### Title page for etd-0704105-132441

URN etd-0704105-132441 Yan-Hsiou Cheng No Public. This thesis had been viewed 5205 times. Download 1831 times. Applied Mathematics 2004 2 Ph.D. English Inverse Problems for Various Sturm-Liouville Operators 2005-06-27 94 Sturm-Liouville problem spectral problem nodal problem Hill's equation In this thesis, we study the inverse nodal problem and inversespectral problem for various Sturm-Liouville operators, inparticular, Hill's operators.We first show that the space of Schr"odinger operators underseparated boundary conditions characterized by \${H=(q,al, e)inL^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}\$ is homeomorphicto the partition set of the space of all admissible sequences \$X={X_{k}^{(n)}}\$ which form sequences thatconverge to \$q, al\$ and \$ e\$ individually. The definition of\$Gamma\$, the space of quasinodal sequences, relies on the \$L^{1}\$convergence of the reconstruction formula for \$q\$ by the exactlynodal sequence.Then we study the inverse nodal problem for Hill's equation, andsolve the uniqueness, reconstruction and stability problem. We dothis by making a translation of Hill's equation and turning itinto a Dirichlet  Schr"odinger problem. Then the estimates ofcorresponding nodal length and eigenvalues can be deduced.Furthermore, the reconstruction formula of the potential functionand the uniqueness can be shown. We also show the quotient space\$Lambda/sim\$ is homeomorphic to the space \$Omega={qinL^{1}(0,1) :int_{0}^{1}q = 0, q(x)=q(x+1)mbox{on} mathbb{R}}\$. Here the space \$Lambda\$ is a collectionof all admissible sequences \$X={X_{k}^{(n)}}\$ which form sequences thatconverge to \$q\$.Finally we show that if the periodic potential function \$q\$ ofHill's equation is single-well on \$[0,1]\$, then \$q\$ is constant ifand only if the first instability interval is absent. The same isalso valid for convex potentials. Then we show that similarstatements are valid for single-barrier and concave densityfunctions for periodic string equation. Our result extends that ofM. J. Huang and supplements the works of Borg and Hochstadt. Chao-Liang Shen - chair Tzon-Tzer Lu - co-chair Jenn-Nan Wang - co-chair Chiu-Ya Lan - co-chair Chung-Tsun Shieh - co-chair Wei-Cheng Lian - co-chair Chao-Nien Chen - co-chair Tzy-Wei Hwang - co-chair Chun-Kong Law - advisor indicate accessible in a year 2005-07-04

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