微分流形上Laplace型算子的主特征值估计

The Principal Eigenvalue Estimations of La-place Type Operators on DifferentialManifold

以特征值估计的Li-猜想与Yang-猜想的提出与发展为基础,分类研究总结典型的黎曼流形上Laplace算子主特征值估计条件改变时估计值上下界的变化,得到最新的更加精确的估计结果。主要研究一般黎曼流形上的p-Laplacian的主特征值估计;将黎曼度量推广到Finsler流形上的主特征值估计;以及引入势函数后特征函数改变而得到新的Laplace型算子——Schrödinger算子的主特征值的估计。特征值估计的研究体现出微分流形与广义相对论的紧密联系,能够促进量子力学中能谱等问题的解决,为量子力学、量子光学和固体物理提供新方法。

Based on the eigenvalue estimations of Li-Conjecture’s and Yang-Conjecture’s proposed and de-veloped ideas, this paper summarized the variation of the upper and lower bounds of the estimated value when the Laplace operator principal eigenvalue estimation conditions were changed on a typical Riemannian manifold, and yielded some precise estimation results. The principal eigenvalue estimation of p-Laplacian on general Riemannian manifolds is studied. The estimation of principal eigenvalues of Finsler manifolds is studied. Since the potential function is introduced, the eigenfunction is changed, then the estimation of the principal eigenvalue of the new Laplace type operator-Schrödinger operator is studied. It shows the close connection between Riemannian manifold and general relativity. It also can simplify the solution of energy spectrum and other problems in quantum mechanics, and provide some new methods for quantum mechanics, quan-tum optics and solid physics.