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Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions
Boundary Value Problems volume 2013, Article number: 180 (2013)
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.
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
while is a solution of equation (1) satisfying the initial conditions
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.
Let the function be a solution of equation (1) under the conditions
For every fixed , the functions , and their derivatives with respect to x are entire in λ. The eigenvalues , of the problem L coincide with the zeros of its characteristic function
where . It is known (see, e.g., ) that the spectrum has the asymptotics
Denote by the multiplicity of the eigenvalue () and put . Note that by virtue of (7) for sufficiently large n, we have . Denote
Hence, for we have
Moreover, for formula (6) yields
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:
We note that the numbers for sufficiently large n coincide with the classical weight numbers (4) for the selfadjoint Sturm-Liouville operator.
Definition 1 The numbers are called the generalized spectral data of L.
Consider the following inverse problem.
Inverse Problem 1 Given the generalized spectral data , find .
Let the functions , be solutions of equation (1) under the conditions
The functions and are called the Weyl solution and the Weyl function for L, respectively. According to (6), we have
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.
Lemma 1 (i) For , the following asymptotics holds
uniformly with respect to .
Fix . Then for sufficiently large(18)
Using (7), (10), (11) and (16), one can calculate
Fix . According to (14), the function has a representation
where , and the function is regular in a vicinity of . The sequence is called the Weyl sequence for L. By virtue of (14), (17) and (18), the following estimate holds
Moreover, according to (6), (14), (16) and (17), for each fixed , we have
The maximum modulus principle together with (7), (20) and (21) give
Choose such that and put
According to (7) and (23), we get
and hence the series
converges absolutely and uniformly in λ on bounded sets.
Theorem 1 The following representation holds
Proof Consider the contour integral
where the contour , has the counterclockwise circuit. According to (7), we have for sufficiently large N and sufficiently small fixed . By virtue of (21), we obtain the estimate
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
Substituting this into (27) and using (26), we obtain . By virtue of (22), we get and arrive at (25). □
Theorem 2 The coefficients and the generalized weight numbers determine each other uniquely by the formula
Proof Using (10), (14) and (20), one can calculate
where . Obviously, , . Moreover, by virtue of (8) and (10), induction gives
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
Since , , by induction we obtain (28). □
According to (19) and (28), we have the asymptotics
Consider the following inverse problems.
Inverse Problem 2 Given the spectra , , construct the function .
Inverse Problem 3 Given the Weyl function , construct the function .
Remark 1 According to (14), (15), (24), (25) and (28), inverse Problems 1-3 are equivalent. The numbers can also be used as spectral data.
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 .
Theorem 3 If , , , then , i.e., , a.e. on . Thus, the specification of the generalized spectral data determines the potential uniquely.
Proof By virtue of (7), we have . According to Remark 1, it is sufficient to prove that if , then . Define the matrix by the formula
Using (13) and (34), we calculate
It follows from (12) and (35) that
By virtue of (16)-(18), this yields
uniformly with respect to . On the other hand, according to (12) and (35), we get
Thus, if , then for each fixed x, the functions and are entire in λ. Together with (37) this yields , . Substituting into (36), we get and consequently . □
4 Main equation. Solution of the inverse problem
Let the spectral data of be given. We choose an arbitrary model boundary value problem (e.g., one can take ). Introduce the numbers , by the formulae
According to (7) and (33), we have
For , put
where , . Analogously, we define , , and , , , replacing S with in the definitions above.
The analogous estimates are also valid for , , .
Lemma 2 The following relation holds
where the series converges absolutely and uniformly with respect to .
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
uniformly with respect to and λ on bounded sets. Calculating the integral in (45) by the residue theorem and using (20), we get
for sufficiently large N. Taking the limit in (45) as , we obtain
Differentiating this with respect to λ, the corresponding number of times and then taking , we arrive at (44). □
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)).
Let w be the set of indices , , . For each fixed , we define the vector
(where T is the sign for transposition) by the formula
Note that if , are given, then , can be found by the formula
Consider also a block-matrix
Analogously, we introduce , and , by the replacement of , in the preceding definitions with , , respectively. Using (41) and (43), we get the estimates
Consider the Banach space B of bounded sequences with the norm . It follows from (49) and (50) that for each fixed , the operators and , acting from B to B, are linear bounded ones and
Theorem 4 For each fixed , the vector satisfies the equation
in the Banach space B, where I is 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.
Theorem 5 For each fixed , the operator has a bounded inverse operator, namely , i.e., the main equation (53) is uniquely solvable.
Proof Acting in the same way as in Lemma 2 and using (37) and (38), we obtain
uniformly with respect to and λ, μ on bounded sets. Calculating the integral by the residue theorem and passing to the limit as , we obtain
According to the definition of , , we arrive at
Further, taking the definition of , into account, we get
which is equivalent to . Symmetrically, one gets
Hence the operator exists, and it is a linear bounded operator. □
Using the solution of the main equation, one can construct the function . Thus, we obtain the following algorithm for solving the inverse problem.
Algorithm 1 Let the spectral data be given. Then
construct , , by solving the linear systems (28);
choose and calculate and ;
find by solving equation (53);
choose (e.g., ) and construct by the formula
Remark 2 In the particular case, when , for (let for definiteness ) according to (44) and the definition of , , the main equation becomes the linear algebraic system
whose determinant does not vanish for any by virtue of Theorem 5.
In the next section for the case , we give another algorithm, which is used in Section 6 for obtaining the necessary and sufficient conditions for the solvability of the inverse problem.
