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A new kind of uniqueness theorems for inverse Sturm-Liouville problems

Boundary Value Problems20172017:79

  • Received: 25 February 2017
  • Accepted: 18 May 2017
  • Published:


We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyan’s theorem.


  • inverse problem
  • Sturm-Liouville operator
  • uniqueness theorem
  • Ambarzumyan theorem

1 Introduction

Let us denote by \(L(q, \alpha, \beta)\) the Sturm-Liouville boundary value problem
$$\begin{aligned}& -y''+q(x)y=\mu y,\quad x\in(0, \pi), \mu \in\mathbb{C}, \end{aligned}$$
$$\begin{aligned}& y(0)\cot\alpha+ y'(0) = 0,\quad\alpha\in(0, \pi), \end{aligned}$$
$$\begin{aligned}& y(\pi)\cot\beta+ y'(\pi) = 0,\quad\beta\in(0, \pi), \end{aligned}$$
where q is a real-valued, summable function, \(q \in L^{1}_{\mathbb{R}}(0, \pi)\). At the same time, \(L(q, \alpha, \beta)\) denotes the self-adjoint operator generated by problem (1.1)-(1.3) (see, e.g., [13]). It is known that under the above conditions the spectrum of operator \(L(q, \alpha, \beta)\) is discrete and consists of real, simple eigenvalues (see, e.g., [2, 4]), which we denote by \(\mu_{n}=\mu_{n}(q,\alpha ,\beta)\), \(n \geq0\), emphasizing the dependence of \(\mu_{n}\) on q, α and β. We assume that eigenvalues are enumerated in the increasing order, i.e.,
$$\mu_{0}(q, \alpha, \beta) < \mu_{1}(q, \alpha, \beta) < \cdots< \mu_{n}(q, \alpha, \beta) < \cdots. $$
Let \(\varphi(x,\mu)\) be a solution of equation (1.1), which satisfies the initial conditions
$$ \varphi(0,\mu)=1,\qquad \varphi'(0,\mu)=-\cot\alpha. $$
The eigenvalues \(\mu_{n}=\mu_{n}(q, \alpha, \beta)\), \(n \geq0\), of \(L(q, \alpha, \beta)\) are the solutions of equation
$$ \varphi(\pi, \mu)\cot\beta+\varphi'(\pi, \mu)=0. $$
It is easy to see that the functions \(\varphi(x, \mu_{n})\), \(n \geq0\), are the eigenfunctions corresponding to the eigenvalue \(\mu_{n}\). The squares of the \(L^{2}\)-norm of these eigenfunctions
$$ a_{n} = a_{n}(q,\alpha,\beta) := \int_{0}^{\pi} \bigl\vert \varphi(x, \mu_{n}) \bigr\vert ^{2} \,dx,\quad n \geq0, $$
are called norming constants. The eigenvalues and norming constants are called spectral data (besides these, there are other quantities, which are also called spectral data). The inverse Sturm-Liouville problem is to reconstruct the quantities \(q, \alpha, \beta\) by some spectral data.

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]

Let \(q \in L^{1}_{\mathbb{R}}(0, \pi)\). If
$$\begin{aligned}& \mu_{n}(q, \alpha, \beta) = \mu_{n}(q_{0}, \alpha_{0}, \beta_{0}), \end{aligned}$$
$$\begin{aligned}& a_{n}(q, \alpha, \beta) = a_{n}(q_{0}, \alpha_{0}, \beta_{0}), \end{aligned}$$
for all \(n \geq0\), then \(\alpha= \alpha_{0}, \beta= \beta_{0}\) and \(q(x) = q_{0}(x)\) almost everywhere.

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

Let \(q' \in L^{2}_{\mathbb{R}}(0, \pi)\). If
$$\begin{aligned}& \mu_{n}(q, \alpha_{0}, \beta) = \mu_{n}(q_{0}, \alpha_{0}, \beta_{0}), \end{aligned}$$
$$\begin{aligned}& a_{n}(q, \alpha_{0}, \beta) \geq a_{n}(q_{0}, \alpha_{0}, \beta_{0}), \end{aligned}$$
for all \(n \geq0\), then \(\beta= \beta_{0}\) and \(q(x) \equiv q_{0}(x)\).

