Research Article | Open Access
I. Dehghani Tazehkand, A. Jodayree Akbarfam, "On Inverse Sturm-Liouville Problems with Spectral Parameter Linearly Contained in the Boundary Conditions", International Scholarly Research Notices, vol. 2011, Article ID 754718, 23 pages, 2011. https://doi.org/10.5402/2011/754718
On Inverse Sturm-Liouville Problems with Spectral Parameter Linearly Contained in the Boundary Conditions
In this paper, we study Sturm-Liouville problems with spectral parameter linearly contained in one of the boundary conditions. We prove uniqueness theorems for the solution of the inverse problems according to the Weyl function, spectral data, and two spectra. Then, we recover the potential function and coefficients of boundary conditions from the spectral data by the method of spectral mappings.
We consider the Sturm-Liouville boundary value problem : where the potential is a real-valued function, , and is the spectral parameter.
Sturm-Liouville problems with the spectral parameter linearly contained in the boundary conditions have been investigated extensively. Fulton  and Walter  have given an operator-theoretic formulation of the problems of the form (1.1)–(1.3). It has been shown that one can associate a self-adjoint operator in adequate Hilbert space with such problems whenever . In the case the problem (1.1)–(1.3) can be associated with a self-adjoint operator in Pontryagin space and not all eigenvalues are necessarily real (see [3, 4]). Binding et al.  have discussed the oscillation theory for these problems under a variety of assumptions on the coefficients. Basic properties and eigenfunction expansions have been studied in [6–9].
Inverse problems with parameter-dependent boundary conditions have also been studied. Browne and Sleeman  have shown that the eigenvalues and appropriately defined norming constants of the problems of the form (1.1)–(1.3) determine the potential in the sense that two such problem with identical spectra and norming constants must have the same potential. Moreover in  Guliyev has proved that the kernel of the operator transforming the function to the corresponding solution of (1.1) satisfies the Gelfand-Levitan-Marchenko-type integral equation. Then by using this equation, he has shown that the boundary value problem (1.1)–(1.3) can uniquely be determined from its spectrum and norming constants. There, he has used a method analogous to that of Gel’fand and Levitan  to reconstruction of the problem from these spectral characteristics.
In this paper we define the Weyl function for this problem and then we prove uniqueness theorems for the solution of the inverse problem of recovering from the Weyl function, spectral data, and two spectra and show connections between the different spectral characteristics. For solving inverse problem we use the method of spectral mappings, which has evolved from the contour integral method. Note that the contour integral method was first used for the study of inverse spectral problems by Levinson  and Leibenzon . Yurko  later developed the ideas of the contour integral method. This method is an effective tool for investigating a wide class of inverse problems not only for Sturm-Liouville operators, but also for other more complicated classes of operators such as differential operators with singularities and/or turning points, pencils of operators, and others (see [16–18]). In the method of spectral mappings, at first, we use Cauchy’s integral formula in the complex plane of the spectral parameter for specially constructed analytic functions having singularities connected with the given spectral characteristics. Then by this we reduce the inverse problem to the so-called main equation which is a linear equation in a corresponding Banach space of sequences. In Section 4 we give a derivation of the main equation and prove its unique solvability. Using the solution of the main equation, we provide an algorithm for the solution of the inverse problem.
Let , , and be the solutions of (1.1) satisfying the initial conditions We define where is the Wronskian of and . We note that if and are solutions of and , respectively, then By virtue of the Liouville’s formula for the Wronskian (see [19, page 83]), does not depend on . The function is called the characteristic function of . Substituting and into (2.3) we get The function is entire in and it has an at most countable set of zeroes .
Lemma 2.1. The zeros of the characteristic function coincide with the eigenvalues of the boundary value problem . The functions and are eigenfunctions, and there exists a sequence such that
Proof. Let be a zero of . Then by virtue of (2.3) we have and . Since , we have , and the functions and satisfy the boundary conditions (1.2) and (1.3). Hence is an eigenvalue, and , are eigenfunctions related to .
Conversely, let be an eigenvalue of , and let be a corresponding eigenfunction. Then . Clearly , otherwise, and by uniqueness theorem for (1.1) (see [19, Chapter 1]), . Without loss of generality, we put . Then . Hence . Thus, from (2.6), is obtained.
Let the inner product in the Hilbert space be defined by where We define an operator acting in such that with It is easy to show that are the eigenvalues of and are eigenelements of .
Lemma 2.2. The eigenvalues and the eigenfunctions and are real.
