Olga Boyko, Olga Martinyuk, and Vyacheslav Pivovarchik

Transcription

1 Opuscula Math. 36, no ), Opuscula Mathematica HIGHER ORDER NEVANLINNA FUNCTIONS AND THE INVERSE THREE SPECTRA PROBLEM Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi Communicated by P.A. Cojuhari Abstract. The three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on [0, a], the Dirichlet-Dirichlet problem on [0, a/] and the Neumann-Dirichlet problem on [a/, a] is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found. Keywords: spectrum, eigenvalue, Dirichlet boundary condition, Neumann boundary condition, Marcheno equation, Nevanlinna function. Mathematics Subject Classification: 34A55, 34B4. 1. INTRODUCTION In [15] the so-called three spectra inverse problem was introduced see also [6], and generalizations in [1 5, 9, 16, ]). There the spectrum of a boundary value problem generated by the Sturm-Liouville equation on a finite interval is given together with the spectra of the boundary value problems generated by the same equation on complementary subintervals and the aim is to find the equation. Let us describe a functional equation which appears in both the Hochstadt- -Liebermann problem see [7, 8, 13, 17, 19 1] for different versions of it) and in the three spectra inverse problem. It is more convenient to measure distance on the right half of the interval in the opposite direction. Then we can write our problem as follows: y j + q j x)y j = y j, x [0, a/], j = 1,, 1.1) y 1 0) = 0, 1.) y 0) = 0, 1.3) y 1 a/) = y a/), 1.4) y 1a/) + y a/) = ) c AGH University of Science and Technology Press, Kraow

2 30 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi Here q j L 0, a/) j = 1, ) are real valued. We denote the eigenvalues of problem 1.1) 1.5) by { }, 0 = ). We introduce s j, x), the solution of the Sturm-Liouville equation 1.1) which satisfies the conditions s j, 0) = 0, s j, 0) = 1. Let us consider also the following problems on the subintervals: y j + q j x)y j = y j, x [0, a/], 1.6) y j 0) = 0, 1.7) y j a/) = 0, 1.8) the spectra {ν j) }, 0 j = 1, ) of which coincide with the sets of zeros of s j, a/) and problem y j + q j x)y j = y j, x [0, a/], 1.9) y j 0) = 0, 1.10) y j a/) = 0, 1.11) the spectra {µ j) }, 0 j = 1, ) of which coincide with the sets of zeros of s j, a/). Let us loo for a solution to problem 1.1) 1.5) in the form y 1 = C 1 s 1, x), y = C s, x) where C j are constants. Then 1.4) and 1.5) imply C 1 s 1, a/) = C s, a/), C 1 s 1, a/) + C s, a/) = 0. This system of equations possesses a nontrivial solution at the zeros of the characteristic function Φ) = s 1, a/)s, a/) + s, a/)s 1, a/). 1.1) The set of zeros { }, 0 of this function is the spectrum of problem 1.1) 1.5). Let us notice that equation 1.1) is a particular case of the second of equations.18) in Theorem.1 of [11] which appears in the spectra theory of quantum graphs. Equation 1.1) plays an important role in the theory of the three spectra problem and in the theory of the Hochstadt-Liebermann problem. Since nowing q 1 x) we can solve equation 1.1) with j = 1 and find s 1, a/) and s 1, a/), nowing { }, 0 we can find Φ), the Hochstadt-Liebermann problem can be formulated as follows: given Φ), s 1, a/) and s 1, a/) find q x). The three spectra problem of [15] can be formulated as follows: given Φ), s 1, a/) and s, a/) find q 1 x) and q x). In the same way as in [15] one can solve the following problem: given Φ), s 1, a/) and s, a/) find q 1 x) and q x). In the present paper we introduce the notion of the Nevanlinna function of higher order and essentially positive Nevanlinna functions of higher order Section ) which we use in Section 3 to describe boundary value problems with conditions at two or more interior points of the interval. There we compare the spectrum of the Dirichlet problem on the whole interval with the union of the spectra of the Dirichlet problems on the complementary subintervals.