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 \(\alpha= \alpha_{0}\).

Remark 1

Some analogues of Theorem 1.2 will be stated in the Appendix.

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., [815] 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 \(L_{0}\).

The problem of describing all the operators L isospectral with \(L_{0}\) first was considered by Trubowitz et al. (see [1619]) for \(q \in L^{2}_{\mathbb{R}}(0, \pi)\). The same problem was considered by Jodeit and Levitan in [6] for q such that \(q' \in L^{2}_{\mathbb{R}}(0, \pi)\). For this aim the Gelfand-Levitan integral equation and transformation operators were used in [6]. They constructed the kernel \(F(x,y)\) of the integral equation as follows. Let \(c_{n}, n \geq0\), be arbitrary real numbers converging to zero, as \(n \rightarrow\infty\), so rapidly that the function
$$ F(x, y) = \sum_{n=0}^{\infty} c_{n} \varphi_{0} \bigl(x,\mu_{n}^{0} \bigr) \varphi_{0} \bigl(y, \mu_{n}^{0} \bigr) $$
is continuous and all the second order partial derivatives are also continuous. The integral equation
$$ K(x, y) + F(x, y) + \int^{x}_{0} K(x, t) F(t, y) \,dt = 0,\quad 0 \leq y \leq x \leq\pi, $$
is called Gelfand-Levitan integral equation.2
They proved that if \(1 + c_{n} a_{n}^{0} > 0\) for all \(n \geq0\), then the integral equation (2.2) has a unique solution \(K(x,y)\) and the function
$$ \varphi(x, \mu) = \varphi_{0}(x, \mu) + \int_{0}^{x} K(x, t) \varphi _{0}(t, \mu) \,dt $$
is a solution of the differential equation (1.1), with potential function
$$ q(x) = q_{0}(x) + 2 \frac{d}{dx} K(x, x), $$
and \(\varphi(x,\mu)\) satisfies the initial conditions
$$ \varphi(0, \mu) = 1, \qquad\varphi'(0, \mu) = -\cot\alpha, $$
$$ \cot\alpha= \cot\alpha_{0} + \sum _{n=0}^{\infty} c_{n}. $$
It means that the function \(\varphi(x, \mu)\) satisfies the boundary condition (1.2) for all \(\mu\in\mathbb{C}\).
Find \(\beta\in(0,\pi)\) such that \(\mu_{n}(q, \alpha, \beta) = \mu _{n}(q_{0}, \alpha_{0}, \beta_{0})\) for all \(n \geq0\), i.e., \(\varphi(x, \mu)\) should satisfy, at the point \(x = \pi\), the boundary condition (1.3)
$$\varphi \bigl(\pi, \mu_{n}^{0} \bigr) \cot\beta+ \varphi' \bigl(\pi, \mu_{n}^{0} \bigr)=0 $$
for this \(\beta\in(0, \pi)\). Such β (in [6]) is being defined from the following relation
$$ \cot\beta= \cot\beta_{0} + \sum _{n=0}^{\infty} \frac{c_{n} \varphi _{0}^{2} (\pi, \mu_{n}^{0} )}{1 + c_{n} a_{n}^{0}}. $$

