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Reconstruction of the solution of inverse Sturm–Liouville problem
Boundary Value Problems volume 2024, Article number: 55 (2024)
Abstract
In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on \([0,1/2]\) and the coefficient at the left end point are known a priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the MittagLeffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue.
1 Introduction
Consider the Sturm–Liouville operator L given by
with the Robin boundary conditions
where \(q\in L^{2}[0,1]\) is realvalued and \(h_{1}\), \(h_{2}\) are real constants.
In 1978, Hochstadt and Lieberman [6] proved the remarkable uniqueness theorem, which proved that, if the potential \(q(x)\in L^{1}[0,1]\) in equation (1.1) together with real constants \(h_{1}\), \(h_{2}\) in (1.2)–(1.3) is known a priori on the halfinterval \((1/2,1)\), then the spectrum \(\sigma =\{\lambda _{n}^{2}\}_{n=0}^{+\infty }\) alone is sufficient for the unique specification of \(q(x)\) on \((0,1/2)\). The Hochstadt–Lieberman problem was the first halfinverse problem. Various aspects of this socalled halfinverse spectral problem were investigated in [3, 7, 12, 13] and other works. In [12, 13], Martinyuk and Pivovarchik studied this type problems with Dirichlet and Robin boundary conditions respectively for a potential \(q\in L^{2}[0,1]\). They proposed a method of recovering the potential on the whole interval and obtained the necessary and sufficient conditions of the Hochstadt–Lieberman problem solvability. Buterin [3] proved the uniqueness theorem for this halfinverse spectral problem for a secondorder differential pencil with spectral parameter dependent boundary conditions by Weyl function. Using the transformation operator and the properties of Riesz basis, the necessary and sufficient conditions of the Hochstadt–Lieberman problem solvability have been obtained in [7] for singular potentials from the space \(W_{2}^{1}(0,1)\). Furthermore, a reconstruction algorithm was provided.
Our main goal in this paper is to provide a new method for reconstructing potentials on the halfinterval \([1/2,1]\) and \(h_{2}\) for the above inverse spectral problem. We also give a necessary and sufficient condition for the existence issue.
Let \(u_{+}(x,\lambda )\) be the solution of equation (1.1) satisfying the initial conditions \(u_{+}(1,\lambda )=1\) and \(u_{+}^{\prime}(1,\lambda )=h_{2}\). In order to solve the halfinverse problem by finding the functions \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )\), the method we use is to employ the MittagLeffler expansion for meromorphic functions, which has been used in [14] to uniquely reconstruct the potential for the interior transmission eigenvalue problem and the Sturm–Liouville problem with the potential function known on the subinterval \((0,a)\) (\(a<1/2\)) [15], respectively. This can help us to use the Levin–Lyubarski interpolation formula to find the unknown \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )\). Moreover, this decomposition also provides a wellsuited situations for utilizing the Levin–Lyubarski interpolation formula to our problem. Let us mention that our reconstructing process also deduces the existence condition of solutions for the above halfinverse spectral problem. In fact, the necessary and sufficient conditions are similar to the conditions of Pivovarchik [12, 13]. The difference is the method by which the functions \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime}(1/2,\lambda )\) are constructed. Let us mention that our method can also be used to treat the case where potential q is known a priori on the interval \([0,a]\) with \(a<1/2\) (see [8, 14] and the references therein).
Throughout this paper, we denote by \(\mathcal{L}_{a}\) the class of entire functions of exponential type ≤a that belong to \(L^{2}(\infty ,\infty )\) for real λ [10].
