The fundamental gap of a kind of two dimensional sub-elliptic operator

TL;DR

This paper investigates the fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators, establishing key theoretical results including existence, spectral properties, and extremizing potentials under constraints.

Contribution

It introduces new existence and characterization theorems for extremizing potentials in sub-elliptic operators with norm constraints, advancing spectral theory in this context.

Findings

Existence of weak solutions established

Spectral theory for sub-elliptic operators developed

Characterization of extremizing potentials provided

Abstract

This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of sub-elliptic operators. We provide the existence and characterization theorems for extremizing potentials $V(x)$ when $V(x)$ is subject to $L^\infty$ norm constraint.