Lower bounds of Dirichlet eigenvalues for degenerate elliptic operators and degenerate Schrödinger operators
Hua Chen, Peng Luo, and Shuying Tian
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Submission date: 19. Aug. 2013
MSC-Numbers: 35P05, 35P20
Keywords and phrases: Lower bounds for Dirichlet eigenvalues, finite degenerate elliptic operators, infinite degenerate elliptic operators, singular potential term
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Let X = (X1,X2,,Xm) be a system of real smooth vector fields defined in an open domain ⊂ ℝn, Ω ⊂⊂ be a bounded open subset in ℝn with smooth boundary ∂Ω, △X = ∑ j=1mXj2. In this paper, if λj is the jth Dirichlet eigenvalue for the degenerate elliptic operator -△X (or the degenerate Schrödinger operator -△X + V ) on Ω, we deduce respectively that the lower estimates for the sums ∑ j=1kλj in both cases for the operator -△X to be finitely degenerate (i.e. the Hörmander condition is satisfied) or infinitely degenerate (i.e. the Hörmander condition is not satisfied).