Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

In this paper, we consider the optimal recovery of potentials for a Sturm-Liouville problem -y′′+qy=λy,y(0)=0=y(1)-hy′(1),0<h<1,q∈L1[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-y''+qy= \lambda y, y(0)=0=y(1)-hy'(1), 0<h<1, q\in L^1[0,1]$$\end{document} with only one given eigenvalue. Denote by λn(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _n(q)$$\end{document} the n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-$$\end{document}th eigenvalue of this problem. For λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {R}$$\end{document}, denote by Ωn(λ)=q:q∈L1[0,1],λn(q)=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{n}(\lambda )=\left\{ q: q \in L^{1}[0,1], \lambda _{n}(q)=\right. $$\end{document}λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left. \lambda \right\} $$\end{document}, n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 1$$\end{document} and En(λ)=inf‖q‖:q∈Ωn(λ).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{n}(\lambda )=\inf \left\{ \Vert q\Vert : q \in \Omega _{n}(\lambda )\right\} .$$\end{document} The optimal recovery of potential function in this paper refers to finding the infimum of the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-norm for potential function in the set Ωn(λ).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{n}(\lambda ).$$\end{document} We will obtain a formula for En(λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{n}(\lambda )$$\end{document} and specify where the infimum can be attained. Our results are closely related to the discontinuity of the eigenvalues with respect to the boundary conditions. Since the optimal recovery problem with only one fixed eigenvalue is just the duality problem to the extremum problem of eigenvalues, we also give the extremum of the n-th eigenvalue of a problem for potentials on a sphere in L1[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1[0, 1]$$\end{document}.

