Three Types Of The Uniqueness Theorem Of Sturm-liouville Problems

It is well known that the Sturm-Liouville problems are derived from solid heat-conduction models which have widely applications, and often apply in mathematics, physics, geophysics and other branches of natural sciences. In particular, it is a basic mathematical method for describing micro-particles in a state exercises in quantum mechanics. More than a century, the spectral and inverse spectral theory of ordinary differential operator step by step to be an important theoretical research branch in the field of mathematics and physics. In particular, the study of inverse spectral problem aroused widespread attention of many mathematicians and physicists and obtained a lot of good theory results. This paper studies the uniqueness theorem of three Sturm-Liouville problems. that is, potential function and boundary conditions of corresponding problem can be determined uniquely by some spectral data. This paper contains three chapters:In chapter 1, the problem of the potential function being determined uniquely for the classic Sturm-Liouville equation-u″(x)+q(x)u(x)=λu(x), subject to the boundary conditions-u′(0)＋hv(0)＝0,u′(1)＋Hu(1)=0 is considered. Explicitly, if knowing partial information on the potential function and infinitely many partially spectra, we will give some sufficient conditions to determine potential function uniquely on the interval. This result promotes the conclusion for N spectra of Miklos Horvath. and makes the choose conditions of spectra from one group to infinite groups.In chapter 2. the problem of integro-differential equation with self-adjoint boundary conditions-y′(0)+hy(0)=0,y′(π)+Hy(π)=0 is discussed, we regard this equation as the perturbation of classical potential equa-tion. Then the inverse eigenvalue problem of Sturm-Liouville operator with afteref-fect is studied. We give some conditions which the common eigenvalues satisfy, when potential function and partial information on the kernel function are obtained. Our aim is to establish Simon theorem and Hochstadt's half inverse spectral theorem.In chapter 3, an inverse eigenvalue problem for a Sturm-Liouville equation-u″(x)+q(x)u(x)=λp(x)u(x), subject to the boundary conditions-u′(0)＋hu(0)＝0.u′(1)＋Hu(1)＝0 is studied. Namely, some sufficient, conditions to determine potential function uniquely are given, when density function and partial information on the potential are avail-able. We get Simon theorem for this system and promote this result to the case of 2 spectra even N spectra.