3 Higher order Nevanlinna functions and the inverse three spectra problem 303 In Section 4 we deal with a direct three spectra problem. We show that Φ z) s 1 z, a/)s z, a/) and s 1 z, a/)s z, a/) Φ z) belong to the class of essentially positive Nevanlinna functions see Definition.3 below), while s 1 z, a/)s z, a/) Φ z) belongs to the class of essentially positive Nevanlinna functions of the second order see Definition.6 below). A consequence of these facts appears in certain mutual order in the location of zeros of Φ) and zeros of the product s 1, a/)s, a/). In Section 5 we consider the following inverse problem. Given the spectra of problem 1.1) 1.5), of problem 1.6) 1.8) with j = 1 and of problem 1.9) 1.11) with j =, in other words given Φ), s 1, a/) and s, a/), find q 1 x) and q x). We give sufficient conditions of solvability and uniqueness of solutions to such a problem.. NEVANLINNA FUNCTIONS In the sequel we will use the notion of the Nevanlinna function, also called the R-function in [10]. In this paper we deal only with meromorphic functions. Definition.1 see e.g. [14, Definition 5.1.0]). The meromorphic function θ is said to be a Nevanlinna function, or an R-function, or an N -function if: i) θ is analytic in the half-planes Im > 0 and Im < 0; ii) θ) = θ) if Im 0; iii) Im Imθ) > 0 for Im 0. Lemma. see e.g. [14, Lemma 5.1,]). If θ is a Nevanlinna function, then so are the functions 1 θ and 1 θ + c) 1 for any real constant c. Proof. This easily follows from Im 1 ) = Imθ) θ) θ) ) 1 1 and Im θ) + c = Im 1 θ) 1 θ) + c. Definition.3 see e.g. [18] or [14, Definition 5.1.6]). 1. The class N ep of essentially positive Nevanlinna functions is the set of all functions θ N which are analytic in C \ [0, ) with the possible exception of finitely many poles.. The class N ep + N ep ) is the set of all functions θ N ep such that for some γ R we have θ) > 0 θ) < 0) for all, γ). Lemma.4 see e.g. [18]). Let θ N ep +. Then the zeros {a } =1 and poles {b } =1 of θ interlace, i.e. < b 1 < a 1 < b < a <....

4 304 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi Lemma.5 see [18] or [14, Theorem ]). θ N ep ± if and only if 1 θ N ep. Proof. Since poles and zeros of θ N ep + interlace by Lemma.4, there is γ 0 such that θ has no poles or zeros in, γ) and we have that θ) > 0 for all, γ). This means that 1 1 θ also has no poles and zeros in, γ) and θ) < 0 for all, γ) Definition.6. A function fz) ep gz) is said to belong to N+,p essentially positive Nevanlinna class of order p) if there exist functions g 1 z), g z),..., g p z) such that fz) g 1 z) N ep +, g 1 z) g z) N ep +,..., g p 1 z) g p z) N ep +, g p z) gz) N ep +. Here N ep ep +,0 =: N+. It is clear that N ep +,r N ep +,p for p > r. Theorem.7. Let fz) gz) N +,p. ep Then zeros {a } m =1 {b } m1 =1 m 1 ) of gz) satisfy the conditions: m ) of fz) and zeros 1. the interval, b 1 ] does not contain elements of {a } ) =1,. for each interval, b ) contains not more than 1 and not less than 1 p elements of {a } ) =1. Proof. Denote by {τ j) } =1 j = 1,,..., p) the sequence of zeros of g jz). Statement 1 is obvious due to < b 1 < τ 1) 1 < τ ) 1 < <τ p) 1 < a 1 < a <.... The interval, b ) containes exactly 1 elements of {τ 1) } =1. The interval τ 1) 1, τ 1) ) ) containes exactly one element of {τ } ) =1, namely τ 1 which can lie either inside τ 1) 1, b ) or outside of it. Thus, the number of elements of {τ ) } =1 belonging to, b ) is either 1 or. Thus, our theorem is proved for p = 1. In the case of p > 1 we consider the following cases: 1. τ ) 1) 1 τ 1, b ). If τ 3) 1 < b, then since τ 3) > τ ) > τ 1) > b we conclude that the interval, b ) contains 1 element of {τ 3) } =1. If τ 3) 1 > b, then since τ 3) < τ ) 1 < b we conclude that, b ) contains element of {τ 3) } =1. 1 [b, τ 1) 3) ). If τ [b, τ ) 3) 1 ), then since τ 3 < τ ) < τ 1) 1 < b we. τ ) conclude that, b ) contains 3 element of {τ 3) } =1. If τ 3) < b, then since τ 3) 1 > τ ) 1 > b and, b ) contains element of {τ 3) } =1. Thus, our theorem is true for p =. The case of any p > can be treated in the same way.

