Lower bounds of Dirichlet eigenvalues for degenerate elliptic operators and degenerate Schrödinger operators
Hua Chen, Peng Luo, Shuying Tian · 2013
Let X = (X1, X2, · · · , Xm) be a system of real smooth vector fields defined in an open domain Ω ⊂ R, Ω ⊂⊂ Ω be a bounded open subset in R with smooth boundary ∂Ω, △X = ∑m j=1 X 2 j . In this paper, if λj is the j th Dirichlet eigenvalue for the degenerate elliptic operator −△X (or the degenerate Schrodinger operator −△X + V ) on Ω, we deduce respectively that the lower estimates for the sums ∑k j=1 λj in both cases for the operator −△X to be finitely degenerate (i.e. the Hormander condition is satisfied) or infinitely degenerate (i.e. the Hormander condition is not satisfied).