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

Consider operators \(L_{0}=L(q_{0}, \alpha_{0}, \beta_{0})\) and \(L=L(q, \alpha _{0}, \beta)\) with the set of norming constants \(a_{n}^{0} = a_{n}(q_{0}, \alpha _{0}, \beta_{0})\) and \(a_{n} = a_{n}(q, \alpha_{0}, \beta)\), \(n \geq0\), respectively. It is known (see, e.g., [6]) that in this case the kernel \(F(x, y)\) of the integral equation (2.2) is
$$ F(x, y) = \sum_{n=0}^{\infty} \biggl( \frac{1}{a_{n}} - \frac{1}{a_{n}^{0}} \biggr) \varphi_{0} \bigl(x,\mu_{n}^{0} \bigr) \varphi_{0} \bigl(y, \mu_{n}^{0} \bigr). $$
Since by the condition of Theorem 1.2 the operators L and \(L_{0}\) are isospectral, then formulae (2.3)-(2.5) hold. If we compare kernels (2.1) and (3.1), we will refer that \(c_{n} = \frac{1}{a_{n}} - \frac{1}{a_{n}^{0}}\). So formulae (2.4) and (2.5) will become
$$\begin{aligned}& \cot\alpha= \cot\alpha_{0} + \sum _{n=0}^{\infty} \biggl( \frac {1}{a_{n}} - \frac{1}{a_{n}^{0}} \biggr), \end{aligned}$$
$$\begin{aligned}& \cot\beta= \cot\beta_{0} + \sum _{n=0}^{\infty} \bigl( a_{n}^{0} - a_{n} \bigr) \frac { \varphi^{2}_{0} (\pi, \mu_{n}^{0} )}{ (a_{n}^{0} )^{2}}. \end{aligned}$$
Thus, we have all the operators \(L(q, \alpha, \beta)\) isospectral with \(L(q_{0}, \alpha_{0}, \beta_{0})\).
We supposed that \(\alpha= \alpha_{0}\), then by formula (3.2) we have
$$ \sum_{n=0}^{\infty} \biggl( \frac{1}{a_{n}} - \frac{1}{a_{n}^{0}} \biggr) = 0. $$
Since \(a_{n} \geq a_{n}^{0}\) for all \(n \geq0 \), thus from equation (3.4) it refers that \(a_{n} = a_{n}^{0}\) for all \(n \geq0 \). Thus, from Marchenko’s uniqueness theorem 1.1 we obtain \(q(x) \equiv q_{0}(x)\) and \(\beta= \beta_{0}\).

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 operator3 \(L(0, \alpha, \pi- \alpha)\).

Levinson proved [20] (see also [21]) that an operator L is even if and only if
$$ \varphi(\pi, \mu_{n})=(-1)^{n}, \quad n \geq0. $$

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 \(L(0, \alpha, \pi- \alpha)\).

Therefore, taking into account that \(q_{0}(x) \equiv0\), \(\alpha_{0} = \alpha\), \(\beta_{0} = \beta= \pi- \alpha\) and (4.1), relations (2.3)-(2.5), which connect these two operators, will become
$$\begin{aligned}& q(x) = 2 \frac{d}{dx} K(x, x), \end{aligned}$$
$$\begin{aligned}& \sum_{n=0}^{\infty} c_{n} = 0, \end{aligned}$$
$$\begin{aligned}& \sum_{n=0}^{\infty} \frac{c_{n}}{1 + c_{n} a_{n}^{0}} = 0. \end{aligned}$$
If we subtract (4.3) from (4.4), we will obtain
$$ \sum_{n=0}^{\infty} \frac{c_{n}^{2} a_{n}^{0}}{1 + c_{n} a_{n}^{0}} = 0. $$
Since \(1 + c_{n} a_{n}^{0} > 0\) and \(a_{n}^{0} > 0\) for all \(n \geq0\), then from equation (4.5) we obtain that \(c_{n} = 0, n \geq0\). Thus, from equations (2.1), (2.2) and (4.2) it follows that \(q(x) \equiv0\).

Remark 3

We will get the classical Ambarzumyan theorem if we take \(\alpha= \pi/ 2\).


The theorem of Marchenko is more general, see e.g. [5, 2325].


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 \(\alpha+ \beta= \pi\).




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 (, 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

Department of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, Yerevan, 0025, Armenia


