1. Introduction

The present paper is concerned with the Sturm–Liouville eigenvalue problem subject to Dirichlet boundary condition:

(1.1)\begin{equation} -y''+qy=\lambda y, \ y=y(x),\ x\in[0,1], \ y(0)=0=y(1), \end{equation}

where the potential $q\in L^1([0,1],\ \mathbb {R})$ and $L^1([0,1],\ \mathbb {R})$ is the space of Lebesgue integrable real-valued functions on $[0,1]$. From the spectral theory of differential equations, it is known that (1.1) has a countable number of eigenvalues, which are algebraically simple, bounded below, and tend to $\infty$. Let $\lambda _n(q)$ be the $n$-th eigenvalue of (1.1). Then

\[ -\infty<\lambda_1(q)<\lambda_2(q)<\cdots<\lambda_n(q)\to\infty,\ n\to\infty. \]

For example, if $q\equiv 0$, then $\lambda _n(0)=n^2\pi ^2, \ n\in \mathbb {N}$. The set of all the eigenvalues of (1.1) is the spectral set, denoted by $\sigma (q)$.

The main objective in spectral theory of differential equations is relations between geometric data: coefficients of the equation, shape of the boundary, etc. and spectral data: eigenvalues and eigenfunctions of the differential equation. The direct problem is to determine spectral data from geometric data. The spectral theory for the direct problem is well developed for regular as well as singular problems, and the reader can refer to the books [Reference Dunford and Schwartz2, Reference Weidmann23, Reference Zettl24]. The inverse problem is to recover the geometric data (or part of it) from some spectral dada. Compared with the direct problem, the inverse problem is hard to solve and the corresponding spectral theory needs to be developed further. One of the reason is that the spectral set is determined uniquely by $q\in L^1[0,1]$, but the converse is not true. It was proved in [Reference Pöschel and Trubowitz13] that for a given spectral set $\sigma (q_0)$ with $q_0\in L^2[0,1]$, the isospectral set $M(q_0)$:

(1.2)\begin{equation} M(q_0)=\left\{q\in L^2[0,1]:\ \sigma(q)=\sigma(q_0)\right\} \end{equation}

is an infinite dimensional real analytic submanifold of $L^2[0,1]$ [Reference Pöschel and Trubowitz13, p. 68], and only the even potentials in an isospectral set can be determined uniquely with the smallest $L^2$-norm, see [Reference Pöschel and Trubowitz13, Corollary 1, p. 77]. In fact, earlier in 1946, Borg [Reference Borg1] gave the fundamental theorem that two sets of eigenvalues uniquely determine the potential. Since then, many scholars have carried out in-depth research and generalization of Borg's results, see [Reference Levinson10, Reference Marchenko11]. Hochstadt and Lieberman [Reference Hochstadt and Lieberman7] showed that the potential which is known on half of the interval can be recovered from a set of eigenvalues and it is also the start of recovering potentials from partial spectral data [Reference Gesztesy and Simon3, Reference Gesztesy and Simon4, Reference Wei and Xu21, Reference Wei and Xu22].

In this paper, we attempt to study the optimal inverse problem of the potentials with fixed finite eigenvalues $\lambda _j\in \mathbb {R},j=1,\ldots,m$ and $\lambda _1<\cdots <\lambda _m$, $m\in \mathbb {N}$. That is, we will estimate the infimum of the norm $\|q\|_{1}$ of $q$ in $\Omega (\lambda _1,\ldots,\lambda _m)\subset L^1[0,1]$, where

\[ \Omega(\lambda_1,\ldots,\lambda_m)=\left\{q:\ q\in L^1([0,1],\ \mathbb{R}),\ \lambda_j\in\sigma(q), \ j=1,\ldots,m \right\}. \]

Since $\Omega (\lambda )$ is an infinite-dimensional submanifold in $L^2[0,1]$ with codimension one and

\[ \Omega(\lambda_1,\lambda_2, \ldots,\lambda_m)=\cap_{k=1}^m\Omega(\lambda_k), \]

the set $\Omega (\lambda _1,\lambda _2, \ldots,\lambda _m)$ must be an infinite set in $L^2[0,1]$, and hence in $L^1[0,1]$ due to $L^2[0,1]\subset L^1[0,1]$. So that, the uniqueness of potential for the recovery problem does not hold. Thus, we recover the optimal potential under the condition that the $L^1$-norm of the potential is the infimum in $\Omega (\lambda _1,\lambda _2, \ldots,\lambda _m)$. For the optimal recovery problem, we also study the existence—whether the infimum is attained in $\Omega (\lambda _1,\lambda _2, \ldots,\lambda _m)$. Furthermore, we will present the quantitative representation of the optimal potential. The optimal recovery problem, or the optimal inverse problem, is very related to the extremal problem of eigenvalues, particularly, the two problems are equivalent to each other provided that $m=1$ [Reference Qi and Chen14].

