Abstract |
The inverse spectral problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of eigenvalues plus some additional spectral data. The theory of transformation operators, first introduced by Marchenko, and then reinforced by Gelfand and Levitan, is a powerful method to deal with the different stages of the inverse spectral problem: uniqueness, reconstruction, stability and existence. In this thesis, we shall give a survey on the theory of transformation operators. In essence, the theory says that the transformation operator $X$ mapping the solution of a Sturm-Liouville operator $varphi$ to the solution of a Sturm-Liouville operator, can be written as $$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$ where the kernel $K$ satisfies the Goursat problem $$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0$$ plus some initial boundary conditions. Furthermore, $K$ is related by a function $F$ defined by 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, all the above relations are bilateral, that is $$qLeftrightarrow KLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.$$ hspace*{0.25in}We shall give a concise account of the above theory, which involves Riesz basis and order of entire functions. Then, we also report on some recent applications on the uniqueness result of the inverse spectral problem. |