A new kind of uniqueness theorems for inverse Sturm-Liouville problems
- Yuri Ashrafyan^{1}Email author
Received: 25 February 2017
Accepted: 18 May 2017
Published: 30 May 2017
Abstract
We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyan’s theorem.
Keywords
1 Introduction
Let \(L=L(q, \alpha, \beta)\) and \(L_{0}=L(q_{0}, \alpha_{0}, \beta_{0})\) be two operators. The following assertion is usually called the uniqueness theorem of Marchenko.^{1}
Theorem 1.1
Marchenko [5]
One of the results of the present paper is the following theorem which, in some sense, is a generalization of Marchenko’s uniqueness theorem.
Theorem 1.2
This kind of uniqueness theorem has not been considered before. The main difference between Theorems 1.1 and 1.2 is that we replace the equality in (1.6) with the inequality in (1.8). Note that we assume \(q' \in L^{2}_{\mathbb{R}}(0, \pi)\) instead of general \(q \in L^{1}_{\mathbb{R}}(0, \pi)\) since our proof is based on the results of Jodeit and Levitan (see [6]). And the parameter α of boundary condition is in advance fixed \(\alpha= \alpha_{0}\).
Historically, the first work in the theory of inverse spectral problems for Sturm-Liouville operators belongs to Ambarzumyan [7]. He proved that if the eigenvalues of Sturm-Liouville operator with Neumann boundary conditions are \(n^{2}\), then the potential q is 0 on \([0, \pi]\). It is known that the eigenvalues \(\mu_{n}(0, \pi/2, \pi/2)\) of operator \(L(0, \pi/2, \pi/2 )\) are \(n^{2}, n \geq0\). The classical Ambarzumyan theorem in our notations will be as follows.
Theorem 1.3
Ambarzumyan [7]
If \(\mu_{n}(q, \pi/2, \pi/2) = \mu_{n}(0, \pi/2, \pi/2)=n^{2}\) for all \(n \geq0\), then \(q(x) \equiv0\).
This was an exception as in general additional information was needed in order to reconstruct the potential q uniquely. There are many generalizations of Ambarzumyan’s theorem in various directions, we mention several of them (see, e.g., [8–15] and the references therein).
Our generalization of Ambarzumyan’s theorem is as follows.
Theorem 1.4
Let \(q' \in L^{2}_{\mathbb{R}}(0, \pi)\).
If \(\mu_{n}(q, \alpha, \pi- \alpha) = \mu_{n}(0, \alpha, \pi- \alpha)\) for all \(n \geq0\), then \(q(x) \equiv0\).
We think that Theorem 1.4 is a natural generalization, because we use only one spectrum to reconstruct the potential q without any additional conditions, as it is in the classical result.
2 Preliminaries
Two operators \(L=L(q, \alpha, \beta)\) and \(L_{0}=L(q_{0}, \alpha_{0}, \beta _{0})\) are called isospectral if they have the same spectra, i.e., \(\mu_{n} (q, \alpha, \beta) = \mu_{n} (q_{0}, \alpha_{0}, \beta_{0}), n \geq0\). In what follows, if a certain symbol γ denotes an object related to L, then \(\gamma_{0}\) (or \(\gamma^{0}\) depending on situation) will denote a similar object related to \(L_{0}\).
Thus Jodeit and Levitan showed that each admissible sequence \(\{c_{n}\} _{n=0}^{\infty}\) generates an isospectral operator \(L(q, \alpha, \beta)\), where \(q, \alpha\) and β are given by formulae (2.3), (2.4) and (2.5), respectively. In this way they obtained all the potentials q, with \(q' \in L^{2}(0, \pi)\), having a given spectrum \(\mu_{n}^{0} = \mu_{n}(q_{0}, \alpha_{0}, \beta _{0}), n \geq0\).
3 Proof of Theorem 1.2
This completes the proof.
Remark 2
From equation (3.4) it follows that the condition \(a_{n} \geq a_{n}^{0}\) can be changed with \(a_{n} \leq a_{n}^{0}\). From relation (3.3) it follows that we can assume \(\beta= \beta _{0}\) instead of \(\alpha= \alpha_{0}\) with the condition \(a_{n} \geq a_{n}^{0}\) or \(a_{n} \leq a_{n}^{0}\), and then we will also obtain \(q(x) \equiv q_{0}(x)\) and \(\alpha= \alpha_{0}\).
4 Proof of Theorem 1.4
Consider an operator \(L(q, \alpha, \pi- \alpha)\) and an even operator^{3} \(L(0, \alpha, \pi- \alpha)\).
The condition of the theorem means that the operator \(L(q, \alpha, \pi - \alpha)\) is isospectral with \(L(0, \alpha, \pi- \alpha)\). Since the method of Jodeit and Levitan has described all the isospectral operators for a potential function q with \(q' \in L^{2}(0, \pi)\), then there exists a sequence \(\{c_{n}\}_{n=0}^{\infty}\) such that \(1 + c_{n} a_{n}^{0} > 0\) for all \(n \geq0\), \(\{c_{n}\}_{n=0}^{\infty}\) has the properties described in Section 2, and formulae (2.3)-(2.5) hold for operators \(L(q, \alpha, \pi- \alpha)\) and \(L(0, \alpha, \pi- \alpha)\).
Remark 3
We will get the classical Ambarzumyan theorem if we take \(\alpha= \pi/ 2\).
Here \(F(x, y)\) is a kernel of integral equation (2.2), where x is a parameter, \(F(x, y)\) is a known function and \(K(x, y)\) is an unknown function, as functions of y.
A problem \(L(q, \alpha, \beta)\) is said to be even if \(q(x) = q(\pi- x)\) and \(\alpha+ \beta= \pi\).
Declarations
Acknowledgements
The author would like to thank the referees for their helpful comments and suggestions. The author is also grateful to professor TN Harutyunyan for valuable remarks and discussions.
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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