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.

We investigate spectral properties of limit-periodic Schrödinger operators in [cursive l] 2 ([Special characters omitted.] ). Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the orbits of a minimal translation of a procyclic group. This perspective was first proposed by Avila and further exploited by the author, which allows one to separate the base dynamics and the sampling function. Starting from this point of view, we conclude that all the spectral types (i.e. purely absolutely continuous, purely singular continuous, and pure point) can appear within the class of limit-periodic Schrödinger operators. We furthermore answer questions regarding how often a certain type of spectrum would occur and discuss the corresponding Lyapunov exponent. In the regime of pure point spectrum, we exhibit the first almost periodic examples that are uniformly localized across the hull and the spectrum.

In this thesis, we study two stochastic reaction diffusion models - the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching the boundary of a properly embedded open set, termed the reaction region (or more generally some fixed lower dimensional sub-manifold). The Doi model uses reaction potentials, supported in the reaction region, whereby two molecules react with a fixed probability per unit time, λ, upon entering the reaction region. The problem considered is that of obtaining estimates for convergence rates, in λ, of the Doi model to the Smoluchowski model. This problem fits into the theory of singular perturbation or optimization, depending on which reactive boundary conditions one considers, and we solve it - at least for the bimolecular reaction with one stationary target.

In this thesis, having in mind applications to the fault-detection/diagnosis of electrical networks, we consider some inverse scattering problems for the Zakharov-Shabat equations and time-independent Schrödinger operators over star-shaped graphs. The first chapter is devoted to describe reflectometry methods applied to electrical networks as an inverse scattering problems on the star-shaped network. Reflectometry methods are presented and modeled by the telegrapher's equations. Reflectometry experiments can be written as inverse scattering problems for Schrödinger operator in the lossless case and for Zakharov-Shabat system for the lossy transmission network. In chapter 2 we introduce some elements of the inverse scattering theory for 1 d Schrödinger equations and the Zakharov-Shabat system. We recall the basic results for these two systems and we present the state of art of scattering theory on network. The third chapter deals with some inverse scattering for the Schrödinger operators. We prove the identifiability of the geometry of the star-shaped graph: the number of the edges and their lengths. Next, we study the potential identification problem by inverse scattering. In the last chapter we focus on the inverse scattering problems for lossy transmission star-shaped network. We prove the identifiability of some geometric informations by inverse scattering and we present a result toward the identification of the heterogeneities, showing the identifiability of the loss line factor.