Consider the Sturm-Liouville system : 8 > > > > > < > > > > > : − y00 + q(x)y = y y(0) cos + y0(0) sin = 0 y(1) cos + y0(1) sin = 0 , where q 2 L 1 (0, 1) and , 2 [0, £¾). Let 0 < x(n)1 < x(n)2 < ... < x(n)n − 1 < 1 be the nodal points of n-th eigenfunction in (0,1). The inverse nodal problem involves the determination of the parameters (q, , ) in the system by the knowledge of the nodal points . This problem was first proposed and studied by McLaughlin. Hald-McLaughlin gave a reconstruc- tion formula of q(x) when q 2 C 1 . In 1999, Law-Shen-Yang improved a result of X. F. Yang to show that the same formula converges to q pointwisely for a.e. x 2 (0, 1), when q 2 L 1 . We found that there are some mistakes in the proof of the asymptotic formulas for sn and l(n)j in Law-Shen-Yang¡¦s paper. So, in this thesis, we correct the mistakes and prove the reconstruction formula for q 2 L 1 again. Fortunately, the mistakes do not affect this result.Furthermore, we show that this reconstruction formula converges to q in L 1 (0, 1) . Our method is similar to that in the proof of pointwise convergence.

In this thesis, we study two new inverse nodal problems introduced by Yang, Shen and Shieh respectively. Consider the classical Sturm-Liouville problem: $$ left{ egin{array}{c} -phi'+q(x)phi=la phi phi(0)cosalpha+phi'(0)sinalpha=0 phi(1)coseta+phi'(1)sineta=0 end{array} ight. , $$ where $qin L^1(0,1)$ and $al,ein [0,pi)$. The inverse nodal problem involves the determination of the parameters $(q,al,e)$ in the problem by the knowledge of the nodal points in $(0,1)$. In 1999, X.F. Yang got a uniqueness result which only requires the knowledge of a certain subset of the nodal set. In short, he proved that the set of all nodal points just in the interval $(0,b) (frac{1}{2}<bleq 1)$ is sufficient to determine $(q,al,e)$ uniquely. In this thesis, we show that a twin and dense subset of all nodal points in the interval $(0,b)$ is enough to determine $(q,al,e)$ uniquely. We improve Yang's theorem by weakening its conditions, and simplifying the proof. In the second part of this thesis, we will discuss an inverse nodal problem for the vectorial Sturm-Liouville problem: $$ left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x) A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f y}(1)+B_{2}{f y}'(1)={f 0} end{array} ight. . $$ Let ${f y}(x)$ be a continuous $d$-dimensional vector-valued function defined on $[0,1]$. A point $x_{0}in [0,1]$ is called a nodal point of ${f y}(x)$ if ${f y}(x_{0})=0$. ${f y}(x)$ is said to be of type (CZ) if all the zeros of its components are nodal points. $P(x)$ is called simultaneously diagonalizable if there is a constant matrix $S$ and a diagonal matrix-valued function $U(x)$ such that $P(x)=S^{-1}U(x)S.$ If $P(x)$ is simultaneously diagonalizable, then it is easy to show that there are infinitely many eigenfunctions which are of type (CZ). In a recent paper, C.L. Shen and C.T. Shieh (cite{SS}) proved the converse when $d=2$: If there are infinitely many Dirichlet eigenfunctions which are of type (CZ), then $P(x)$ is simultaneously diagonalizable. We simplify their work and then extend it to some general boundary conditions.

In this thesis, we study the inverse nodal problem and inverse spectral problem for various Sturm-Liouville operators, in particular, Hill's operators. We first show that the space of Schr"odinger operators under separated boundary conditions characterized by ${H=(q,al, e)in L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic to the partition set of the space of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge 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 exactly nodal sequence. Then we study the inverse nodal problem for Hill's equation, and solve the uniqueness, reconstruction and stability problem. We do this by making a translation of Hill's equation and turning it into a Dirichlet Schr"odinger problem. Then the estimates of corresponding nodal length and eigenvalues can be deduced. Furthermore, the reconstruction formula of the potential function and the uniqueness can be shown. We also show the quotient space $Lambda/sim$ is homeomorphic to the space $Omega={qin L^{1}(0,1) : int_{0}^{1}q = 0, q(x)=q(x+1) mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q$. Finally we show that if the periodic potential function $q$ of Hill's equation is single-well on $[0,1]$, then $q$ is constant if and only if the first instability interval is absent. The same is also valid for convex potentials. Then we show that similar statements are valid for single-barrier and concave density functions for periodic string equation. Our result extends that of M. J. Huang and supplements the works of Borg and Hochstadt.

The inverse nodal problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of the nodal data ( zeros of eigenfunction ). This problem was first defined by McLaughlin[12]. Up till now, the problem on finite intervals has been studied rather thoroughly. Uniqueness, reconstruction and stability problems are all solved. In this thesis, I investigate the inverse nodal problem on semi-infinite intervals q(x) is real and continuous on [0,1) and q(x)!1, as x!1. we have the following proposition. L is in the limit-point case. The spectral function of the differential operator in (1) is a step function which has discontinuities at { k} , k = 0, 1, 2, .... And the corresponding solutions (eigenfunction) k(x) = (x, k) has exactly k zeros on [0,1). Furthermore { k} forms an orthogonal set. Finally we also discuss that density of nodal points and a reconstruction formula on semiinfinite intervals.