5 Higher order Nevanlinna functions and the inverse three spectra problem 305 Theorem.8. Let hz) fz) N ep + and hz) gz) N ep +. Then zeros {a } m =1 m ) of fz) and zeros {b } m1 =1 m 1 ) of gz) satisfy the condition: each interval, b ) contains 1 or elements of {a } ) =1. Proof. Denote by {d } =1 the zeros of hz). Then and a 1 < d 1 < b < d < a +1 d 1 < a < d. If a d 1, b ), then, b ) contains elements of {a } ) =1. If a [b, d ), then, b ) contains 1 elements of {a } ) =1. 3. SPECTRAL PROBLEM OF VIBRATIONS OF A SMOOTH STRING CLAMPED AT MORE THAN ONE INTERIOR POINT We obtain an example where functions of N ep +,p can be used considering the problem of vibrations of a smooth string clamped at a few interior points. Let 0 = a 0 < a 1 < a <... < a n = a and consider the Dirichlet problems { y + qx)y = y, 3.1) ya 0 ) = ya n ) = 0, { y + qx)y = y, where j = 1,,..., n, 3.) ya j 1 ) = ya j ) = 0, generated by the same real q L 0, a). Denote by s j, x) the solution of the differential equation in 3.) which satisfies s j, a j 1 ) = s j, a j 1) 1 = 0 j = 1,,..., n + 1). Then s j, a j a j 1 ) j = 1,,..., n) are the characteristic functions of problems 3.). The case of n = was considered in [15]. Theorem 3.1. For n, n j=1 s j z, a j a j 1 ) s 1 z, a) Proof. Let us consider the boundary value problems { y + qx)y = y, ya j ) = ya n ) = 0. N ep +,n. 3.3) The characteristic function of problem 3.1) is s 1 z, a), the characteristic function of problems 3.) are s j z, a j a j 1 ) j = 1,,..., n), the characteristic function of problem 3.3) is s j+1 z, a). Evidently, s 1 z, a 1 )s z, a) s 1 z, a) N ep +,

6 306 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi s 1 z, a 1 )s z, a a 1 )s 3 z, a) s 1 z, a 1 )s z, a) s 1 z, a 1 )s z, a a 1 )s 3 z, a 3 a )s 4 z, a) s 1 z, a 1 )s z, a a 1 )s 3 z, a) = s z, a a 1 )s 3 z, a) s N ep +, z, a) = s 3 z, a 3 a )s 4 z, a) s 3 N ep +. z, a) Theorems.7 and 3.1 imply the following corollary. Corollary 3.. Let n. The spectrum { }, 0 of problem 3.1) is related with the union {η }, 0 = n j=1 {νj) }, 0 of the spectra of problems 3.) in the sence that each interval, ) contains not more than 1 and not less than + 1 n elements of {η } =1. Lemma 3.3. If any two of the three equalities s 1, a) = 0, s 1, a j ) = 0, s j, a) = 0 are true, then the third of them is true also. Proof. This follows from the identity s 1, a) = s 1, a j )s n+1, a j ) + s 1, a j )s n+1, a j ) which is an analogue of 1.1) because the sets of zeros of s n+1, a j ) and of s j, a) coincide Corollary 3.4. Let n > and > n 1. Then if = η n+1 = η n+ =... = η 1 then = η. Proof. Condition = η n+1 = η n+ =... = η 1 means that among s j, a j a j 1 ) j = 1,,..., n) at least n 1 are equal to zero. Suppose s p, a p a p 1 ) 0. Then, by Lemma 3.3, s 1, a) = 0 and s 1, a p 1 ) = 0 imply s p, a) = 0. Now s p, a) = 0 and s p+1, a) = 0 imply s p, a p a p 1 ) = 0, a contradiction. 4. DIRECT THREE SPECTRA PROBLEM Here we compare the spectrum of problem 1.1) 1.5) with the union of the spectrum of problem 1.6) 1.8) with j = 1 and the spectrum of problem 1.9) 1.11) with j =. We will use the following simple result. Lemma 4.1. Let Φ be given by 1.1). Then the function s z, a/)s 1 z, a/) Φ z) is an essentially positive Nevanlinna function.

