### Title page for etd-0704105-182611

URN etd-0704105-182611 YU-HAO LEE m922040010@student.nsysu.edu.tw This thesis had been viewed 5220 times. Download 1492 times. Applied Mathematics 2004 2 Master English The theory of transformation operators and its application in inverse spectral problems 2005-06-03 63 eigenvalue Transformation operator spectral problem eigenvector The inverse spectral problem is the problem ofunderstanding the potential function of the Sturm-Liouvilleoperator from the set of eigenvalues plus some additionalspectral data. The theory of transformation operators, firstintroduced by Marchenko, and then reinforced by Gelfand andLevitan, is a powerful method to deal with the different stagesof the inverse spectral problem: uniqueness, reconstruction,stability and existence. In this thesis, we shall give a surveyon the theory of transformation operators. In essence, the theorysays that the transformation operator \$X\$ mapping the solution ofa Sturm-Liouville operator \$varphi\$ to the solution of aSturm-Liouville operator, can be written as \$\$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,\$\$ where thekernel \$K\$ satisfies the Goursat problem\$\$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0\$\$ plus some initial boundaryconditions. Furthermore, \$K\$ is related by a function \$F\$ definedby the spectral data \${(lambda_{n},alpha_{n})}\$ where\$alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2})^{frac{1}{2}}\$through the famous Gelfand-Levitan equation\$\$K(x,y)+F(x,y)+int_{o}^{x}K(x,t)F(t,y)dt=0.\$\$ Furthermore, allthe above relations are bilateral, that is \$\$qLeftrightarrowKLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.\$\$hspace*{0.25in}We shall give a concise account of the abovetheory, which involves Riesz basis and order of entire functions.Then, we also report on some recent applications on theuniqueness result of the inverse spectral problem. none - chair none - co-chair C.K.Low - advisor indicate access worldwide 2005-07-04

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