Probabilistic Method to Fundamental gap problems on the sphere

TL;DR

This paper extends probabilistic methods to prove the fundamental gap estimate for Schrödinger operators on convex domains on the sphere, generalizing previous Euclidean results using reflection coupling on Riemannian manifolds.

Contribution

It introduces a probabilistic proof for the fundamental gap on spherical domains, expanding the scope of prior Euclidean and Laplacian results with reflection coupling techniques.

Findings

Probabilistic proof of the fundamental gap estimate on the sphere

Extension of Euclidean results to spherical convex domains

Application of reflection coupling on Riemannian manifolds

Abstract

We provide a probabilistic proof of the fundamental gap estimate for Schr\"odinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei, as well as by C. He, G. Wei, and Qi S. Zhang. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.