For the case $m=1$, the optimal inverse problem has been solved in [Reference Qi and Chen14] by using the generalized Lyapunov-type inequality together with Rayleigh–Ritz principle, and this method has been used successfully to solve the norm estimating of the optimal potentials for Sturm–Liouville problem with general separated boundary conditions (see [Reference Guo and Qi5]) and with Dirac distribution weights (see [Reference Guo and Qi6]). However, the above technique is hardly applicable to solve the problem directly for the cases $m\ge 2$ since the (generalized) Lyapunov-type inequality involves only one eigenvalue.

Another efficient method to solve the optimal inverse problem is the critical equations in $L^p[0,1]$ for $p>1$, which were early used by the authors in [Reference Wei, Meng and Zhang20, Reference Zhang25] for Sturm–Liouville operators in studying the extremal problems of eigenvalues, in which the similar results for the optimal potentials with Dirichlet and Neumann boundary conditions have been obtained based on the critical equations. Recently, the critical equations are also constructed in [Reference Sh. Ilyasov and Valeeva18] for elliptic operators with the case $m=1$, then the authors in [Reference Ilyasov and Valeev8, Reference Valeev and Ilyasov19] extended the case to any finite $m$. Such method is also used in [Reference Qi and Xie16] to obtain the optimal weight of vibrating string equations for the case $m=1$.

For $m\geq 2$, the optimal inverse problem can be turned into finding the solution of a boundary value problem for a system of nonlinear differential equations. For example, the authors in [Reference Valeev and Ilyasov19] used this method to find the optimal potential $q\in L^2$ which is the nearest to given $q_0$ with prescribed partial trace. And the authors in [Reference Ilyasov and Valeev8] extended to estimate the infimum of the norm $\|q-q_0\|_{2}$ for fixed finite eigenvalues of $q$. In both of the above papers, the existence of the optimal potentials is proved and the expressions of such potentials are given by the solutions of boundary value problems of nonlinear differential equations.

In the present paper, the similar problem for $m=2$ is considered in the space $L^1[0,1]$ with $q_0=0$. Since the norm in $L^1[0,1]$ is not differentiate, the critical equation method could not be applied directly. Besides, when $m\ge 2$, it is difficult to obtain the explicit form of the optimal potential by the critical equation method due to the nonlinearity of the corresponding problems.

Similar to the present problem, the inverse optimal problem of weights for the problem

\[{-}y''=\lambda w y, \ y(0)=0=y(1) \]

has been investigated in [Reference Qi, Li and Xie15] when the first two eigenvalues are known. To date, few results of estimating the extremal norm of potentials for the cases $m\ge 2$ are available.

In this paper, we will study the case for $m=2$. For fixed $\lambda _1, \lambda _2\in \mathbb {R}$ and $\lambda _1<\lambda _2$, the paper considers the infimum:

(1.3)\begin{equation} E(\lambda_1, \lambda_2)=\inf\left\{\|q\|_{1}:\ q\in \Omega(\lambda_1, \lambda_2)\right\} \end{equation}

of the $L^1$-norm of $q$ in the set:

(1.4)\begin{equation} \Omega(\lambda_1, \lambda_2)=\left\{q:\ q\in L^1[0,1], \lambda_1(q)=\lambda_1, \lambda_2(q)=\lambda_2\right\}. \end{equation}

For the purpose of the clear statement of the methods, we consider only the special case, where

(1.5)\begin{equation} \left\{\begin{array}{@{}l} \lambda_1\in(-\infty,\pi^2),\ \lambda_2\in(\lambda_1, 4\pi^2),\ \rho_j=\sqrt{\lambda_j},\ j=1,2,\\ \displaystyle\rho_1\left(\cot\dfrac{\rho_1}{4}-\tan\dfrac{\rho_1}{4}\right)\ge 2\rho_2\cot\dfrac{\rho_2}{4}. \end{array}\right. \end{equation}

Note that the functions $\sqrt {\lambda }\cot (\sqrt {\lambda }/4)$ and $\sqrt {\lambda }\tan \left (\sqrt {\lambda }/4\right )$ are analytic in the half-plane Re $\lambda <4\pi ^2$ except for the only removable singularity $\lambda =0$, and hence for $\sqrt {\lambda }\cot (\sqrt {\lambda }/4)$, we use $\sqrt {\lambda }\cot (\sqrt {\lambda }/4)|_{\lambda =0}=4$ and for $\lambda <0$:

\[ \sqrt{\lambda}\cot\sqrt{\lambda}=\sqrt{|\lambda|}\coth\sqrt{|\lambda|}, \quad \sqrt{\lambda}\tan\sqrt{\lambda}={-}\sqrt{|\lambda|}\tanh\sqrt{|\lambda|}. \]

