Estimates On Eigenvalues Of F-Laplacian And Schrodinger Operators

In this paper,we mainly study eigenvalue estimates of two kinds of operators:the first one is the eigenvalue estimates of f-Laplacian operator;the second one is the eigenvalue estimates of Schr(?)dinger operator.For f-Laplacian operator,we firstly obtain the lower bound estimate of the first Dirichlet eigenvalue on compact smooth Riemannian manifold with boundary.Then,by changing the condition of curvature,we get a better lower bound estimate of the first Dirichlet eigenvalue.Both estimates extend the results of Hongcang Yang.Finally,we obtain the lower bound estimate of the first Robin eigenvalue on compact smooth Riemannian manifold with boundary,and this extends the result of Hongcang Yang.For Schr(?)dinger operator,we mainly study the upper bound estimate of the gap between the first and second eigenvalues for the one-dimensional case.For Dirichlet eigenvalue problem,based on the study of Shing-Tung Yau et al,we obtain an inequality of the gap by Rayleigh-Ritz inequality.For Neumann eigenvalue problem,we obtain an estimate of the second eigenvalue by proving the monotonicity of the eigenvalue on the interval and the potential function.