  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.
