applications to Charged Positive Mass Theorems Outside Horizon(s)

## TL;DR

This paper investigates conditions for the existence of integrating factors for Dirac-Schrodinger operators on spin manifolds, leading to improved eigenvalue estimates and applications to charged positive mass theorems in mathematical physics.

## Contribution

It introduces criteria for when Dirac-Schrodinger operators admit integrating factors, extending previous results to higher dimensions with applications in general relativity.

## Key findings

- Derived conditions for integrating factors of Dirac-Schrodinger operators.
- Extended positive mass theorems to higher-dimensional manifolds with electric fields.
- Generalized Herzlich's theorem to dimensions n ≥ 3 with additional geometric structures.

## Abstract

Let $(M^{n}, g)$ denote a Riemannian spin manifold of dimension $n$ with Dirac operator $D$ induced from the Levi-Cevita connection acing on the spinor bundle, $S$ ($D$ is also called the Atiyah-Singer Operator). Let $c: Cl(TM^{n}) \rightarrow End(S)$ be the standard representation of the Clifford Algebra as endomorphisms of the spinor bundle. Let $B \in End(S)$ be a zeroth-order endomorphism of the spinor bundle; given an in an orthonormal frame, $e_{j} \in TM^{n}$ by the expression $B=f^{\alpha}c(e_{\alpha})$ where the sum is taken over multi-indices, $\alpha = (i_{j_{m}}), \ m = 1, \, 2, \, 3 \, ... ,\ k$, $j_{1}< j_{2} < ... < j_{k}$ and each $f^{\alpha} \in C^{\infty}(M^{n})$. The purpose of this paper is investigate when the Dirac-Schrodinger operator $D + B$ has an integrating factor, i.e. when does there exist an invertible endomorphism $A \in End(S)$ such that $D(AB)=AD+AB$. This has applications to improving eigenvalue estimates for Dirac-Schrodinger operators and proving positive charged positive mass theorems where such operators appear on the boundary. Of particular interest is the case $n = 2$, for boundary Dirac operators of this form appear in charged positive mass theorems based on the initial data formulation in mathematical general relativity. It allows us to generalize a theorem of M. Herzlich (set-forth in his attempt to prove the Riemannian Penrose-inequality using spinors, cf. [1]) to a manifold of dimension $n \geq 3$ containing an electric field and symmetric two-tensor representing the second-fundamental form.