The paper is organized as follows. In Sect. 2 we give some preliminaries that will be needed subsequently. Section 3 presents our main results for inverse problems.
2 Preliminaries
In this section, we recall the spectral characteristics of the operator L and give some theorems we will use.
Let \(u_{}(x,\lambda )\) be the solution of equation (1.1) satisfying the initial conditions \(u_{}(0)=1\) and \(u_{}^{\prime }(0)=h_{1}\). According to [11], one knows that
where
and \(\tilde{K}_{1}(x,t)\) is the solution of the integral equation
The solution \(\tilde{K}_{1}(x,t)\) possesses partial derivatives of first order with \(\frac{\partial}{\partial t}\tilde{K}_{1}(x,t)\in L^{2}[0, 1/2]\) and \(\frac{\partial}{\partial x}\tilde{K}_{1}(x,t)\in L^{2}[0, 1/2]\). Moreover,
By using (2.1) we infer
where \(K_{1}=\tilde{K}_{1}(1/2,1/2)\) defined by (2.3) and \(\psi _{,j}\in \mathcal{L}_{1/2}\) for \(j=0,1\).
On the other hand, we denote by \(u_{+}(x,\lambda )\) the solution of (1.1) satisfying the initial conditions \(u_{+}(1,\lambda )=1\) and \(u_{+}^{\prime}(1,\lambda )=h_{2}\). One knows that \(u_{+}(x,\lambda )\) has a similar representation as (2.1):
where the function \(K_{2}(x,t,h_{2})\) satisfies the expression similar to (2.2). This gives the asymptotics of \(u_{+}(x,\lambda )\) by
where \(K_{2}=\frac{1}{2}\int _{1/2}^{1}q(t)\,dt \) and \(\psi _{+,j} \in \mathcal{L}_{1/2}\) for \(j=0,1\).
The eigenvalues \(\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) of problem (1.1)–(1.3) coincide with the zeros of
which are called the characteristic function of (1.1)–(1.3). From (2.1), \(\Delta (\lambda )\) has the following representation:
where \(\hat{\psi}\in \mathcal{L}_{1}\). It is known that the function \(\Delta (\lambda )\) is entire in λ of type 1. The eigenvalues \(\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) behave asymptotically as follows:
as \(n\rightarrow \infty \), where \(\{\alpha _{n}\}_{n=0}^{+\infty}\in l^{2}\), which implies that
Moreover, the specification of the spectrum \(\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) uniquely determines the characteristic function \(\Delta (\lambda )\) by the formula [4, Theorem 1.1.4]:
We write the MittagLeffler theorem [1, Theorem 3.6.2] for the case of simple poles as follows.
Theorem 2.1
Assume that \(F(z)\) is a meromorphic function and has only simple poles \(\{z_{j}\}_{j\in \mathbb{Z}}\) with \(z_{j}\) distinct and \(z_{j}\rightarrow \infty \) as \(j\rightarrow \infty \). Let \(c_{j}\) be the residues of poles \(z_{j}\) of \(F(z)\). If
then there exists an entire function \(f(z)\) such that
where the series in the righthand side of (2.13) converges uniformly on every bounded set of \(\mathbb{C}\) not containing the points \(\{z_{j}\}_{j\in \mathbb{Z}}\).
The following theorem [9, Theorem A] is corresponding to sine type functions, which plays an important role in our paper.
Theorem 2.2
(Levin–Lyubarski interpolation formula) Let f be a sine type function with indicator diagram of width 2a, and \(\{z_{k}\}_{k\in \mathbb{Z}}\) be the set of its zeros. Then, for any sequence \(\{c_{k}\}_{k\in \mathbb{Z}}\in l^{p}\) with \(1< p<\infty \), the interpolation series
converges uniformly on any compact subsets in \(\mathbb{C}\) and also in the norm of \(L^{p}(\mathbf{\infty ,\infty })\) on the real axis, which belongs to \(\mathcal{L}_{a}\).
3 Inverse spectral problem
In this section, we show the way of recovering q on \([1/2,1]\) and give conditions of the existence of the solution in an implicit form.
Denote by \(v_{}(x,\lambda )\) the solution of (1.1) satisfying the initial conditions \(v_{}(0,\lambda )=0\) and \(v_{}^{\prime}(0,\lambda )=1\). We infer
where \(\varphi _{+,j}\in \mathcal{L}_{1/2}\) for \(j=0,1\).
We denote by \(\{\mu _{n}\}_{n\in \mathbb{Z}}\) the zeros of \(u_{}(1/2,\lambda )\), then
