Uniqueness Properties of The Solution of The Inverse Problem for The Sturm-Liouville Equation With Discontinuous Leading Coefficient

The present paper studies uniqueness properties of the solution of the inverse problem for the SturmLiouville equation with discontinuous leading coefficient and the separated boundary conditions. It is proved that the considered boundary-value is uniquely reconstructed, i.e. the potential function of the equation and the constants in the boundary conditions are uniquely determined by given Weyl function or by the given spectral data.


INTRODUCTION
This paper is concerned with the uniqueness theorems for the solution of some inverse spectral problems for the boundary value problem −y + q(x)y = λ 2 ρ(x)y , 0 ≤ x ≤ π (1) where q(x) is real-valued function in L 2 (0, π) , λ is a complex parameter, h ,h 1 are real numbers, with a ∈ (0, π) , α = 1.
Inverse spectral problems consist in recovering differential operators from their spectral characteristics (see Marchenko 2011, Levitan 1987).Such problems arise in many areas of science and engineering (see Hald 1980, Krueger 1982, Willis 1984).The goal of this work is to prove the uniqueness theorems for the solution of the inverse problem which determines the potential function q(x) and the constants h, h 1 by the Weyl function or by the spectral data of the boundary value problem (1) − (2).
The spectral analysis of the boundary value problem (1) − (2) was examined in Adiloglu and Amirov (2013) where useful integral representations for two linearly independent solutions of equation (1) were constructed (see also Akhmedova and Huseynov 2010), the asymptotic formulas for the eigenvalues and eigenfunctions were obtained, completeness and expansion theorems for the system of the eigenfunctions were proved.Using the results of Adiloglu and Amirov (2013), in the present paper, we investigate uniqueness properties of the solution of the inverse problems for the problem (1) − (2) and study some other types boundary value problems related with the equation (1).In section 2 we prove that the boundary-value problem (1) − (2) is uniquely reconstructed, i.e. the potential function q(x) and the constants h, h 1 are uniquely determined by given Weyl function or by the given spectral data.We also show that in the special case the potential q(x) and the coefficient h can be determined by the one spectrum only.In section 3 we investigate the properties of the spectral characteristics of two boundary-value problems related to Eq. (1).

UNIQUENESS OF THE SOLUTION OF THE INVERSE PROBLEM
Let s (x, λ ), c (x, λ ) be solutions of Eq. (1) with initial conditions respectively (see Adiloglu and Amirov 2013).
Denote by Φ(x, λ ) the solution of equation (1) satisfying conditions We set M(λ ) = Φ(0, λ ) and consider the linearly independent solutions s(x, λ ) and w 1 (x, λ ) of equation (1).We have where Therefore, Then Eq. (4) can be written as Additionally we see that the solution w 2 (x, λ ) also satisfies the second one of the conditions (3).
Consequently we obtain We also have The functions Φ(x, λ ) and M(λ ) are called the Weyl solution and the Weyl function of the boundary value problem (1) − (2), respectively.From equation (5) we have that the Weyl function M(λ ) is a meromorphic function with simple poles at the points λ = λ n , n ≥ 0, where λ 2 n are eigenvalues of the boundary value problem (1) − (2) (see Adiloglu and Amirov 2013).
We recall that the normalized numbers (see Adiloglu and Amirov 2013) α n of the boundary value problem (1) − (2) are defined as the set λ 2 n , α n n≥0 is the spectral data of the problem (1) − (2) .Note that there exists the sequence β n such that Adiloglu and Amirov 2013).

Theorem 1. The following formula holds
Further using the Lemma 1 (Adiloglu and Amirov ( 2013)) (see also the formula (44) there) we find Let Then by virtue of (9) we have lim N→∞ I N (λ ) = 0. On the other hand by the residue theorem which gives the desired results as N → +∞.Theorem is proved.Now let the Weyl function M(λ ) of the boundary value problem (1) − (2) is given.In the following theorem we prove that the boundary problem (1) − (2) is uniquely reconstructed, i.e. the potential function q(x) and the constants h, h 1 are uniquely determined by given Weyl function.
Let us denote the boundary value problem (1) − (2) by L = L(q(x), h, h 1 ) and the similar boundary value problem with the potential q(x) and boundary constants h, h 1 by L = L( q(x), h, h 1 ).Then the following theorem is satisfied.

2553
where where W (x,t) = W 1 (x,t) + h is continuous kernel.Then we can see the equation (26) as a Volterra integral equation with respect to cos λ x.From the theory of the Volterra integral equations, we know that the equation ( 26) is then uniquely solvable and the solution is where W 1 (x,t) is a continuous kernel.
Let now x > a.In this case the equation ( 24) is written as where α ± = 1 2 1 ± 1 α .Here the kernel W (x,t) has a jump discontinuity at t = µ − (x).Clearly, µ + (x) > a and 0 < µ − (x) < a when x > a and therefore (28) takes the form where Now using (27) we obtain that An Acad Bras Cienc (2017) 89 (4) where U(x, µ + (t)) is a continuous kernel.Therefore the equation ( 29) is written as where the kernel W (x, µ + (t)) is continuous.Hence, we obtain the Volterra integral equation where Solving the Volterra integral equation (33) with respect to α + cos λ µ + (x), we find Taking into account, the expression for h(x, λ ), we have where W (x,t) is a kernel with jump at t = µ − (x).Consequently, we have where H(x,t) is a kernel with jump at t = µ − (x).
Let q(x) = α 2 q(απ − αx) for x > a and h 1 = αh.We now show that in this case the potential q(x) and coefficient h can be determined by the spectrum λ 2 n n≥0 only.
Proof.If q(x) = α 2 q(απ − αx), x > a and y(x) is a solution of the Eq.(1) for 0 ≤ x ≤ a, then y 1 (x) := y(απ − αx) is a solution of (1) for x > a. Indeed, it is easy to check that In particular, if we take the solution is the solution of (1) satisfying the initial conditions and Amirov 2013).
Consequently we have β n = (−1) n+1 , hence from the formula 2λ n we obtain Since λ n = λ n we have α n = α n .Then by the previous theorem q(x) = q(x) a.e. on (0, π) and h = h.Theorem is proved.
BOUNDARY VALUE PROBLEMS L 0 AND L 1 (i) Consider the boundary value problem L 1 = L 1 (q(x), h) for equation (1) with the boundary conditions The characteristic function of the problem L 1 is d(λ ) = w 1 (π, λ ) and the eigenvalues of L 1 are the squares of zeros of the equation w 1 (π, λ ) = 0. Note that as in the case of the problem L we can prove that the eigenvalues µ 2 n of the problem L 1 are real and simple. Since we have for the eigenvalues µ 2 n the following asymptotic formula: where are the roots of the equation c 0 (π, λ ) = 0 and ε n = o(1), n → ∞.
Further it is easy to show that (see Adiloglu and Amirov 2013 also) the solution w 1 (x, λ ) satisfies the asymptotic relation Therefore we obtain where 40), and (41) imply that and consequently which implies where Moreover if we define the normalized numbers (α n 1 ) for the problem L 1 as then we have where Since d(λ ) is an entire function of order one by the H'Adamard's theorem the function d(λ ) is uniquely determined up to a multiplicative constant C by its zeros: We also have Since We have proved the following two theorems: Theorem 5.The boundary value problem L 1 = L 1 (q(x), h) has a countable set of eigenvalues µ 2 n n≥1 and for sufficiently large values of n, the asymptotic formula are satisfied,where where s(x, λ ) is the solution of Eq.( 1) with the initial conditions s(0, λ ) = 0, s (0, λ ) = 1.