  1. Naimark, MA: Linear Differential Operators. Nauka, Moscow (1969) (in Russian) MATHGoogle Scholar
  2. Marchenko, VA: Sturm-Liouville Operators and Its Applications. Naukova Dumka, Kiiv (1977) (in Russian) MATHGoogle Scholar
  3. Levitan, BM, Sargsyan, IS: Sturm-Liouville and Dirac Operators. Nauka, Moskva (1988) (in Russian) MATHGoogle Scholar
  4. Yurko, VA: An Introductions to the Theory of Inverse Spectral Problems. Fizmatlit, Moskva (2007) (in Russian) Google Scholar
  5. Marchenko, VA: Concerning the theory of a differential operator of the second order. Dokl. Akad. Nauk SSSR 72, 457-460 (1950) (in Russian) MathSciNetGoogle Scholar
  6. Jodeit, M, Levitan, BM: The isospectrality problem for the classical Sturm-Liouville equation. Adv. Differ. Equ. 2(2), 297-318 (1997) MathSciNetMATHGoogle Scholar
  7. Ambarzumyan, VA: Über eine frage der eigenwertsththeori. Z. Phys. 53, 690-695 (1929) View ArticleGoogle Scholar
  8. Kuznezov, NV: Extensions of VA Ambarzumyan theorem. Dokl. Akad. Nauk 146, 1259-1262 (1962) Google Scholar
  9. Chakravarty, NK, Acharyya, SK: On an extension of the theorem of VA Ambarzumyan. Proc. R. Soc. Edinb. A 110, 79-84 (1988) View ArticleMATHGoogle Scholar
  10. Chern, H-H, Law, CK, Wang, H-J: Extension of Ambarzumyan’s theorem to general boundary conditions. J. Math. Anal. Appl. 263, 333-342 (2001) MathSciNetView ArticleMATHGoogle Scholar
  11. Chern, H-H, Law, CK, Wang, H-J: Corrigendum to “Extension of Ambarzumyan’s theorem to general boundary conditions”. J. Math. Anal. Appl. 309, 764-768 (2005) MathSciNetView ArticleGoogle Scholar
  12. Yang, C-F, Huang, Z-Y, Yang, X-P: Ambarzumyan’s theorems for vectorial Sturm-Liouville systems with coupled boundary conditions. Taiwan. J. Math. 14(4), 1429-1437 (2010) MathSciNetMATHGoogle Scholar
  13. Yang, Y, Wang, F: New Ambarzumyan’s theorems for differential operators with operator coefficient. Adv. Math. 40(6), 749-755 (2011) MathSciNetGoogle Scholar
  14. Yurko, VA: On Ambarzumyan-type theorems. Appl. Math. Lett. 20, 506-509 (2013) MathSciNetView ArticleMATHGoogle Scholar
  15. Yilmaz, E, Koyunbakan, H: Ambarzumyan type theorem for a matrix valued quadratic Sturm-Liouville problem. Comput. Model. Eng. Sci. 99(6), 463-471 (2014) MathSciNetMATHGoogle Scholar
  16. Isaacson, EL, Trubowitz, E: The inverse Sturm-Liouville problem, I. Commun. Pure Appl. Math. 36, 767-783 (1983) View ArticleMATHGoogle Scholar
  17. Isaacson, EL, McKean, HP, Trubowitz, E: The inverse Sturm-Liouville problem, II. Commun. Pure Appl. Math. 37, 1-11 (1984) MathSciNetView ArticleMATHGoogle Scholar
  18. Dahlberg, BEJ, Trubowitz, E: The inverse Sturm-Liouville problem, III. Commun. Pure Appl. Math. 37, 255-267 (1984) MathSciNetView ArticleMATHGoogle Scholar
  19. Poshel, J, Trubowitz, E: Inverse Spectral Theory. Academic Press, New York (1987) Google Scholar
  20. Levinson, N: The inverse Sturm-Liouville problem. Mat. Tidsskr., B 1949, 25-30 (1949) MathSciNetMATHGoogle Scholar
  21. Harutyunyan, TN: On a uniqueness theorem in the inverse Sturm-Liouville problem. Mat. Vesn. 61, 139-147 (2009) MathSciNetMATHGoogle Scholar
  22. Harutyunyan, TN: Representation of the norming constants by two spectra. Electron. J. Differ. Equ. 2010, 159 (2010) MathSciNetMATHGoogle Scholar
  23. Marchenko, VA: Concerning the theory of a differential operator of the second order. Tr. Mosk. Mat. Obŝ. 1, 327-420 (1952) (in Russian) Google Scholar
  24. Levitan, BM: Generalized Translation Operators and Some of Their Applications. Fizmatgiz, Moskva (1962) (in Russian) MATHGoogle Scholar
  25. Freiling, G, Yurko, VA: Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, New York (2001) MATHGoogle Scholar


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