For λ ∈ R, denote by the set of all potentials in L 1 [0, 1] with the same n-th eigenvalue λ. From [1, p.68], one knows that n (λ) is infinite and unbounded subsets of L 1 [0, 1]. Let E n (λ) = inf { q : q ∈ n (λ)} , (1.4) where · stands for the L 1 -norm. Sturm-Liouville spectral theory originated from the Fourier method for solving the solid heat conduction model. It is an important mathematical tool in the fields of mathematical physics, earth meteorology and biological science. For example, in quantum mechanics, Sturm-Liouville spectral theory is an important method for describing the motion of microscopic particles. Driven by many practical problems, the inverse spectral problem and extremum problem in Sturm-Liouville spectral theory have also attracted the interest of many scholars [2][3][4][5]. For classical inverse spectrum problems, it is generally necessary to know two infinite sets of eigenvalues to uniquely determine the potential function. However, in practical problems, we can only measure a finite number of eigenvalues. For example, in quantum mechanics, eigenvalues are only basic physical quantities that can be observed, and even only the first eigenvalue can be measured. Therefore, under the condition of known finite eigenvalues, how to determine the optimal potential function is a problem with both application background and theoretical research value.
In this paper, we will give explicit expression of E n (λ) in terms of λ and the parameter h for n ≥ 1. Such problem is called the optimal recovery problem of potentials in spectral theory. The reason why the optimal potential function is proposed here is that the potential function is not unique at this time, so the potential function can only be restored under the optimal objective, and the uniqueness of the potential function is guaranteed under the optimal objective.
The optimal recovery of potential function for Sturm-Liouville problem can be regarded as a new type of inverse spectral problem. The classical inverse spectral theory study the existence and uniqueness of the potentials from suitable spectral data.
In 1946, Borg [6] gave the fundamental theorem that two infinite sets of eigenvalues uniquely determine the potential. Since then, many scholars have carried out in-depth research and generalization of Borg's results in [7,8]. In 1978, Hochstadt gave the semi-inverse spectrum theorem in [9], which states that the potential can be uniquely determined from a potential on a half interval and one set of eigenvalues. Recently, A Sinan Ozkan andİbrahim Adalar consider a half-inverse Sturm-Liouville problem on a time scale in [10], and give a Hochstadt-Lieberman-type theorem for a Sturm-Liouville dynamic equation with Robin boundary conditions. Other studies on semi inverse spectral problems can be also founded in [1,[11][12][13][14].
Il'yasov and Valeev [3,4,15] dealt with inverse optimization spectral problem: for a given q 0 and {λ i } m i=1 , m < +∞, find a potentialq closest to q 0 in a prescribed norm, such that λ i = λ i (q) for all i = 1, ..., m. The optimal recovery of potentials for Sturm-Liouville eigenvalue problem with different boundary conditions were considered in [16,17]. Qi and Chen in [17] give an expression for E n (λ) for the problem (1.1) under Dirichlet boundary condition by using the generalized Lyapunov-type inequalities. Qi and Guo [16] obtained an expression for E n (λ) for the problem (1.1) subject to boundary conditions y(0) + k 1 y (0) = y(1) − k 2 y (1) = 0, where k 1 , k 2 ≤ 0 by using Mercer theorem.
On the other hand, the optimal recovery of potentials for Sturm-Liouville eigenvalue problem is closely related to the extremum problem of eigenvalues. In fact, the optimal recovery problem for only one fixed eigenvalue is just the duality problem to the extremum problem of eigenvalues.
The extremum problem of eigenvalues goes back to the famous problem of Lagrange problem [18]. In 1955, Krein [19] studied the following problem −y = λwy, y(0) = 0 = y(1), where w ≥ 0, bounded and 1 0 w = 1. It gives maximum and minimum values of eigenvalues. In this period, the extremum problem of eigenvalues mainly focus on the study of extremum of eigenvalues for potentials belonging to a certain class of functions, such results can be found in [2,[20][21][22]. In 2009, Zhang and his cooperators [23,24] studied the eigenvalue problem (1.1) with Dirichlet boundary condition, Neumann boundary condition and periodic boundary condition. The extremum problem of the n-th eigenvalues for potentials in L 1 [0, 1] space is solved by using critical equations. Applying this method, Qi and Xie [25] gave the concrete expression of infimum of the weights of the problem −y = λwy with boundary condition (1.2). Since the set (λ 1 , λ 2 , . . . , λ n ) be an infinite set in L 2 [0, 1], and hence in L 1 [0, 1] due to 1]. So that, the uniqueness of potential in the optimal recovery problem does not hold. Therefore, the recovery of potential function in this paper refers to finding the infimum of the L 1 -norm for potential function in the set (λ 1 , λ 2 , . . . , λ n ).
For the optimal recover problem, we also study the uniqueness and the existencewhether or not the infimum is attained in (λ 1 , λ 2 , . . . , λ n ). Furthermore, we will give the quantitative representation of the optimal potential. Our method in this paper is different from that used in [3,4,[15][16][17]. In this paper, the Sturm-Liouville problem with separated boundary conditions has negative eigenvalues. At this time, the positivity of Green's function is no longer satisfied, so it is impossible to directly apply Mercer's theorem to study this problem. Considering that the structure of the critical equation corresponding to problem (1.1) subject to (1.2) is very complex, it is difficult to obtain the desired conclusion by analyzing the properties of the critical equation. Therefore, we turn to consider the method of selecting suitable distribution functions and then verifying it with Rayleigh-Ritz principle of quadratic form to give the infimum and the quantitative representation of the optimal potential. Since the method to select potential function is also used in [16,17], some differences should be pointed out. Firstly, the potential functions selected are different. It belongs to L 1 in [16,17], while we choose the potential functions with Dirac distributions not in L 1 space. In addition, the method used in [16,17] is the maximum and minimum principle of operators, nevertheless the eigenfunction we obtained is not in the definition domain of operators. Thus, the most difficult point in this paper is how to select the potential function and what method we need to get the final results.
This paper is organised as follows. Section 2 contains some preliminaries knowledge that will be needed to develop this work. Section 3, we set up a formula for E n (λ) and specify where the infimum can be attained. We first obtain the concrete expression of the infimum of the L 1 norm for the potential function when the first eigenvalue of (1.1) is known. Then, we use the discontinuity of eigenvalues on the boundary conditions to explain why our results are different from previous results. Furthermore, by using the zero property of the characteristic function, the optimal recovery problem of the potential function with the known n-th eigenvalue is transformed into the optimal recovery problem of the potential function with the known first eigenvalue. Finally, the maximum and minimum values of the n-th eigenvalue of the potential function on the L 1 sphere are obtained by using the obtained results.

Preliminary
In this section, we summarize some of the basic results needed later.
By the definition of λ n (q), λ n (0) is the n-th eigenvalue of the following problem where λ * 1 is the solution to the equation tanh(

Results
In this section, the main result is given by theorems 3.1 and 3.2, which gives the answer to the extremal problem (1.4).  1) with (1.2). Let 1 (λ) and E 1 (λ) be defined as in (1.3) and (1.4) , respectively. Then