Proof. Let , be a nonreal eigenvalue with an eigenfunction . Since ), , , , and are real, we get that is also the eigenvalue with the eigenfunction . By vitue of (2.5) we get and hence with the help of (1.2) Also, by virtue of (1.3) we can write Now, from (2.14) and (2.15) we get hence , which is a contradiction. Thus, all eigenvalues are real, and consequently the eigenfunctions and are real too.
Lemma 2.3. The eigenelements of related to different eigenvalues are orthogonal in .
Proof. Let be eigenelements of related to eigenvalues , respectively. Since and , we have and hence Also, since and , from (2.5) we get Thus,
Lemma 2.4. The following relation holds: where the numbers are defined by (2.7) and .
Lemma 2.6. For, , the following asymptotic formulae hold: uniformly with respect to . Here and in the sequel and .
Proof. The asymptotic formulae (2.27) have been proved in [20, Lemma ]. We prove (2.28). Let us show that
Since satisfies (1.1), we have Substituting this in the following integral and twice integrating by parts the term involving , we obtain (2.29).
Differentiating (2.29), we calculate In (2.29), we put . Then Let . Then using the inequalities we obtain For sufficiently larg , this gives Hence , as , and therefore , uniformly with respect to as . Substituting this estimate into right-hand sides of (2.29) and (2.32), we get (2.28).
Theorem 2.7. The boundary value problem has a countable set of eigenvalues . Moreover, for , where
Proof. (1) By virtue of the following asymptotic formulae (see [20, page 7])
and using (2.6) we get
Denote , . Using (2.43) we get for
Now, since (see [20, page 6])
for sufficiently large .
Let . By virtue of (2.43), It follows from (2.46) that , , for sufficiently large . Hence by Rouché’s theorem [21, page 125], the number of zeros of inside coincides with the number of zeros of , that is, it equals . Thus, in the circle there exist exactly eigenvalues of . Analogously, by using Rouché’s theorem one can prove that for sufficiently large vales of , every circle contains exactly one zero of , namely, . Since is arbitrary, we must have Since are zeros of , from (2.43) we get and consequently where . Hence , that is, . Thus, . Using (2.51) once more we get more precisely , where .
(2) Substituting (2.37) into (2.41), we get (2.38) where Therefore, , and (2.38) is proved.
(3) From (2.38) we calculate Also, using (1.3), (2.37), and (2.38) we obtain Thus, and Theorem 2.7 is proved.
Theorem 2.8. The specification of the spectrum uniquely determines the characteristic function by the formula
Proof. It follows from (2.6), (2.27), and (2.28) that is an entire function of of order , and hence by Hadamard’s factorization theorem [21, page 289], is uniquely determined up to a multiplicative constant by its zeros: The case requires minor modifications. We consider the function Then With the help of (2.37) and (2.43), we calculate and hence Substituting this into (2.57), we get (2.56).
Remark 2.9. Analogous results are valid for boundary value problems with other types of spectral parameter-dependent boundary conditions. Let us state some of these results for one of them which will be used below.
Consider the boundary value problem for (1.1) with the boundary conditions . The eigenvalues of are simple and coincide with the zeros of the characteristic function and where
3. Inverse Problems
In this section, we study three inverse problems of recovering from its spectral characteristics, namely,(i)from the Weyl function,(ii)from the so-called spectral data,(iii)from two spectra.
For each class of inverse problems we prove the corresponding uniqueness theorems and show connection between the different spectral characteristics.
3.1. The Inverse Problem from the Weyl Function
Let be the solution of (1.1) under the conditions and . We set . The functions and are called the Weyl solution and the Weyl function for the boundary value problem , respectively. The notion of the Weyl function introduced here is a generalization of the Weyl function for the classical Sturm-Liouville operators (see [20, 22]). Clearly Since and have no common zeros, it follows from (3.2) that is a meromorphic function with poles and zeros . We consider the following inverse problem.
Inverse Problem 1
Given the Weyl function , construct , , , , and .
Let us prove the uniqueness theorem for the solution of the Inverse Problem 1. For this purpose we agree that together with we consider a boundary value problem of the same form but with different coefficients , , , , and . Everywhere below if a certain symbol denotes an object related to , then the corresponding symbol with tilde denotes the analogous object related to .
Theorem 3.1. If , then a.e. on , , , and . Thus, the specification of the Weyl function uniquely determines .
Proof. Let us define the matrix by the formula Using (3.3) and (3.4) we calculate for : It follows from (3.1), (3.3), and (3.5) that By virtue of (2.27), (2.28), and (2.47), this yields Uniformly with respect to . Similarly, we have Uniformly with respect to . On the other hand according to (3.1) and (3.5), Since , it follows that for each fixed , the functions and are entire in . With the help of (3.8), this yields , . Substituting into (3.6), we get , for all and . Hence from (1.1) and (2.1) we get a.e. on , , , , and . Consequently, .
