Title page for etd-0911104-022942


[Back to Results | New Search]

URN etd-0911104-022942
Author Hsien-kuei Huang
Author's Email Address jaga_1@pchome.com.tw
Statistics This thesis had been viewed 4719 times. Download 1366 times.
Department Applied Mathematics
Year 2003
Semester 2
Degree Master
Type of Document
Language English
Title Optimal estimates of the eigenvalue gap and eigenvalue ratio with variational
Date of Defense 2004-06-11
Page Count 48
Keyword
  • optimal estimates
  • eigenvalue
  • 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.
    Advisory Committee
  • W. C. Lian - chair
  • Tzon-Tzer Lu - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0911104-022942.pdf
  • indicate access worldwide
    Date of Submission 2004-09-11

    [Back to Results | New Search]


    Browse | Search All Available ETDs

    If you have more questions or technical problems, please contact eThesys