7 Higher order Nevanlinna functions and the inverse three spectra problem 307 Proof. It is nown that s1 z,a/) s 1 z,a/) and s z,a/) s z,a/) functions. Therefore, such is also s z, a/)s 1 z, a/) Φ z) = s1 ) 1 z, a/) s 1 + z, a/) by arguments similar to the proof of Lemma. are essentially positive Nevanlinna s ) 1 ) 1 z, a/) s z, a/) Theorem 4.. The sequences { }, 0 and {ζ } def, 0 = {ν 1) }, 0 {µ ) }, 0 are interlaced as follows: each interval, ) ) contains or 1 elements of the sequence {ζ ) } =1. Proof. This follows from s 1 z, a/)s z, a/) Φ z) N ep +, s 1 z, a/)s z, a/) s 1 z, a/)s z, a/) N ep + and Theorem.8. In the next section we will use the following nown result. Lemma 4.3 see, e.g. [1, Theorem 3.4.1] with a = π or [14, Corollaries and 1.5.]). If q j L 0, a/), then the sequences { }, 0, {µj) }, 0, {ν j) }, 0, which are the sets of zeros of the functions s, a) = where A 0 = A 1 + A, A j = 1 sin a A 0 cos a a/ 0 q j x)dx, ψ L a, + ψ), s j, a/) = cos a + A j sin a + ψ j ), s j, a/) = sin a cos a A j + ψ j) with ψ j L a/, ψ j L a/, behave asymptotically as follows: = π a + A 0 π + β, 4.1) µ j) π 1) = + A j a π ν j) = π a j) β +, 4.) + A j π + βj), 4.3) where {β }, 0 l, {β j) }, 0 l j), { β }, 0 l j = 1, ).

8 308 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi 5. THREE SPECTRA INVERSE PROBLEM Now we consider a three spectra inverse problem in which the spectrum of problem 1.1) 1.5) is given together with f the spectrum of problem 1.6) 1.8) with j = 1 and the spectrum of problem 1.9) 1.11) with j =. The aim is to recover q j j = 1, ). In other words, in this case the sets of zeros of Φ), s 1, a/) and s, a/) are given. Definition 5.1. An entire function of exponential type a is said to belong to the Paley-Wiener class L a if its restriction onto the real axis belongs to L, ). Lemma 5.. The set of zeros of the function Φ) = cos a sin a cos a A sin a + A 1 + ψ), 5.1) where A j are real constants, ψ L a can be given as the union of two sets denote them by {ζ ) }, 0 and {ζ1) }, 0 ) such that ζj) = ζj) j = 1, ) and ζ ) ζ 1) = where {γ }, 0 l, { γ }, 0 l. Proof. The function Φ) can be presented as follows: = π a + A π + γ, 5.) π 1) + A 1 a π + γ, where Φ) = Φ 0 ) + ψ 1), Φ 0 ) = cos a sin a = sin a cos a A ) cos a A and Ψ 1 L a. Let us consider circles C ρ) of radii ρ ) sin cos π + aa π + aρ eiχ π + aa π + aρ eiχ Φ 0 π a + A π + ρ eiχ sin a + A 1 cos a + A 1 A 1 A cos a ) sin a centered at π a = 1) 1 aa π + ρa ) ) 1 = 1) 1 + O, ) ) = ρa 1 π eiχ + O 3. eiχ sin a 3 + A π. If χ [0, π), then ) + O ) 1 3,

