Imm.az

In the paper an inverse problem by two given spectrum for second order differential operator with singularity of type δ constant), is studied. It is well known that the two spectrum {λn} and {µ } uniquely determine the potential function q (x) in a singular Sturm-Liouvilleequation defined on interval [0,π] .
One of the aims of the paper is to prove the generalized degeneracy of the kernel of integral equation in inverse problem. In particular we obtain a newproof of Hochstadt’s theorem concerning the structure of the difference ˜ 1. IntroductionWe consider the two singular Sturm-Liouville problems where δ = constant 1 < p < 2, q (x) = δ + q qo (x) are assumed to be real valued and square integrable We denote the spectrum of the first problem by {λn}∞, and the spectrum of the Next, we denote by ϕ (x, λ) the solution of (1), and we denote by ˜ solution of (4) satisfying the initial condition (2).
It is well known [1] that there exists a function K (x, s) such that The function K (x, s) satisfies the equation ρ (λ) is called the spectral function of the problem (1)-(3), (4)-(6). Problem (1)-(3) will be regarded as an unperturbed problem, while (4)-(6)will be considered to be a perturbation of (1)-(3).
It is a known [8] fact that knowledge of two spectrum for a given singular Sturm- Liouville equation makes it possible to recover its spectral function, i.e., to find thenumbers {cn}. More exactly, suppose that in addition to the spectrum of the problem(1)-(3) [(4)-(6)] we also know the spectrum {µ } n , we can calculate the numbers {cn}. Similarly, for (4), if it then follows that we can determine the numbers {˜ 2. Statement of ResultsThis section will be devoted to a statement of the theorems, whose proofs will where q is square integrable on [0, π] and spectrum of L subject to (16) and (17).
If (17) is replaced by a new boundary conditions, a new operator and a new spectrum, say {µ } q is square integrable on [0, π]. Suppose that, under the boundary conditions real numbers which are not infinite.
We shall denote by Λ0 the finite index set for which ˜ µn = µn. Under the about assymptions, it follows that the kernel K (x, s) is degenerated in the extended sense ϕn are suitable solutions of (1) and (4).
µn differ in a finite number of their terms, then the integral equation is degenerated in the extended sense. In (22) differ in a finite number of their terms, i.e., ˜ ϕn are suitable solutions of (1) and (4).
3. Proof of Theorem 1. From (7) it follows that ϕ (x, λ) = ϕ (x, λ) + K (x, x) ϕ (x, λ) + Putting x = π, λ = λn into the last equation and using boundary conditions (17);(20), we obtain h ϕ (π, λn) + K (π, π) ϕ (π, λn) + As n → ∞, λn · ϕ (π, λn) → (−1)n the integral on the right tends to zero.
Since the system of functions ϕ (s, λn) is complete, it follows from the last equa- We now use the condition imposed on the second-named spectrum. Using (7) Putting x = π and λ = µn (n ∈ Λ) and using (18); (21), we obtain h1 ϕ (π, µn) + K (π, π) ϕ (π, µn) = 0.
In the last equation, as n → ∞, the left side tends to zero and ϕ (π, µn) → o (1) .
Therefore Comparing (25) and (29), we obtain h − ˜ The function K (x, s) satisfies (10). Therefore, from the initial conditions (33) s (x, λ) are solutions of (1)-(3) and (4)-(6) respectively satisfying Proof of Theorem 2. From (22) we obtain, for x = π , In this equation, if the expansions (33) and (23) are taken in places of K (π, s) andF (π, s) (in which λn and ˜ orthogonality of the functions ϕ (s, µn) , the equality Next, from (4) we easily obtain, for k < N k ) ϕ (π, µn) − ϕ (π, µn) ϕ (π, ˜ To calculate Fx (π, s) we differentiate (22) with respect to x ; we then obtain Putting x = π here and replacing K (π, π) by (29), F (π, s) by (36), and Kx (π, s)by (34), we find that where the bn are constants which we shall not write out.
From (23), F (x, s) satisfies the equation Therefore, from the boundary conditions (36) and (37), we find that where c (x, λ) and s (x, λ) are solutions of (4) satisfying the boundary conditions It is also evident from (23) that F (r, s) satisfies the boundary condition The same boundary condition is satisfied, obviously, by the sum (38).
Proof of Teorem 3. We obtain from (11) Differentiating (35) and putting s = r , we obtain ψn , ϕ (x, µn) = ϕn (x, µn) and ˆcn = This completes the proof of Theorem 3. We note that similar problems are investi-gated in [3], [11]-[16].
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Etibar PanakhovInstitute of Mathematics and Mechanics of NAS of Azerbaijan.
9, F. Agayev str., AZ-1141, Baku, Azerbaijan.
Tel: (99412) 439 47 20 (off.)E-mail: [email protected] Department of MathematicsFirat University23119 Elazı˘ Tel: (99412) 439 47 20 (off.)E-mail: [email protected]

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