Uniqueness results for inverse Sturm-Liouville problems with partial information given on the potential and spectral data
- Zhaoying Wei^{1, 2}Email author and
- Guangsheng Wei^{2}
Received: 15 May 2016
Accepted: 10 October 2016
Published: 11 November 2016
Abstract
We consider the inverse spectral problem for a Sturm-Liouville problem on the unit interval \([0,1]\). We obtain some uniqueness results, which imply that the potential q can be completely determined even if only partial information is given on q together with partial information on the spectral data, consisting of the spectrum and normalizing constants. Moveover, we also investigate the problem of missing both eigenvalues and normalizing constants in the situation where the potential q is \(C^{2k-1}\) near a suitable point.
Keywords
1 Introduction
The uniqueness problem of determining the potential q in terms of one of the above-mentioned three sets of spectral data \(\Gamma _{j}\) (\(j=1,2,3\)) is well known (see, e.g., [2, 3]). A comprehensive review for the inverse problem in these cases is presented by McLaughlin [4].
This paper is related immediately to a earlier paper [5] by the second author and Xu in that it provides some uniqueness results, which imply that the potential q and \(h_{1}\) can completely be determined even if only partial information is given on q together with partial information on the spectral data, consisting of either one full spectrum and a subset of ratios \(\kappa_{n}\) or a subset of pairs of eigenvalues and the corresponding ratios \(\kappa_{n}\). In the present paper, we consider the same uniqueness problem under the same circumstances but with the ratios \(\{\kappa_{n}\}\) replaced by normalizing constants \(\{\alpha_{n}\}\). In other words, we mainly investigate the uniqueness problem when only partial information on q, on the eigenvalues \(\{\lambda_{n}\}_{n\in \mathbb{N}}\), and on the normalizing constants \(\{\alpha _{n}\}_{n\in\mathbb{N}}\) is available.
Our original motivation for the above works is theorems of Hochstadt-Lieberman [6] and Gesztesy-Simon [7]. Specifically, in 1978, Hochstadt and Lieberman [6] proved that the whole spectrum uniquely determines q when it is already known on \([0,1/2]\). In 2000, Gesztesy and Simon [7] gave several important generalizations of the Hochstadt-Lieberman theorem to the case where the \(L^{1}[0,1]\) potential q is known on a larger interval \([0,a]\) with \(a\in[1/2,1)\) and the set of common eigenvalues is sufficiently large. Another result in [7] is obtained under the assumption that the potential q belongs to \(C^{2k}\) for some \(k\in\mathbb{N}_{0}\) near \(1/2\) so that \(C^{2k}\)-smoothness can replace the knowledge of some \(k+1\) eigenvalues, that is, \(k+1\) eigenvalues may be missing. These results have been generalized and improved in a variety of ways; see [8–14]. Our aim here is to realize that, for the question of uniqueness for the Sturm-Liouville problem, normalizing constants play an equal role with eigenvalues. In other words, the number of normalizing constants is, in a sense, equivalent to the number of eigenvalues.
Here is one of the main results of this paper.
Theorem 1.1
Remark 1.2
Under the assumption that \(\sigma(L)\) is known, from (1.6) it follows that the result of Theorem 1.1 remains valid if the condition of unknown normalizing constants \(\{\alpha_{j_{i}}\}_{i=1}^{k+1}\) is replaced with the condition of unknown ratios \(\{\kappa_{j_{i}}\}_{i=1}^{k+1}\). This is the same as [5], Thm. 1.1. However, Theorem 1.1 here shows that both some eigenvalues and normalizing constants may be missing.
Theorem 1.3
Remark 1.4
Remark 1.5
Comparing the result of Theorem 1.3 with that of [7], Thm. 1.4, we can see that the lack of a certain number of normalizing constants can be reduced to the situation of lack the same number of eigenvalues.
This result is related to another paper by the authors [15], where we consider an analog of Theorem 1.3 for finite tridiagonal (Jacobi) matrices. Moreover, as a particular case of Theorem 1.3, we have the following corollary, which is parallel with [5], Thm. 4.2, where the problem of the partial information on the subset of the pair of sequences \(\Gamma _{2}:=\{\lambda_{n},\kappa_{n};n\in\mathbb{N}_{0}\}\) is concerned.
