Inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with two discontinuities
© Güldü; licensee Springer 2013
Received: 18 June 2013
Accepted: 26 August 2013
Published: 11 September 2013
In this paper, we consider a discontinuous Sturm-Liouville operator with parameter-dependent boundary conditions and two interior discontinuities. We obtain eigenvalues and eigenfunctions together with their asymptotic approximate formulas. Then, we give some uniqueness theorems by using Weyl function and spectral data, which are called eigenvalues and normalizing constants for solution of inverse problem.
MSC:34A55, 34B24, 34L05.
It is well known that the theory of Sturm-Liouville problems is one of the most actual and extensively developing fields of theoretical and applied mathematics, since it is an important tool in solving many problems in mathematical physics (see [1–4]). In recent years, there has been increasing interest in spectral analysis of discontinuous Sturm-Liouville problems with eigenvalue-linearly and nonlinearly dependent boundary conditions [1, 5–12]. Various physics applications of such problems can be found in [1, 3, 4, 13–19] and corresponding bibliography cited therein.
Some boundary value problems with discontinuity conditions arise in heat and mass transfer problems, mechanics, electronics, geophysics and other natural sciences (see  also [20–29]). For instance, discontinuous inverse problems appear in electronics for building parameters of heterogeneous electronic lines with attractive technical characteristics [20, 30, 31]. Such discontinuity problems also appear in geophysical forms for oscillations of the earth [32, 33]. Furthermore, discontinuous inverse problems appear in mathematics for exploring spectral properties of some classes of differential and integral operators.
Inverse problems of spectral analysis form recovering operators by their spectral data. The inverse problem for the classical Sturm-Liouville operator was studied first by Ambarsumian in 1929  and then by Borg in 1945 . After that, direct and inverse problems for Sturm-Liouville operator have been extended to so many different areas.
In the present paper, we construct a linear operator T in a suitable Hilbert space such that problem (1)-(7) and the eigenvalue problem for operator T coincide. We investigate eigenvalues and eigenfunctions together with their asymptotic behaviors of operator T. Besides, we study some uniqueness theorems according to Weyl function and spectral data, which are called eigenvalues and normalizing constants.
2 Operator formulation and spectral properties
such that for and also - are satisfied for f.
Thus, we can rewrite the considered problem (1)-(7) in the operator form as .
Theorem 1 The operator T is symmetric in H.
Thus, we have , i.e., T is symmetric. □
Lemma 1 Problem (1)-(7) can be considered as the eigenvalue problem of the symmetric operator T.
Lemma 2 Let , .
Lemma 3 Let , .
and (12). □
Corollary 2 The eigenvalues of problem L are simple.
Lemma 5 
where is a bounded sequence.
We can see that non-zero roots, namely of the equation , are real and analytically simple.
are valid for .
which are independent of and are entire functions such that , , .
Example Let , , , , , , , , , , , , , , , .
3 Inverse problems
In this section, we study the inverse problems for the reconstruction of the boundary value problem (1)-(7) by Weyl function and spectral data.
We consider the boundary value problem with the same form of L but with different coefficients , , , , , , , .
If a certain symbol α denotes an object related to L, then the symbol denotes the corresponding object related to .
The Weyl function Let be a solution of equation (1), which satisfies the condition and transmissions (4)-(7).
Assume that the function is the solution of equation (1) that satisfies the conditions , and the transmission conditions (4)-(7).
is called the Weyl function.
Theorem 3 If , then , i.e., , a.e. and , , , , , , .
Thus, if , then the functions and are entire in λ for each fixed x.
and similarly, . Thus, we have .
As in (22), we contradict . Therefore, , . Thus, , and . Hence, from equation (1) and transmission conditions (4)-(7), , a.e., , , , and from (9) and (10), , , , . □
Theorem 4 If and for all n, then , i.e., , a.e., , , , , , , . Hence, problem (1)-(7) is uniquely determined by spectral data .
Proof If and for all n, then by Lemma 6. Therefore, we get by Theorem 3.
Theorem 5 If and for all n, then , , , .
Hence, the problem L is uniquely determined by the sequences and , except coefficients and .
where C and are constants dependent on and , respectively. Therefore, when and for all n, and . Hence, . As a result, we get by (16). So, the proof is completed by Theorem 3. □
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