Optimal Recovery Of Potentials Of Sturm-Liouville Problems

Sturm-Liouville theory originated from Fourier method for solving solid heat conduction models.Since its emergence,it has been a vital mathematical tool in mathematical physics,earth meteorology,biological science and other fields.Especially in quantum mechanics,it is an important means to describe the motion of microscopic particles.Driven by the application in many fields,the research on the extreme value problems and inverse spectral problems in S-L theory have attracted the interest of many scholars at home and abroad.For classical inverse spectral problems,we generally need to know two sets of eigenvalues to uniquely determine the potential function.But in real life,we can only measure a finite number of eigenvalues.Especially in quantum mechanics,the eigenvalue is the basic physical quantity that can only be observed,and we can only measure the first eigenvalue.This paper mainly studies the optimal recovery of potentials with finite eigenvalues,in other words,we find the optimal potential function with known finite eigenvalues.Here,the optimization is proposed because the potential function is not unique at this time,so we can only recover the potential function under the optimal target,which ensures the uniqueness of the potential function.We give the concrete expression of infimum of the L1-norm of potentials in the case of a single eigenvalue.An outline of this paper is as follows:The first chapter is the introduction,which mainly introduces the background,significance,current situation of the research on the optimal recovery of potentials and the main content and innovation points of this paper.The second chapter is the preliminary knowledge,which gives the related theory of optimal recovery of potentials and the preparation knowledge needed in this paper.In the third chapter,the optimal recovery problem of potentials under separated boundary conditions is studied.Since boundary conditions are different from previous studies,negative eigenvalues appear in the problem we study,and the positivity of Green’s function is not satisfied.So Mercer’s theorem cannot be directly applied to our problem.Of course,we can solve the problem by critical equations,but the structure of the corresponding critical equations will be more complicated,since the potential q(x)is involved in critical equations.It is difficult for us to draw conclusions by analyzing its properties.However we find that optimal potentials are attained by the distributions in most of the optimal recovery problems,so that our method is to select the suitable potential function with distributions,and then verify it by Rayleigh-Ritz principle.By this way,we first get the specific expression of infimum of the L1-norm of potentials when the first eigenvalue is known.Then we use the zero-point property of the eigenfunctions to transform the optimal recovery problem of potentials with the n-th eigenvalue known into the optimal recovery problem of potentials with the first eigenvalue known.And the main results of this paper are obtained.After that,we give the maximum and minimum eigenvalues of the potential function on L1 sphere.Finally,we find that our results are quite different from the previous ones,and we give a reasonable explanation by the discontinuity of the eigenvalues on the boundary conditions.