The admissible set of the above restrictions on $\lambda _1$ and $\lambda _2$ in (1.5) is not empty, see remark 2 in §3. The reason why we consider condition (1.5) is that for general situations, it follows from the existing results, see [Reference Qi and Chen14, p. 11], the optimal potential is either the Dirac-delta function or the bathtub function. We will prove that condition (1.5) guarantees that the optimal potential is a Dirac-delta function.

Since the Dirac-delta function is not Lebesgue integral, then we introduce the measure space $\mathcal {M}_{0}$ [Reference Meng and Zhang12, Reference Zhang, Wen, Gang, Qi and Xie26] with the norm $\|\mu \|_V$ for $\mu \in \mathcal {M}_{0}$, see the definitions in §2. Then, we consider the eigenvalue problem of the second-order measure differential equations with the given measure $\mu \in \mathcal {M}_{0}$:

(1.6)\begin{equation} \left\{\begin{array}{@{}l} -{\rm d}\dot{y}(x)+y(x)\,{\rm d}\mu(x)=\lambda y\,{\rm d}x,\ x\in [0,1],\\ y(0)=0=y(1), \end{array}\right. \end{equation}

where $\dot {y}$ expresses the generalized derivative of the solution $y(x)$ and (1.6) has a countable number of eigenvalues, which are algebraically simple, bounded below, and tend to $\infty$ [Reference Meng and Zhang12].

Therefore, we can introduce the similar optimal recovery problem as (1.3) and (1.4) in $\mathcal {M}_{0}$. Let $\lambda _1(\mu ),\ \lambda _2(\mu )$ be the first and second eigenvalues of (1.6) and define:

(1.7)\begin{align} E_{0}(\lambda_1, \lambda_2)=\inf\left\{\|\mu\|_V:\ \mu\in \Omega_{0}(\lambda_1, \lambda_2)\right\}, \end{align}

(1.8)\begin{align} \Omega_{0}(\lambda_1, \lambda_2)=\left\{\mu:\ \mu\in \mathcal{M}_{0},\ \lambda_1(\mu)=\lambda_1, \ \lambda_2(\mu)=\lambda_2\right\}. \end{align}

Applying the spectral shifting lemma, see lemma 2.4, we prove in §2 that

Theorem 1.1 indicates that the above optimal recovery problems are well posed and they are equivalent to each other. This enables us to look for the optimal solution $\mu _0\in \Omega _0(\lambda _1, \lambda _2)$ such that $\|\mu _0\|_V=E_0(\lambda _1, \lambda _2)$ in $\mathcal {M}_{0}$. The following theorem is the main result of this paper.

Theorem 1.2 Consider the eigenvalue problem (1.1) with symmetric potential $q$. Let $E(\lambda _1, \lambda _2)$ and $\Omega (\lambda _1, \lambda _2)$ be defined in (1.3) and (1.4), respectively. If $\lambda _1$ and $\lambda _2$ satisfy (1.5), then

\[ E(\lambda_1, \lambda_2)=2r,\ r=\rho_1\left[\cot(\rho_1 a)-\tan\rho_1(1/2-a)\right]>0, \]

where $a\in [1/4, 1/2)$ is the unique root of the equation:

\[ \rho_1\left[\cot(\rho_1 a)-\tan\rho_1(1/2-a)\right]=\rho_2\left[\cot(\rho_2 a)+\cot\rho_2(1/2-a)\right]. \]

Furthermore, $E(\lambda _1, \lambda _2)$ is attained in $\mathcal {M}_{0}$ by $\mu (x)=-r(H_{a}(x)+H_{1-a}(x))$ in (1.6), where $H_{a}(x)$ is Heaviside function, see definition (2.2) in §2.

The arrangement of this paper is as follows. Section 2.1 provides some preliminary knowledge about the real measure space on $[0,1]$. In §2.2, we introduce the inverse spectral theory in $L^2[0,1]$, and then provide the proof of theorem 1.1. We will introduce the min-max principle of quadratic form in §2.3 and provide the generalized Lyapunov inequality with modification in §2.4. The proof of theorem 1.2 is given in §3. In the Appendix, we will supplement some concepts of quadratic form and add the proof of the correspondence between the quadratic form and the problem proposed in §2.3 and 2.4.