3.2. The Inverse Problem from the Spectral Data
Let and be the eigenvalues and norming constants of , respectively. We consider the following inverse problem.
Inverse Problem 2
Given the spectral data , construct , and .
Lemma 3.2. The following representation holds:
Proof. Consider the contour integral where the contour is assumed to have the counterclockwise circuit. Since , it follows from (2.28) that . Then, using (3.1) and (2.47), we get for sufficiently large , Moreover, using (3.2) and (2.23), we calculate In view of (3.13), . By virtue of (3.14) and residue theorem [21, page 112], we have and consequently (3.11) is proved.
Let us prove a uniqueness theorem for the solution of Inverse Problem 2.
Theorem 3.3. If and for all , then a.e. on , , , and . Thus, the specification of the spectral data , uniquely determines .
Remark 3.4. By virtue of (3.11), the specification of the Weyl function is equivalent to the specification of the spectral data , that is, the Inverse Problem 1 is equivalent to the Inverse Problem 2.
3.3. The Inverse Problem from Two Spectra
Let and be the eigenvalues of the problems and , respectively. We consider the following inverse problem.
Inverse Problem 3
Given two spectra , construct , , , , and .
Let us prove a uniqueness theorem for the solution of Inverse Problem 3.
Theorem 3.5. If and , then a.e. on , , , , and . Thus, the specification of two spectra , uniquely determines .
Proof. According to Lemma 2.1 and Remark 2.9 the sets and coincide with the set of zeros of the functions and , respectively. Using (2.56) and (2.62), we get and . Together with (3.2) this yields . By Theorem 3.1 we get a.e. on , , , , and .
Remark 3.6. It follows from Theorems 3.1 and 3.5 that the specification of Weyl function is equivalent to the specification of two spectra , that is, the Inverse Problem 1 is equivalent to the Inverse Problem 3.
4. Solution of the Inverse Problem
In this section, we give a constructive procedure for the solution of the inverse problem of recovering from the given spectral data by the method of spectral mappings and state necessary and sufficient solvability conditions.
Let be the spectral data of . Denote
Let us choose a model boundary value problem with real , , , , , and such that (take, e.g., , , ). Let be the spectral data of . Denote Since , it follows from (2.37), (2.39), and analogous formulae for and that Denote where , . Let , , . It follows from (2.27) and (2.37) that Moreover, for a fixed Applying Schwarz’s lemma [21, page 130] in the -plane to the circle and to the function with fixed , , , and , we get In particular, this yields By similar arguments (see [20, page 48]) one gets that the following estimates are valid for , , : The analogous estimates are also valid for and .
Lemma 4.1. The following relations hold: Both series converge absolutely and uniformly with respect to and , on compact sets.
Proof. (1) Denote and take a fixed .
Let , and let be the boundary of . Denote , , . In the -plane we consider closed contours (with counterclockwise circuit), (with clockwise circuit). Let be the matrix defined by (3.4). It follows from (3.5) and (3.1) that for each fixed , the functions are meromorphic in with simple poles and . By Cauchy’s integral formula [21, page 84], where is the Kronecker delta. Hence where is used with counterclockwise circuit. Substituting into (3.6) we obtain where By virtue of (3.8) we have that uniformly with respect to and on compact sets. Using (3.5) we calculate This, in combination with (3.1), implies that where , since the terms with vanish by Cauchy’s theorem [21, page 85]. It follows from (3.14) that Applying residue theorem to the integral in (4.18) and using (4.16) we obtain (4.10).
(2) Since we have by Cauchy’s integral formula Acting in the same way as above and using (3.8) and (3.9), we obtain where , .
It follows from (3.5) and (3.3) that for any . Using (4.22) and (4.24), we get where . From (3.4) and (4.23), we get Hence, for , (4.25) gives where . By virtue of (3.1), (3.14), and the residue theorem, we get (4.11).
It follows from the definition of , , and from (4.10) and (4.11) that For each , the relation (4.28) can be considered as a system of linear equations with respect to , , . But the series in (4.28) converges only “with brackets.” Therefore, it is not convenient to use (4.28) as a main equation of the inverse problem. Below we will transfer (4.28) to a linear equations in corresponding Banach space of sequences (see (4.38) or (4.40)).
Let be a set of indices , , . For each fixed , we define the vector by the formulae where We also define the block matrix by the formulae Analogously we define , by replacing in the previous definitions, by and by . It follows from (4.5)–(4.9) that Similarly
Let us consider the Banach space of bounded sequences with the norm