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
Lemma 3 The series in (57) converges absolutely and uniformly on and allows termwise differentiation. The function is absolutely continuous, and .
Proof It is sufficient to prove for the case , , . We rewrite to the form , where
It follows from (33), (41) and (55) that the series in (59) converges absolutely and uniformly on , and
Furthermore, using the asymptotic formulae (7), (16) and (33), we calculate
Hence . Similarly, we get , and consequently . □
Lemma 4 The following relation holds
Proof Differentiating (46) twice with respect to x and using (57) and (58), we get
Using (1) and (8), we replace here the second derivatives, and then replace using (46). This yields
Using (1) and (8) for , , , we calculate
Applying this relation, we get
which together with (57) and (61) gives (60). □
Thus, we obtain the following algorithm for solving the inverse problem.
Algorithm 2 Let the spectral data be given. Then
construct , , by solving the linear systems (28);
choose so that and calculate and ;
find by solving equation (53), and calculate , , , by (48);
calculate by formulae (56), (57) and (60).
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.
Theorem 6 For complex numbers to be the spectral data of a certain boundary value problem with , it is necessary and sufficient that
the relations (7) and (19) hold with ;
for all ;
(Condition S) for each , the linear bounded operator , acting from B to B, has a bounded inverse one. Here is chosen so that .
The boundary value problem can be constructed by Algorithms 1 and 2.
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
Let be the solution of the main equation (53). Denote
Similarly to Lemma 1.6.7 in  using (51) and (53), one can prove the following assertion.
Lemma 5 For , , , the following relations hold
where Ω is defined in (55) and
We define the functions by formulae (48), and according to (64), we get (41). Then (44) is also valid. By virtue of (48), (65) and Lemma 5, we have
Furthermore, we construct the functions and via (46) and (47) and the function by formulae (56), (57) and (60). Clearly,
Analogously to Lemma 1.6.8 in  using (41) and (69), one can prove the following assertion.
Lemma 6 .
Lemma 7 For , , the following relations hold
Proof (1) According to the estimates (42), the series in (46) is termwise differentiable with respect to x, and hence , . By virtue of (70), we have , . Thus, formula (47) gives .
In order to prove (71) and (72), we first assume that(74)
Differentiating (63) twice, we obtain
It follows from (50), (66) and (67) that the series in (75) converges absolutely and uniformly for , , and
Solving the main equation (53), we infer
According to (50) and (66), the series in (77) converges absolutely and uniformly for . Further, using (77), we calculate
where according to (50), (67), (68) and (76), the series converges absolutely and uniformly for and
On the other hand, it follows from the proof of Lemma 3 and from (74) that ; hence
Together with (48) this implies that
Using (44), (57) and (60), we get
Similarly, using (46) and (47), we calculate
For and , it follows from (78) that
and, consequently, we arrive at
where for ,
Using (81), we get
Since , , we have
It follows from (50), (74), (82) and (83) that . Then, by virtue of Condition S in Theorem 6, , and consequently . Thus, we obtain (71).
formula (79) gives
From this, by virtue of (46), it follows that . Analogously, using (80) we obtain . Thus, (71) and (72) are proved for the case when (74) is fulfilled.
Denote . It follows from (46) and (47) for that
where , , . Differentiating (84) with respect to λ an appropriate number of times and substituting , we get
Let us show that
Indeed, for , , we have
Moreover, according to (9), we have , and hence
where for negative α or β. For solving the system (89), we obtain , which together with (88) gives for . If , then (11) and (89) give
According to (88) and (90), we have , . Moreover, using (88), (90) and (28), we calculate
and arrive at (87). Using (86) and (87), we get
Then, by virtue of Condition S, , . Substituting this into (85) and using the relation , , we obtain .
Let us now consider the general case when instead of (74) only (55) holds. Put
We agree that if the symbol γ denotes an object constructed with the help of the numbers , then the symbol denotes the corresponding object, constructed with the help of . Then for all , we have
For each fixed , we solve the corresponding main equation
and construct the functions and the boundary value problem . Using Lemma 1.5.1 in , one can show that
Hence , i.e., . Similarly, we get .
Notice that we additionally proved that , , , i.e., is a spectrum of L. □
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.
The selfadjoint case. It is known that in the selfadjoint case, i.e., when the function is real-valued, the spectral data are real numbers, and(92)
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.
Lemma 8 For each fixed , the operator , acting from B to B, has a bounded inverse operator. Thus, the main equation (53) has a unique solution .
By virtue of Theorem 6 and Lemma 8, the following theorem holds.
Theorem 7 For real numbers to be the spectral data of a certain selfadjoint boundary value problem with , it is necessary and sufficient to satisfy the asymptotics (7) and (19) with and condition (92).
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.
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 8 Let be given. There exists (which depends on ) such that if complex numbers satisfy the condition , then there exists a unique boundary value problem with , for which the numbers are the spectral data, and
where C depends only on .
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
where the numbers , are determined by the formulae
for , , , and for other n. According to (7), (19) and (28), we have , , and hence (93) is equivalent to the estimate
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 .
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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).
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.
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Buterin, S.A., Shieh, C. & Yurko, V.A. Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions. Bound Value Probl 2013, 180 (2013). https://doi.org/10.1186/1687-2770-2013-180
- non-selfadjoint Sturm-Liouville operators
- inverse spectral problems
- method of spectral mappings
- generalized spectral data
- generalized weight numbers