as \(n\rightarrow \infty \), where \(\{\kappa _{n}\}_{n\in \mathbb{Z}}\in l^{2}\). It is easy to say that \(\frac{v_{}(1/2,\lambda )\Delta (\lambda )}{u_{}(1/2,\lambda )}\) is meromorphic and has only simple poles \(\{\mu _{n}\}_{n\in \mathbb{Z}}\). Let \(e_{n} \) be the residues of \(\frac{v_{}(1/2,\lambda )\Delta (\lambda )}{u_{}(1/2,\lambda )}\) at \(\mu _{n}\). One has
with \(\dot{u}_{}=\partial u_{}/\partial \lambda \). By virtue of (2.4), (2.8), and (2.9), we have
where \(\{\zeta _{n}\}_{n\in \mathbb{Z}}\in l^{2}\), which together with (3.2) yields \(\{e_{n}/\mu _{n}\}_{n\in \mathbb{Z}}\in l^{1}\).
By MittagLeffler expansion, there exists an entire function \(a_{0}(\lambda )\) such that
Denote by
Substituting (3.6) into (3.5), we arrive at
It is easy to see that \(a_{0}(\lambda )\) and \(b_{0}(\lambda )\) are realvalued functions when \(\lambda \in \mathbb{R}\).
Lemma 3.1
Let \(a_{0}(\lambda )\) and \(b_{0}(\lambda )\) be defined by (3.5) and (3.6), respectively. If we assume that
then \(\varphi _{0}(\lambda )\in \mathcal{L}_{1}\) and \(\varphi _{1}(\lambda )\in \mathcal{L}_{1/2}\).
Proof
Recall that \(\{\mu _{n}\}_{n\in \mathbb{Z}}\) are the zeros of \(u_{}(1/2,\lambda )\), then (3.7) yields
By virtue of the second equation of (3.8) together with (3.9) and using the asymptotic formulae (3.1) and (2.8) we get
Using (3.2) we get
where \(\{\alpha _{n}\}_{n\in \mathbb{Z},n\neq 0}\), \(\{\xi _{n}\}_{n\in \mathbb{Z},n\neq 0}\), \(\{\vartheta _{n}\}_{n\in \mathbb{Z},n\neq 0}\), and \(\{\delta _{n}\}_{n\in \mathbb{Z},n\neq 0}\) all belong to \(l^{2}\). Substituting (3.11) into (3.10), we obtain
The function \(u_{}(1/2,\lambda )\) is of sine type, i.e., there exist positive numbers m, M, and p such that
for \(\mathrm{Im} \lambda >p\). Taking into account (3.12), we use the Levin–Lyubarski interpolation theorem (see Theorem 2.2 for details) and choose \(\{\mu _{n}\}_{n\in \mathbb{Z}}\) as the nodes of interpolation for finding the function \(\varphi _{1}(\lambda )\):
Note that \(u_{}(1/2,\lambda )\) is a sine type function with the indicator diagram of width 1, thus \(\varphi _{1}(\lambda )\in \mathcal{L}_{1/2}\) according to Theorem 2.2.
On the other hand, note that \(\{\lambda _{n}\}_{n\in \mathbb{Z}}\) are the zeros of \(\Delta (\lambda )\). In virtue of (3.7) we have
It should be noted from (2.9) that
and
Substituting (3.15)–(3.16) into (3.14), one obtains
Let \(\Phi (\lambda )=\Delta (\lambda )/(\lambda _{0}\lambda )\). It is easy to see that \(\Phi (\lambda )\) belongs to sine type functions. We choose \(\{\lambda _{n}\}_{n\in \mathbb{Z}}\) as the nodes of interpolation for finding the function \(\varphi _{0}(\lambda )\):
From (2.8), one knows that \(\Phi (\lambda )\) is a sine type function with the indicator diagram of width 2, thus \(\varphi _{0}(\lambda )\in \mathcal{L}_{1}\) by Theorem 2.2. Moreover, from (3.8) we have
□
Lemma 3.2
Let \(a_{0}(\lambda )\) and \(b_{0}(\lambda )\) be defined by (3.5) and (3.6), respectively. If we write
then
Proof
It should be noted that
From (3.20) and (3.7), by simple computation we have
Note that
and \(b_{0}(\lambda )<u_{}(1/2,\lambda )\). (3.24) together with (3.23) yields that there exists \(h(\lambda )\) satisfying
By virtue of (2.4) and (2.6), for \(\lambda \mu _{n}>0\), we have
thus \(h(\lambda )=1\). It follows that (3.21) remains true from (3.25). This completes the proof. □
By the above arguments, we have recovered the functions \(b_{0}(\lambda )\), \(a_{0}(\lambda )\) and then \(b_{1}(\lambda )\) in terms of the given mixed spectral data consisting of q on \([0,1/2]\), \(h_{1}\), and the set σ of eigenvalues of Sturm–Liouville problems. Thus we can reconstruct \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )\) by (3.21), and hence q on \((1/2,1)\) via the Gelfand–Levitan–Marchenko method [11]. The method of reconstructing the potential \(q(x)\) on the halfinterval \([1/2,1]\) and constant \(h_{2}\) can be summarized as follows.
Algorithm
Let the input data set \(\mathcal{D}=\{q(x)\in L^{2}[0,1/2],\sigma =\{\lambda _{n}^{2}\}_{n=0}^{+ \infty},h_{1}\}\) be given.