Corollary 1.6
All the results obtained concern mainly with a spectrum with \(h_{0}\in\mathbb{R }\cup\{\infty\}\) being fixed. Furthermore, in Section 4, we generalize these results to more general circumstances associated with the spectral data of different operators \(L(q,h_{0,n},h_{1})\), where \(h_{0,n}\) are allowed to belong to different values.
Moreover, note that the case of Dirichlet boundary condition at \(x=1\) demands a separate treatment. Nevertheless, we expect that the method of the paper can be applied in this case.
The results presented in this paper are based on the uniqueness theorem of the Weyl m-function developed by Marchenko [16] and introduced to deal with inverse problems with partial information by Gesztesy, Simon, and del Rio [7, 17–19]. Our proof in the paper is based on two multiple zeros of the Wronskian of two Sturm-Liouville problems. At this point, we note that our proof here is different from that of the results in [5] for dealing with the unique determination problem of q and \(h_{1}\) in terms of eigenvalues and ratios, where the known ratios are transformed to known eigenvalues by a particular solution of the equation \(Lu=\lambda u\) such that the Weyl m-function technique can be used.
The paper is organized as follows. In the next section, we recall the uniqueness theorem of Marchenko [16] and give a proof of Theorem 1.1. The proof of Theorem 1.3 is presented in Section 3. In Section 4, we extend Theorem 1.3 to a more general case, associated with different boundary conditions at the endpoint \(x=0\), and further establish some new uniqueness results.
2 Preliminaries and proof of Theorem 1.1
In this section, we first recall the uniqueness theorem of Marchenko and formulate some asymptotic expansions of m-functions and solutions of Eq. (2.1), which will be used later to prove our principal results.
Throughout this paper, by the statement “q on \([0,a]\), eigenvalues \(\lambda_{n}\), and normalizing constants \(\alpha_{n}\) determine uniquely q and \(h_{1}\)” we mean that there are no two distinct potentials \(q_{1}\) and \(q_{2}\) on \([0,1]\) with the two properties: (i) \(q_{1}=q_{2}\) a.e. on \([0,a]\), and (ii) \(\lambda_{n}\) and \(\alpha_{n}\) are common eigenvalues and normalizing constants for \(q_{1}\) and \(q_{2}\).
Unless explicitly stated otherwise, \(h_{0}\) will be known, and all potentials q, \(q_{1}\), and \(q_{2}\) will be real valued and in \(L^{1}[0,1]\) for the rest of this paper.
Lemma 2.1
Theorem 2.2
\(m_{+}(a,z)\) uniquely determines \(h_{1}\) and q (a.e.) on \([a,1]\).
Lemma 2.3
Proof
According to the preliminaries, we now prove Theorem 1.1.
Proof of Theorem 1.1
3 Proof of Theorem 1.3
Our goal in this section is to prove Theorem 1.3. We first establish a lemma, which will be used later to prove the theorem and its generalizations (see Section 4 for details).
Lemma 3.1
Proof
We now are in position to prove Theorem 1.3.
Proof of Theorem 1.3
4 Uniqueness results for a more general case
In this section, we extend Theorem 1.3 by that the spectral data \(\{ \lambda _{m},\alpha_{m}\}_{m\in\mathbb{N}_{0}}\) can be selected in terms of different Sturm-Liouville operators \(L(q,h_{0},h_{1})=:L(h_{0})\) with \(h_{0}\) being different numbers in the boundary condition (1.2).
It is well known (see [1]) that Borg proved the famous two-spectra theorem that the spectra for two boundary conditions of a regular Sturm-Liouville operator uniquely determine the potential q. Later refinements (see, e.g., [12, 13, 19]) show that the knowing eigenvalues associated with a number of different boundary conditions can also determine the potential uniquely. In particular, McLaughlin and Rundell [22] used fixed jth eigenvalues \(\lambda _{j}(q,h_{0,l},h_{1})\) with \(l\in\mathbb{N}_{0}\) for a countable number of different boundary conditions at \(x=0\) to establish the uniqueness of q. Moreover, Horváth considered the same uniqueness problem when the known eigenvalues are taken from finite different spectra, which are corresponding to a finite number of boundary conditions at \(x=0\).
