The uniqueness of the solution of dual equations of an inverse indefinite Sturm-Liouville problem

In this paper we consider a linear second-order equation of Sturm-Liouville type:

(I)     

An inverse Sturm-Liouville problem for an impedance

A numerical method for reconstructing an impedance in a Sturm-Liouville operator from finitely many eigenvalues is investigated. The method constructs an impedance that has the given eigenvalues by finding a zero of a nonlinear finite dimensional map. A Newton scheme is investigated and numerical examples are considered.     

A finite-difference algorithm for an inverse Sturm-Liouville problem

We study a method for approximating a potential q(x) in y(0)=y()=0 from finite spectral data. When the potential is symmetric,the data are the first M Dirichlet eigenvalues. In the generalcase, the first M terminal velocities are also specified. Acentred finite-difference scheme reduces the inverse Sturm-Liouvilleproblem to a matrix inverse eigenvalue problem. Our approachis motivated by the work of Paine, de Hoog and Anderssen, whoinvestigated the discrepancy between continuous and matrix eigenvaluesunder finite differences. Our modified Newton scheme is basedon choosing the number of interior mesh points in the discretizationto be 2M. The modified Newton scheme is shown to be convergentfor both the case of a symmetric and general potential. Somenumerical experiments are given. Supported in part by Institute for Scientific Computation,Texas A&M University.     

On an inverse problem for the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We derive sufficient conditions for an entire analytic function to be the characteristic determinant of this boundary value problem.     

三组谱的Sturm-Liouville反问题

考虑与三组谱关联的逆Sturm-Liouville问题,证明了若对于给定的两组数列,在一定条件下,可划分为三组数列,使其分别为区间[0,a]上三个Sturm-Liouville问题的部分特征值,则通过三组谱的部分特征值能唯一确定区间[0,a]上的势函数q(x).     

On a local uniqueness result for the inverse Sturm-Liouville problem

A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associatedm-function near −∞. The proof given is based on relations between the classical transformation operators and them-function.     

On inverse problem for singular Sturm-Liouville operator from two spectra

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ _n } and {λ _n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference . Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 132–138, January, 2006.     

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

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.     

On the inverse Sturm-Liouville problem for spatially symmetric operators,III

In this note it is proved that x(·) a boundary trajectory of a Lipschitz-continuous differential inclusion ? ? F(t, x), x(0) = 0, the tangent cone to F(t, x(t)) at ?(t) that of attainable set E(t) at x(t) coincide for almost every t provided that ?F(t, x) is smooth (similar results with more stringent assumptions were obtained by H. Hermes (J. Differential Equations3 (1967), 256–270) and S. ?ojasiewicz, Jr. (Asterisque75–76 (1980), 187–197)). It is also proved that the outward normal to these cones along the trajectory is Lipschitz-continuous (in t). Moreover, using the lower, one-side, directional derivative instead of F. H. Clarke's generalised gradient, first-order necessary conditions are obtained, which can be stronger than those of Clarke (in “International Symposium on the Calculus of Variation and Optimal Control, University of Wisconsin, Madison, Wisconsin, September 1975”). The main ideas of this paper were presented in J. Hale's seminar at Brown University (March 1976).     

An inverse problem for Sturm-Liouville operators on arbitrary compact spatial networks

An inverse spectral problem is studied for Sturm-Liouville differential operators on arbitrary compact graphs (spatial networks). A uniqueness theorem of recovering operators from their spectra is proved, and a constructive procedure for the solution of the inverse problem is provided.     

On the inverse Sturm-Liouville problem for spatially symmetric operators,II

Using a third order Picard-Fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle, which is born from a Hopf bifurcation and dies in a saddle connection. This removes a superfluous hypothesis in Theorem 3.2, Chapter 13 of S. N. Chow and J. K. Hale (“Methods of Bifurcation Theory,” Springer-Verlag, New York, 1982).     

Calculating the eigenvalues of the Sturm-Liouville problem with a fractal indefinite weight

An efficient method is proposed for calculating the eigenvalues of the boundary value problem ?y″ ? λρy = 0, y(0) = y(1) = 0, where ρ ? \(W_2^{^\circ - 1} \) [0, 1] is the generalized derivative of a self-similar function P ? L ₂[0, 1].