We study the inverse problem for non-selfadjoint Sturm-Liouville operators on a finite interval with possibly multiple spectra. We prove the uniqueness theorem and obtain constructive procedures for solving the inverse problem along with the necessary and sufficient conditions of its solvability and also prove the stability of the solution.
MSC: 34A55, 34B24, 47E05.
Keywords:non-selfadjoint Sturm-Liouville operators; inverse spectral problems; method of spectral mappings; generalized spectral data; generalized weight numbers
Inverse spectral problems consist of recovering operators from given spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see, for example, monographs [1-7] and the references therein). We study the inverse problem for the Sturm-Liouville operator corresponding to the boundary value problem of the form
where is a complex-valued function. The results for the non-selfadjoint operator (1), (2) that we obtain in this paper are crucial in studying inverse problems for Sturm-Liouville operators on graphs with cycles. Here also lies the main reason of considering the case of Dirichlet boundary conditions (2) and arbitrary length T of the interval.
For the selfadjoint case, i.e., when is a real-valued function, the inverse problem of recovering L from its spectral characteristics was investigated fairly completely. As the most fundamental works in this direction we mention [8,9], which gave rise to the so-called transformation operator method having become an important tool for studying inverse problems for selfadjoint Sturm-Liouville operators. The inverse problems for non-selfadjoint operators are more difficult for investigation. Some aspects of the inverse problem theory for non-selfadjoint Sturm-Liouville operators were studied in [10-14] and other papers.
In the present paper, we use the method of spectral mappings , which is effective for a wide class of differential and difference operators including non-selfadjoint ones. The method of spectral mappings is connected with the idea of the contour integration method and reduces the inverse problem to the so-called main equation of the inverse problem, which is a linear equation in the Banach space of bounded sequences. We prove the uniqueness theorem of the inverse problem, obtain algorithms for constructing its solution together with the necessary and sufficient conditions of its solvability. In general, by sufficiency one should require solvability of the main equation. Therefore, we also study those cases when the solvability of the main equation can be proved or easily checked, namely, selfadjoint case, the case of finite perturbations of the spectral data and the case of small perturbations. The study of the latter case allows us to prove also the stability of the inverse problem.
In the next section, we introduce the spectral data, study their properties and give the formulation of the inverse problem. In Section 3, we prove the uniqueness theorem. In Section 4 we derive the main equation and prove its solvability. Further, using the solution of the main equation, we provide an algorithm for solving the inverse problem. In Section 5, we obtain another algorithm, which we use in Section 6 for obtaining necessary and sufficient conditions of solvability of the inverse problem and for proving its stability.
2 Generalized spectral data. Inverse problem
Let be the spectrum of the boundary value problem (1), (2). In the self-adjoint case, the potential is determined uniquely by specifying the classical discrete spectral data , where are weight numbers determined by the formula
In the non-selfadjoint case, there may be a finite number of multiple eigenvalues and, hence, for unique determination of the Sturm-Liouville operator, one should specify some additional information. In the present section, we introduce the so-called generalized weight numbers, as was done for the case of operator (1) with Robin boundary conditions (see [11,12]) and study the properties of the generalized spectral data.
where . It is known (see, e.g., ) that the spectrum has the asymptotics
Thus, , are complete systems of eigen- and the associated functions of the boundary value problem L. Together with the eigenvalues we consider generalized weight numbers , , determined in the following way:
Consider the following inverse problem.
The function is the characteristic function of the boundary value problem for the equation (1) with the boundary conditions . Let be its spectrum. Clearly, . Thus, is a meromorphic function with poles in , , and zeros in , . Moreover, (see, e.g., )
Let and put . Using the known method (see, e.g., ), one can prove the following asymptotics.
Using (7), (10), (11) and (16), one can calculate
The maximum modulus principle together with (7), (20) and (21) give
According to (7) and (23), we get
and hence the series
converges absolutely and uniformly in λ on bounded sets.
Theorem 1The following representation holds
Proof Consider the contour integral
uniformly with respect to λ in bounded subsets of ℂ, and hence
On the other hand, using the residue theorem , we calculate
Further, we calculate
Proof Using (10), (14) and (20), one can calculate
and (4), (5) and (6) yield
and we calculate
Using (8) and (10) and integrating by parts, we obtain
Substituting (30) in (31) and taking (11) into account, we arrive at
Finally, substituting (32) in (29), we get
According to (19) and (28), we have the asymptotics
Consider the following inverse problems.
3 The uniqueness theorem
We agree that together with L we consider a boundary value problem of the same form but with another potential. If a certain symbol γ denotes an object related to L, then this symbol with tilde denotes the analogous object related to and .
Using (13) and (34), we calculate
It follows from (12) and (35) that
By virtue of (16)-(18), this yields
4 Main equation. Solution of the inverse problem
According to (7) and (33), we have
Lemma 2The following relation holds
Proof Let real numbers a, b be such that , , , . In the λ-plane consider a closed contour (with a counterclockwise circuit), where . By the standard method (see ), using (12), (35)-(37) and Cauchy’s integral formula , we obtain the representation
Analogously to (46), one can obtain the following relation
For each fixed , the relation (44) can be considered as a system of linear equations with respect to , , . But the series in (44) converges only with brackets, i.e., the terms in them cannot be dissociated. Therefore, it is inconvenient to use (44) as a main equation of the inverse problem. Below, we will transfer (44) to a linear equation in the Banach space of bounded sequences (see (53)).