(1)
Compute \(h_{2}+K_{2}\) in virtue of (2.10) and construct \(\Delta (\lambda )\) in terms of (2.11).

(2)
Compute the functions \(u_{}(1/2, \lambda )\), \(u^{\prime}_{}(1/2, \lambda )\), \(v_{}(1/2, \lambda )\), and \(v^{\prime}_{}(1/2, \lambda )\).

(3)
Determine the sequences \(\varphi _{1}(\mu _{n})\) by (3.10), then construct the function \(\varphi _{1}(\lambda )\) in virtue of (3.13).

(4)
Construct \(b_{0}(\lambda )\) in virtue of the second formula of (3.19) and compute the sequence \(b_{0}(\lambda _{n})\).

(5)
Determine the sequence \(\varphi _{0}(\lambda _{n})\) by (3.14), then construct the function \(\varphi _{0}(\lambda )\) in virtue of (3.18).

(6)
Construct \(a_{0}(\lambda )\) in terms of the first formula of (3.19).

(7)
Construct the function \(b_{1}(\lambda )\) by (3.20).

(8)
Reconstruct \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )\) by (3.21).

(9)
Reconstruct the function q on \((1/2,1)\) from the zeros of \(u_{+}(1/2,\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )\) via the Gelfand–Levitan–Marchenko method [11].

