Solvability of a class of evolution operators on compact Lie groups

TL;DR

This paper establishes conditions for solving certain first-order evolution operators on compact Lie groups, with detailed analysis on the three-sphere and extensions to product groups, advancing understanding of their spectral properties.

Contribution

It provides new sufficient conditions for the solvability of Vekua-type evolution operators on compact Lie groups, including explicit criteria for the three-sphere case.

Findings

Derived explicit solvability criteria for the three-sphere

Extended results to finite products of compact Lie groups

Analyzed spectral behavior of invariant vector fields

Abstract

This paper provides sufficient conditions for the solvability of a class of first-order evolution operators of Vekua-type on the product of a one-dimensional torus and a compact Lie group. The conditions are expressed in terms of the time-dependent coefficients and the spectral behavior of a normalized left-invariant vector field on the group. The three-sphere case is discussed in detail, leading to more explicit criteria, and the main results are further extended to operators defined on finite products of compact Lie groups.