|Author's Email Address
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|Type of Document
||Optimal lower estimates for eigenvalue ratios of Schrodinger operators and vibrating strings|
|Date of Defense
modified Prufer substitution
vibrating string problems
||The eigenvalue gaps and eigenvalue ratios of the Sturm-Liouville systems have been studied in many papers. Recently, Lavine proved an optimal lower estimate of first eigenvalue gaps for Schrodinger operators with convex potentials. His method uses a variational approach with detailed analysis on different integrals. In 1999, (M.J.) Huang adopted his method to study eigenvalue ratios of vibrating strings. He proved an optimal lower estimate of first eigenvalue ratios with nonnegative densities. In this thesis, we want to generalize the above optimal estimate.|
The work of Ashbaugh and Benguria helps in attaining our objective. They introduced an approach involving a modified Prufer substitution and a comparison theorem to study the upper bounds of Dirichlet eigenvalue ratios for Schrodinger
operators with nonnegative potentials. It is interesting to see that the counterpart of their result is also valid.
By Liouville substitution and an approximation theorem, the vibrating strings with concave and positive densities can be transformed to a Schrodinger operator with nonpositive potentials. Thus we have the generalization of Huang's result.
||Jhishen Tsay - chair|
Chu-Hui Huang - co-chair
Chun-Kong Law - advisor
indicate access worldwide|
|Date of Submission