(10)
Compute \(h_{2}=K_{2}+h_{2}\int _{1/2}^{1}q(x)\,dx\).
Let us mention that \((u_{+}/u^{\prime}_{+})(\sqrt{\lambda} )\) belongs to the Nevanlinna class, i.e., \((u{+}/u^{\prime}_{+})(\sqrt{\lambda} )\): \(\mathbb{C}_{+}\rightarrow \mathbb{C}_{+}\) is analytic with \(\mathbb{C}_{+}\) being the open complex upper halfplane [10]. Our reconstructing process also deduces the following conclusion for the existence problem.
Theorem 3.3
Assume that a real function \(q_{}\in L^{2}[0,1/2]\) is known together with the real constant \(h_{1}\). Let a set of numbers \(\{\lambda _{n}^{2}\}_{n=0}^{\infty}\) be given and satisfy the following asymptotics:
where \(A\in \mathbb{R}\) and \(\{\alpha _{n}\}_{n=0}^{\infty}\in l^{2}\). Let \(u_{}(x,\lambda )\) be the solution of (1.1) with the potential \(q=q_{}\) on \([0,1/2]\), which satisfies the initial conditions \(u_{}(0)=1\), \(u_{}^{\prime}(0)=h\), and let \(u_{+}(\lambda )\) and \(\hat{u}_{+}(\lambda )\) be given by
with \(b_{0}(\lambda )\) and \(b_{1}(\lambda )\) being defined by (3.19) and (3.20), respectively.
Then there exists a unique realvalued function \(q_{+}\in L^{2}[1/2,1]\) and a real constant \(h_{2}\) such that the spectrum σ of problem (1.1)–(1.3) with potential \(q=q_{}\) on \([0,1/2]\) and \(q=q_{+}\) on \([1/2,1]\) coincides with the sequence \(\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) if and only if the function \(u_{+}/\hat{u}_{+}(\sqrt{\lambda})\) belongs to the Nevanlinna class.
Proof
Suppose that there exists a realvalued function \(q\in L^{2}(0,1)\) such that \(\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) is the spectrum of the Sturm–Liouville operator defined by (1.1)–(1.3). Then, by the above discussion, \(u_{+}(1/2,\lambda )=u_{+}(\lambda )\) and \(u_{+}^{\prime }(1/2,\lambda )=\hat{u}_{+}(\lambda )\). In this situation, it is known [5, 11] that \((u_{+}/\hat{u}_{+})(\sqrt{\lambda })\) is the Weyl mfunction [5] of Sturm–Liouville equation (1.1), which ensures that the function \((u_{+}/\hat{u}_{+})(\sqrt{\lambda })\) belongs to the Nevanlinna class.
Since the spectrum \(\sigma =\{\lambda _{n}^{2}\}_{n=0}^{+\infty}\) of the operator L is given, by (2.10) and (2.11) one obtains \(K_{2}+h_{2}\) and \(\Delta (\lambda )\). If a realvalued function \(q_{}\in L^{2}(0,1/2)\) is known a priori, then both functions \(u_{}(1/2,\lambda )\) and \(u_{}^{\prime }(1/2,\lambda )\) are also known. Thus by (3.13) and (3.19) we obtain \(b_{0}(\lambda )\) and from Lemma 3.2 we obtain \(b_{1}(\lambda )\). We therefore obtain \(u_{+}(\lambda )\) and \(\hat{u}_{+}(\lambda )\) from (3.27):
and
Here one knows that \(\psi _{+,j}(\lambda )\in \mathcal{L}_{1/2}\) for \(j=0,1\) by computing from (3.20) and above formulae since \(\psi _{,j}(\lambda )\in \mathcal{L}_{1/2}\), \(\varphi _{0}(\lambda )\in \mathcal{L}_{1}\), and \(\varphi _{1}(\lambda )\in \mathcal{L}_{1/2}\). It is easy to see that their zeros, denoted by \(\{\alpha _{n,D}\}_{n\in \mathbb{Z}}\) and \(\{\alpha _{n,N}\}_{n\in \mathbb{Z}}\), satisfy the following conditions:
where \(\{\beta _{n}\}_{n\in \mathbb{Z}}\) and \(\{\hat{\beta}_{n}\}_{n\in \mathbb{Z}}\) belong to \(l^{2}\). Furthermore, if \((u_{+}/\hat{u}_{+})(\sqrt{\lambda })\) belongs to the Nevanlinna class, then its zeros \(\{\alpha _{n,D}^{2}\}_{n=0}^{+\infty}\) and poles \(\{\alpha _{n,N}^{2}\}_{n=0}^{+\infty}\) are interlacing:
Moreover, by (3.28) it is easy to check that the sequences \(\{ (\frac{\alpha _{n,D}}{2\pi} )^{2}\}_{n=0}^{+\infty}\) and \(\{ (\frac{\alpha _{n,N}}{2\pi} )^{2}\}_{n=0}^{+\infty}\) satisfy the conditions of Theorem 3.4.3 in [11]. By Borg’s twospectra theorem [2] there exists a unique realvalued function \(q_{+}\in L^{2}(1/2,1)\) such that \(\{\alpha _{n,D}^{2}\}_{n=0}^{+\infty}\) and \(\{\alpha _{n,N}^{2}\}_{n=0}^{+\infty}\) are exactly the Dirichlet–Dirichlet spectrum (under the boundary conditions \(y(1/2)=0=y(1)\)) and the Dirichlet–Neumann spectrum (under the boundary conditions \(y(1/2)=0=y^{\prime}(1)\)) of two Sturm–Liouville operators defined on \((1/2,1)\) with potential \(q_{+}\). On the other hand, it is easy to see that the known σ is the spectrum of Sturm–Liouville operators defined by (1.1)–(1.3) with potential \(q=q_{}\) on \((0,1/2)\) a.e. and \(q=q_{+}\) on \((1/2,1)\). This completes the proof. □
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Acknowledgements
The authors would like to thank the referees for their insightful suggestions and comments, which improved and strengthened the presentation of this manuscript.
Funding
The research was supported by the National Natural Science Foundation of China (No. 11971284), Natural Science Foundation of Shaanxi Province (No. 2020JM537), and Shaanxi Province training program for innovation (No. 202001070116).
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Zhaoying Wei and Zhijie Hu wrote the main manuscript text and Yuewen Xiang checked calculation results and English. All the authors read and approved the final manuscript.
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Appendix
Appendix
In this section, we supply the details of Marchenko’s uniqueness theorem and Borg’s two spectra theorem.
Let us introduce the Weyl–Titchmarsh mfunction for the operator \(L(q,h_{1},h_{2})\) as
Denote by \(\tilde{m}(x,\lambda )\) by the Weyl–Titchmarsh mfunction for the operator \(L(\tilde{q},h_{1},\tilde{h}_{2})\).
Theorem A.1
(Marchenko’s uniqueness theorem) If \(m(a,\lambda )=\tilde{m}(a,\lambda )\), then \(q(x)=\tilde{q}(x)\) on \([a,1]\).
Theorem A.2
(Borg’s two spectra theorem) Let \(h_{2}\neq h_{3}\). If the two spectra \(\sigma (q,h_{1},h_{2})\) and \(\sigma (q,h_{1},h_{3})\) are known a priori, then \(q(x)\) on \([0,1]\) is uniquely determined.
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Wei, Z., Hu, Z. & Xiang, Y. Reconstruction of the solution of inverse Sturm–Liouville problem. Bound Value Probl 2024, 55 (2024). https://doi.org/10.1186/s13661024018604
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DOI: https://doi.org/10.1186/s13661024018604