Higher Approximation of Eigenvalues by the Sampling Method

We study the rate of convergence of the sampling method, which deals with the computation of eigenvalues of regular Sturm-Liouville problems. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Numerical examples are also provided.     

Eigenvalues of s-l systems using sampling theory

A new method based on Shannon's sampling theorem is introduced for the localization and approximation of the eigenvalues of regular Sturm-Liouville problems. Several examples have been presented to illustrate the effectiveness of the method.     

A uniqueness theorem for indefinite Sturm-Liouville operators

In this paper, we discuss the inverse problem for indefinite Sturm-Liouville operators on the finite interval [a, b]. For a fixed index n(n = 0, 1, 2, ··· ), given the weight function ω(x), we will show that the spectral sets {λ n (q, h a , h k )} +∞ k=1 and {λ-n (q, h b , h k )} +∞ k=1 for distinct h k are sufficient to determine the potential q(x) on the finite interval [a, b] and coefficients h a and h b of the boundary conditions.     

Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method

The purpose in this paper is to compute the eigenvalues of Sturm-Liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

     

左定与定型Sturm-Liouville问题间特征值不等式

应用左定和定型Sturm-Liouville问题特征值的Prüfer角刻画,以及其特征值对边界和边值条件的单调依赖关系,本文建立了左定Sturm-Liouville与两个相关的定型Sturm-Liouville问题之间的特征值不等式关系.     

Extremal Eigenvalues of the Sturm-Liouville Problems with Discontinuous Coefficients

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method.     

Matrix Riccati equations and matrix Sturm-Liouville problems

The asymptotic behavior of determinants of unitary solutions of matrix Riccati differential equations containing a large parameter is determined. The result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix Sturm-Liouville problems.     

左定Sturm-Liouville算子的特征值计算(英文)

研究了定义在有限区间[a,b]上的具有分离型和混合型边界条件的左定正则Sturm-Liouville算子的特征值问题.把具有混合型边界条件的左定正则Sturm-Liouville问题转化成二维的、具有分离型边界条件的右定正则Sturm-Liouville问题,给出了具有混合型边界条件的左定正则Sturm-Liouville算子的特征值的数值计算方法.     

Banach空间中非线性Sturm-Liouville问题的解

利用M(o)nch不动点定理,在更广泛的的条件下,证明了Banach空间中Sturm-Liouville问题解的存在性.     

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation

A partial inverse Sturm-Liouville problem: Matching a step function potential to a finite eigenvalue list

We investigate a certain inverse problem involving the Sturm-Liouville equation. In particular, given a finite list of target values, when can a potential function of a given form be found that produces these numbers as eigenvalues?     

一类正则Sturm—Liouville问题特征的渐近分析

考虑[0,π]上一类带一般分离型边界条件的正则Sturm—Liouville问题特征的渐近表示,利用Frechet导数,对特征值进行精细的分析,清楚地给出了方程系数q（x）及边界条件中常数cot α,col β对特征值的影响．使结论更具一般化．     

On sampling theory associated with the resolvents of singular Sturm-Liouville problems

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

     

Dependence of eigenvalues of regular Sturm-Liouville operators on the boundary condition

In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

Floquet theory for left-definite Sturm-Liouville problems

We determine the essential spectrum of left-definite Sturm-Liouville problems with periodic coefficients and a weight function which changes sign.     

Left-definite spaces of singular Sturm-Liouville problems

The left-definite Hilbert spaces for singular Sturm-Liouville problems of the limit-circle type are explicitly constructed. The construction only uses an explicit form of the left-definite boundary conditions, together with a principle solution and a non-principle solution of the differential equation involved.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

Left-definite boundary conditions of Sturm-Liouville problems when the interval shrinks to a point

On the condition that the interval of the problem shrinks to a point, we investigated the separated boundary conditions S α,β of left-definite Sturm-Liouville problem, and answered the following question: Is there a c 0 ∈ J such that S α,β is always left-definite or semi-left-definite for the Sturm-Liouville equation for each c ∈ (a, c 0 )?