Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions
© Buterin et al.; licensee Springer 2013
Received: 26 June 2013
Accepted: 23 July 2013
Published: 6 August 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.
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
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.
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 and put . Using the known method (see, e.g., ), one can prove the following asymptotics.
- (ii)Fix . Then for sufficiently large(18)
converges absolutely and uniformly in λ on bounded sets.
Substituting this into (27) and using (26), we obtain . By virtue of (22), we get and arrive at (25). □
Since , , by induction we obtain (28). □
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.
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
where , . Analogously, we define , , and , , , replacing S with in the definitions above.
The analogous estimates are also valid for , , .
where the series converges absolutely and uniformly with respect to .
Differentiating this with respect to λ, the corresponding number of times and then taking , we arrive at (44). □
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)).
in the Banach space B, where I is the identity operator.
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.
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.
construct , , by solving the linear systems (28);
choose and calculate and ;
find by solving equation (53);
- (iv)choose (e.g., ) and construct by the formula
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
Lemma 3 The series in (57) converges absolutely and uniformly on and allows termwise differentiation. The function is absolutely continuous, and .
Hence . Similarly, we get , and consequently . □
which together with (57) and (61) gives (60). □
Thus, we obtain the following algorithm for solving the inverse problem.
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.
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.
Similarly to Lemma 1.6.7 in  using (51) and (53), one can prove the following assertion.
Analogously to Lemma 1.6.8 in  using (41) and (69), one can prove the following assertion.
Lemma 6 .
- (2)In order to prove (71) and (72), we first assume that(74)
It follows from (50), (74), (82) and (83) that . Then, by virtue of Condition S in Theorem 6, , and consequently . Thus, we obtain (71).
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.
- (3)Let us now consider the general case when instead of (74) only (55) holds. Put
Hence , i.e., . Similarly, we get .
Notice that we additionally proved that , , , i.e., is a spectrum of L. □
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
- (1)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.
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.
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). □
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 .
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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