Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains

TL;DR

This paper proves the non-degeneracy and uniqueness of solutions to a broad class of singular Toda systems on bounded domains, using spectral analysis of Cartan matrices, covering all simple Lie algebras.

Contribution

It introduces the first comprehensive results on non-degeneracy and uniqueness for Toda systems associated with any simple Lie algebra, including multiple singular sources.

Findings

Proves non-degeneracy of solutions for Toda systems.

Establishes uniqueness of solutions for these systems.

Applies spectral analysis of Cartan matrices to linearized problems.

Abstract

In this note we show non-degeneracy and uniqueness results for solutions of Toda systems associated to general simple Lie algebras with multiple singular sources on bounded domains. The argument is based on spectral properties of Cartan matrices and eigenvalue analysis of linearized Liouville-type problems. This seems to be the first result for this class of problems and it covers all the Lie algebras of any rank.