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.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

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).     

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.     

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.     

Integro-differential properties of the singular Sturm-Liouville operator

We consider the properties of the positiveness, fractional power, boundedness, and separation of the singular Sturm-Liouville operator in a weighted space depending on the behavior of its coefficients. We derive necessary and sufficient conditions for its positiveness, trace class property, boundedness, and separation.     

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.     

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).     

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).     

Necessary and sufficient conditions for the solvability of the inverse problem for the matrix Sturm-Liouville operator

The matrix Sturm-Liouville operator on a finite interval with Dirichlet boundary conditions is studied. Properties of its spectral characteristics and the inverse problem of recovering the operator from these characteristics are investigated. Necessary and sufficient conditions on the spectral data of the operator are obtained. Research is conducted in the general case, with no a priori restrictions on the spectrum. A constructive algorithm for solving the inverse problem is provided.     

On the Sturm-Liouville operator

We consider the homogeneous Sturm-Liouville differential equation, prove the existence of two linearly independent solutions that form a fundamental system, and obtain some of their properties.     

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.