9 Higher order Nevanlinna functions and the inverse three spectra problem 309 Therefore, for any ɛ 0, ρa ) and any χ [0, π) there exists ɛ N such that for > ɛ π Φ 0 a + A π + ρ ) eiχ > ρa ɛ π. By Lemma of [1] or Lemma 1..1 of [14], π ψ 1 a + A π + ρ ) eiχ 0 uniformly with respect to χ [0, π). Thus, we conclude that for large enough π a + A π + ρ ) π eiχ ψ 1 a + A π + ρ ) eiχ < ρa ɛ π < Φ 0 π a + A π + ρ ) eiχ. It is clear that the set of zeros of Φ 0 consists of two subsequences and one of these subsequences due to its asymptotics has exactly one element in each circle C ρ) for > ɛ. Hence by Rouche s theorem we conclude that for large enough each such circle contains exactly one zero of Φ). Since ρ can be chosen arbitrary small we conclude that there is a subsequence of zeros of Φ) of the form ζ ) = π a + A π +, 5.3) where = o1). Substituting 5.3) into Φζ ) ) = 0 and using 5.1) we obtain 5.). In the same way we obtain the asymptotics for {ζ 1) }, 0. Theorem 5.3. Let three sequences {ν 1) }, 0 ν1) µ ) = µ) = ν1) ), {µ) }, 0 ) and { }, 0 = ) satisfy the following conditions: 1. {ν 1) }, 0 satisfy 4.3) with j = 1, {µ) }, 0 satisfy 4.) with j = and { }, 0 satisfy 4.1), where A 0 = A 1 + A,. < 1 ) < ζ 1 ) = µ ) 1 ) < ) < ζ ) <..., where {ζ }, 0 := {ν 1) }, 0 {µ ) }, 0. Then there exists a unique pair of real valued functions q j x) L 0, a/) j = 1, ) which generate problem 1.6) 1.8) with j = 1 and the spectrum {ν 1) }, 0, problem 1.9) 1.11) with j = and the spectrum {µ ) }, 0 and problem 1.1) 1.5) with the spectrum { }, 0.

10 310 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi Proof. Let us construct We consider the function P ) := a =1 φ 1 ) := a =1 φ ) := =1 a ) π ), a ) 1) ν π ) ), a π 1) Since see, e.g. [1, Lemma 3.4.], [14, Lemma 1.3.3]) where τ belongs to L a, and ) µ ) ) ), Φ) := P ) φ 1 )φ ). 5.4) P ) = sin a A 0 cos a φ ) = cos a + A sin a φ 1 ) = sin a cos a A 1 + τ), + τ ), + τ 1) with τ L a/, τ 1 L a/, we conclude that Φ) given by 5.4) can be represented as in 5.1). Therefore, Φ satisfies the conditions of Lemma 5. and the set of zeros of Φ) consists of two subsequences which we denote by {ν ) }, 0 and {µ1) }, 0 which behave asymptotically as follows: ν ) µ 1) = = π a + A π + γ, π 1) a + A 1 π + γ, where {γ }, 0 l and { γ }, 0 l. Let us notice that Φ) > 0 for. Since Φ ) = φ 1 )φ ), condition implies Φ ) 1) > 0. Since Φζ ) = P ζ ), condition implies Φζ ) 1) = P ζ ) 1) > 0. Taing into account the asymptotics above) we conclude that each interval, 1), ζ 1, ), ζ, 3),..., contains exactly one zero of Φ z). Now we denote the zero of

11 Higher order Nevanlinna functions and the inverse three spectra problem 311 Φ z) lying in, 1) by µ ) 1 ). We denote the zero of Φ z) lying in ζ, +1 ) by ν r ) ) if ζ = µ ) r and by µ 1) r+1 ) if ζ = ν r 1). Therefore, {ν ) ) } =1 interlace with {µ ) ) } =1 : < µ ) 1 ) < ν ) 1 ) < µ ) ) < ν ) ) <... Thus the sets {ν ) }, 0 and {µ) }, 0 satisfy the conditions of Theorem in [1] with a = π see also Theorem 1.6. in [14]) and there exists a unique real-valued function q x) L 0, a ) which generates the Dirichlet-Dirichlet and the Dirichlet-Neumann problems on [0, a ] with the spectra {ν) }, 0 and {µ) }, 0, respectively. We can find q x) via procedure [1, Theorem 3.4.1, p. 48] or [13, Theorem 1.6.]) described below. Without loss of generality let us assume that µ 1 > 0, otherwise we apply a shift. Using the functions φ ) and we construct φ ) := a =1 a ) ) ν π ) ), e) = φ ) + i φ ))e i a which is the so-called Jost-function of the corresponding prolonged Sturm-Liouville problem on the semi-axis: y + qx)y = y, x [0, ), with qx) = y0) = 0 { lq x) for x [0, a/], 0 for x a/, ). Then we construct the S-function of the problem on the semi-axis: and the function F x) = 1 π Solving the Marcheno equation S) = K x, t) + F x + t) + e) e ) 1 S))e ix dx, x K x, s)f s + t)dp = 0