In our uniqueness results to be given further, the known eigenvalues and normalizing constants are of problem (1.1)-(1.3) where a countable number of different boundary conditions at \(x=0\) may be involved. These results not only generalize the results of [13, 19, 22] but also give some new uniqueness results for the inverse Sturm-Liouville problems through normalizing constants instead of eigenvalues. It is essential that, roughly speaking, for the unique determination problem of the potentials q and \(h_{1}\), the number of normalizing constants is, in a sense, equivalent to the number of eigenvalues.
We mention some properties of these eigenvalues, which we need further. For their proofs, we refer to [23, 24].
Lemma 4.1
- (i)
If \(h_{0,l_{1}}\neq h_{0,l_{2}}\), then \(\lambda (h_{0,l_{1}})\neq\lambda(h_{0,l_{2}})\), where \(\lambda (h_{0,l_{j}})\) are any eigenvalues of \(L(h_{0,l_{j}})\) for \(j=1,2\).
- (ii)Let \(m\in\mathbb{N}_{0}\). Then \(\lambda _{m}(h_{0})\) is strictly decreasing in \(h_{0}\in\mathbb{R}\) for any fixed q and \(h_{1}\). Furthermore, for \(m\geq1\), we havewhere \(\{\lambda_{m}(\infty)\}_{m=0}^{\infty}=\sigma(L(\infty ))\) with \(h_{0}=\infty\).$$ \lim_{h_{0}\rightarrow\infty}\lambda_{m}(h_{0})= \lambda_{m-1}(\infty ),\qquad\lim_{h_{0}\rightarrow-\infty} \lambda_{m}(h_{0})=\lambda _{m}(\infty), $$
Here is our main result of this section.
Theorem 4.2
Proof
Remark 4.3
As a particular case of Theorem 4.2, we have the following corollary, which concerns the uniqueness problem of q and \(h_{1}\) in terms of eigenvalues and normalizing constants associated with a (countable) number of different boundary conditions.
Corollary 4.4
Proof
As another particular case of Theorem 4.2, we also have the following corollary, which concerns our uniqueness problem in terms of eigenvalues only, associated with a countable number of different boundary conditions.
Corollary 4.5
Proof
The corollary is a generalization of the Borg’s two-spectra theorem. In fact, if \(h_{0,l}=h_{0}\) and \(h_{0,l}^{\prime }=h_{0}^{\prime}\) for all \(l\in\mathbb{N}_{0}\), and \(\{\lambda _{l}(h_{0})\}_{l=0}^{\infty}\) and \(\lambda_{l}(h_{0}^{\prime})\), except any one of them, are known, then q on \([0,1]\) and \(h_{1}\) are determined uniquely. Furthermore, this can be also viewed as a generalization of two-thirds spectra theorem by del Rio, Gesztesy, and Simon [19], Cor. 3.3. However, Corollary 4.5 here shows that the known eigenvalues are allowed to belong to a countable number of different spectra.
Finally, we give a generalization of half-inverse theorem of Hochstadt and Lieberman, which is involved in a countable number of different boundary conditions.
Corollary 4.6
Under the assumptions of Theorem 4.2, if \(a=1/2\), then \(\{h_{0,l}\}_{l=0}^{ \infty}\) and the eigenvalues \(\{\lambda_{l}(h_{0,l})\}_{l=0}^{\infty}\) uniquely determine \(h_{1}\) and q on \([0,1]\).
Proof
It should be noted that if all the \(h_{0,l}=h_{0}\) where \(h_{0}\in\mathbb{R}\) or \(h_{0}=\infty\), then \(h_{1}\) and q on \([0,1]\) are uniquely determined. This is the half-inverse theorem of Hochstadt and Lieberman.
Declarations
Acknowledgements
The authors would like to thank the referees for their helpful comments and suggestions, which improved and strengthened the presentation of this manuscript. The research was supported in part by the National Natural Science Foundation of China (No. 11571212), Youth Innovation Fund of Xi’an Shiyou University (No. Z15135), and Youth Innovation Team Fund of Xi’an Shiyou University (No. 2015QNKYCXTD03).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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