Green's Formulas and Poisson's Equation for Bosonic Laplacians

TL;DR

This paper introduces Green's formulas and solves Poisson's equation for bosonic Laplacians, a class of conformally invariant differential operators, revealing their self-adjointness and providing representation formulas for solutions.

Contribution

It develops Green's formulas and solution techniques for Poisson's equation specifically for bosonic Laplacians, connecting higher spin operators with classical potential theory.

Findings

Green's formulas for bosonic Laplacians derived

Poisson's equation solved for these operators

Bosonic Laplacians shown to be self-adjoint

Abstract

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given $L^2$ inner product on certain compact supported function spaces.