The Critical Equation Method For Optimal Recovery Of Weights In Sturm-Liouville Eigenvalue Theory

The Sturm-Liouville theory originated from boundary value problems of ordinary differential equations.It has a wide range of applications in mathematical physics,numerical methods,geophysics,meteorology,and other fields.For the classical inverse spectral theory,since Borg discovered in 1946 that two sets of eigenvalues uniquely determine the potential function,many scholars have done extensive work on inverse spectral problems.However,Sturm-Liouville problems have infinitely countable eigenvalues,and it is difficult to obtain all the spectral information in reality.Therefore,this paper mainly studies the extremal problem of the L1-norm of weights in the case of given two eigenvalues,obtains the expression of the weight function at the optimal point,and establishes the relationship between the first two eigenvalue ratios and the extremal problem of the L1-norm of weights.The article utilizes the method of establishing the critical equation to analyze the intrinsic properties of the optimal point,thereby accomplishing the theorem’s proof.The main research work is arranged as follows:Chapter 1 introduces the signifi1ance of the extremal problem of the L1-norm of weights,the current state of research and the content and innovations of the study.Chapter 2 provides the relevant theoretical knowledge of the Sturm-Liouville problem and the related conclusions needed in this paper.Chapter 3 focuses on investigating the relationship between the optimal recovery problem of the weight function and the first two eigenvalue ratios in Sturm-Liouville problems.The proof is divided into three parts.In the first part,we utilize Mercer’s theorem and relevant theorems on the number of eigenvalues of a measure differential equation to show how the ratio of the first two eigenvalues affects the optimal recovery problem.Because the L1-norm is not Frechet differentiable,the second part of the study focuses on the extremal problem of the L1-norm of weights in both the LP space and the measure space in the case of given two eigenvalues.By using Pontrayagin’s method,a critical equation for the optimal points of weight functions in the Lp space is established,and the existence of the optimal points is demonstrated.Because the coefficient signs in the critical equation cannot be determined.Therefore,we first discuss the properties of optimal points,and then use the relevant results of weak*convergence in the measure space,as well as the uniqueness of optimal points and the strong continuity of the eigenvalues with respect to the weight function in the weak topology space to demonstrate that the optimal points in Lp space weak*converge to the optimal points in L1 space.The sign of the coefficients in the critical equation is determined based on the properties of the optimal points.The third part demonstrates the impact of the optimal recovery problem on the ratio of characteristic values.By establishing the continuity of the characteristic functions and their derivatives with respect to the weight function,the proof of the problem is completed by analyzing the critical equation as p→1.