Abstract |
The optimal estimates of the eigenvalue gaps and eigenvalue ratios for the Sturm-Liouville operators have been of fundamental importance. Recently a series of works by Keller [7],Chern-Shen [3], Lavine [8], Huang [4] and Horvath [6] show that the first eigenvalue gap of the Schrodinger operator under Dirichlet boundary condition and the first eigenvalue ratio（λ2/λ1）of the string equation under Dirichlet boundary condition are dual problems of each other. Furthermore the problems when the potential functions and density functions are restricted to certain classes of functions can all be solved by a variational calculus method (differentiating the whole equation with respect to a parameter t to find λn'(t)) together with some elementary analysis. In this thesis, we shall give a short survey of these result. In particular, we shall prove $3$ pairs of theorems. First when q is convex (ρ is concave), then λ2-λ1≧3 （λ2/λ1≧4）.If q is a single well and its transition point is π/2 (ρ is a single barrier and its transition point is π/2), then λ2-λ1≧3（λ2/λ1≧4）.All these lower bounds are optimal when q(ρ) is constant. Finally when q is bounded (ρ is bounded), then λ2-λ1 is minimized by a step function (λ2/λ1 is minimized by a step function), after some additional conditions. We shell give a unified treatment to the above results. |