12 31 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi we find K x, t) and the potential q x) = dk x, x), dx which is a real-valued function and belongs to L 0, a/). This potential generates the Dirichlet-Neumann problem with the characteristic function s, a/) φ ) and the spectrum {µ ) }, 0 and Dirichlet-Dirichlet problem with the characteristic function s, a/) φ ). In the same way we construct q 1 x) using the sequences {µ 1) }, 0 and {ν 1) }, 0. It is clear that the obtained q 1x) generates the Dirichlet-Neumann problem with the characteristic function s 1, a/) φ 1 ) =: =1 a π 1) ) µ 1) ) ), and the Dirichlet-Dirichlet problem with the characteristic function s 1, a/) φ 1 ). Now if we solve problem 1.1) 1.5) with obtained q 1 and q, we find the characteristic function φ) =: s 1, a/)s, a/)+s, a/)s 1, a/) = φ 1 ) φ )+φ ) φ 1 ) = P ) with the set of zeros { }, 0. REFERENCES [1] O. Boyo, O. Martynyu, V. Pivovarchi, On a generalization of the three spectral inverse problem, Methods of Functional Analysis and Topology, accepted for publication. [] O. Boyo, V. Pivovarchi, Ch.-F. Yang, On solvability of the three spectra problem, Mathematische Nachrichten, accepted for publication. [3] M. Drignei, Uniqueness of solutions to inverse Sturm-Liouville problems with L 0, a) potential using three spectra, Advances in Applied Mathematics 4 009), [4] M. Drignei, Constructibility of an L R0, a) solution to an inverse Sturm-Liouville using three Dirichlet spectra, Inverse Problems 6 010), 05003, 9 pp. [5] S. Fu, Z. Xu, G. Wei, Inverse indefinite Sturm-Liouville problems with three spectra, J. Math. Anal. Appl ), [6] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential, I, the case of discrete spectrum, Trans. Am. Math. Soc ), [7] O. Hald, Inverse eigenvalue problem for the mantle, Geophys. J. R. Astr. Soc ), [8] H. Hochstadt, B. Lieberman, An inverse Sturm-Liouville problems with mixed given data, SIAM J. Appl. Math ),

13 Higher order Nevanlinna functions and the inverse three spectra problem 313 [9] R.O. Hryniv, Ya.V. Myytyu, Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra, J. Math. Anal. Appl ), [10] I.S. Kac, M.G. Krein, R-functions analytic functions mapping the upper half-plane into itself, Amer. Math. Transl. ) ), [11] C.-K. Law, V. Pivovarchi, Characteristic functions of quantum graphs, J. Phys. A: Math. Theor ) 3, 03530, 11 pp. [1] V.A. Marcheno, Sturm-Liouville Operators and Applications, Birhäuser OT, Basel, [13] O. Martynyu, V. Pivovarchi, On Hochstadt-Lieberman theorem, Inverse Problems 6 010) 3, , 6 pp. [14] M. Möller, V. Pivovarchi, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birhäuser OT, 46, Basel, 015. [15] V. Pivovarchi, An inverse Sturm-Liouville problem by three spectra, Integral Equations and Operator Theory ), [16] V. Pivovarchi, Reconstruction of the potential of the Sturm-Liouville equation from three spectra of boundary value problem, Funct. Anal. Appl ) 3, [17] V. Pivovarchi, On Hald-Gesztesy-Simon, Integral Equations and Operator Theory 73 01), [18] V. Pivovarchi, H. Worace, Sums of Nevanlinna functions and differential equations on star-shaped graphs, Operators and Matrices 3 009) 4, [19] L. Sahnovich, Half-inverse problem on the finite interva, Inverse Problems ), [0] T. Suzui, Inverse problems for heat equations on compact intervals and on circles, I, J. Math. Soc. Japan ) 1, [1] G. Wei, H.-K. Xu, On the missing eigenvalue problem for an inverse Sturm-Liouville problem, J. Math. Pures Appl ), [] G. Wei, X. Wei, A generalization of three spectra theorem for inverse Sturm-Liouville problems, Appl. Math. Lett ), Olga Boyo boyohelga@gmail.com South-Urainian National Pedagogical University Staroportofranovsaya 6, Odesa, Uraine, 6500

14 314 Olga Boyo, Olga Martinyu, and Vyacheslav Pivovarchi Olga Martinyu South-Urainian National Pedagogical University Staroportofranovsaya 6, Odesa, Uraine, 6500 Vyacheslav Pivovarchi corresponding author) South-Urainian National Pedagogical University Staroportofranovsaya 6, Odesa, Uraine, 6500 Received: October 9, 015. Accepted: November 16, 015.