(where T is the sign for transposition) by the formula
Consider also a block-matrix
in the Banach spaceB, whereIis the identity operator.
Proof We rewrite (44) in the form
Substituting here (48), and taking into account our notations, we arrive at
which is equivalent to (53). □
For each fixed , the relation (53) can be considered as a linear equation with respect to . This equation is called the main equation of the inverse problem. Thus, the nonlinear inverse problem is reduced to the solution of the linear equation. Let us prove the unique solvability of the main equation.
Proof Acting in the same way as in Lemma 2 and using (37) and (38), we obtain
5 Algorithm 2
Here and in the sequel, we assume that . It is known that then in formulae (7), (19) and (33). We agree that in the sequel one and the same symbol denotes different sequences in . Let us choose the model boundary value problem , so that (for example, one can take ). Then besides (40), according to (7), (33) and (39), we have
It is obvious that
Furthermore, using the asymptotic formulae (7), (16) and (33), we calculate
Lemma 4The following relation holds
Proof Differentiating (46) twice with respect to x and using (57) and (58), we get
Applying this relation, we get
which together with (57) and (61) gives (60). □
Thus, we obtain the following algorithm for solving the inverse problem.
6 Necessary and sufficient conditions
In the present section, we obtain necessary and sufficient conditions for the solvability of the inverse problem. In the general non-selfadjoint case, they must include the requirement of the solvability of the main equation. In Section 7, some important cases will be considered when the solvability of the main equation can be proved by sufficiency, namely, the selfadjoint case, the case of finite-dimensional perturbations of the spectral data and the case of small perturbations.
The necessity part of the theorem was proved above; here, we prove the sufficiency. We note that sufficiency condition (ii) of the theorem allows to solve linear systems (28) for finding , , which are used for constructing the main equation. Moreover, we have
Similarly to Lemma 1.6.7 in  using (51) and (53), one can prove the following assertion.
where Ω is defined in (55) and
Analogously to Lemma 1.6.8 in  using (41) and (69), one can prove the following assertion.
(2) In order to prove (71) and (72), we first assume that
Differentiating (63) twice, we obtain
Solving the main equation (53), we infer
Together with (48) this implies that
Using (44), (57) and (60), we get
Similarly, using (46) and (47), we calculate
and, consequently, we arrive at
Using (81), we get
formula (79) gives
Let us show that
and arrive at (87). Using (86) and (87), we get
(3) Let us now consider the general case when instead of (74) only (55) holds. Put
and construct the functions and the boundary value problem . Using Lemma 1.5.1 in , one can show that
Denote by the solution of equation (1) under the initial conditions , . According to Lemma 1.5.3 in , we obtain
Proof of Theorem 6 According to (72) and (73), the function is the Weyl function for the constructed boundary value problem L. Choose so that , , and put. Differentiating (47) with respect to x and then substituting , we obtain
where the series converges uniformly with respect to λ in bounded sets. From (62) and (91), it follows that for each , the number is a pole of the function of order . Thus, is the spectrum, and is the Weyl sequence of L. Consequently, are the spectral data of L. □
7 Spacial cases and stability of the solution
The requirement that the main equation is uniquely solvable (Condition S in Theorem 6) is essential and cannot be omitted (see Example 1.6.1 in ). Condition S is difficult to check in the general case. We point out three cases, for which the unique solvability of the main equation can be proved or checked.
Let real numbers having the asymptotics (7) and (19) with and satisfying (92) be given. Choose , construct , and consider the equation (53). Similarly to Lemma 1.6.6 in , one can prove the following assertion.
By virtue of Theorem 6 and Lemma 8, the following theorem holds.
(2) Finite-dimensional perturbations of the spectral data. Let a model boundary value problem with the spectral data be given. We change a finite subset of these numbers. In other words, we consider numbers such that , , for certain and arbitrary in the rest. Then for such spectral data, the main equation becomes the linear algebraic system (54), and Condition S is equivalent to the condition that the determinant of this system does not equal zero for each . Such perturbations are very popular in applications. We note that for the selfadjoint case the determinant of the system (54) is always nonzero.
(3) Local solvability of the main equation. For small perturbations of the spectral data, Condition S is fulfilled automatically. Let us for simplicity consider the case of simple spectra, i.e., . The following theorem is valid.
Theorem 8Letbe given. There exists (which depends on) such that if complex numberssatisfy the condition, then there exists a unique boundary value problemwith, for which the numbersare the spectral data, and
Proof Let C denote various constants, which depend only on . Since , the asymptotical formulae (7) and (19) are fulfilled. Choose such that if then , . According to (52), we have . Choose such that if , then for . In this case, there exists . Thus, all conditions of Theorem 6 are fulfilled, and hence there exists a unique , such that the numbers are the spectral data of . Moreover, (41) and (69) are valid. Using (57), one can get (93). □
Theorem 8 gives the stability of Inverse Problem 1. Denote
Similarly to , one can obtain the stability of the solution in the uniform norm and also the necessary and sufficient conditions of the solvability for the inverse problem, when is in or in .
The authors declare that they have no competing interests.
All authors contributed to each part of this work equally and read and approved the final version of the manuscript.
This research was supported in part by the Russian Foundation for Basic Research (project 13-01-00134) and Taiwan National Science Council (project 99-2923-M-032-001-MY3).
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