Direct and Inverse Problems for Sturm-Liouville Operator Which Has Discontinuity Conditions and Coulomb Potential
We give a derivation of the main equation for Sturm-Liouville operator with Coulomb potential and prove its unique solvability. Using the solution of the main equation, we get an algorithm for the solution of the inverse problem.
In this study, we consider the boundary value problem : where is a spectral parameter, , , , is a real valued bounded function, and .
In spectral theory, the inverse problem is the usual name for any problem in which it is required to ascertain the spectral data that will determine a differential operator uniquely and a method of construction of this operator from the data. This kind of problem was first formulated and investigated by Ambarzumian in 1929 . Since 1946, various forms of the inverse problem have been considered by numerous authors, Borg , Levinson , Levitan , and so forth, and now there exists extensive literature in [5–9]. Later, the inverse problems having specified singularities were considered by a number of authors [10–12].
The method of spectral mappings is an impressive device for investigating a profound class of inverse problems not only for Sturm-Liouville operators, but also for other more complicated classes of operators such as differential operators of arbitrary orders, differential operators with singularities, and pencils of operators. We apply the method of spectral mappings to the solution of the inverse problem for the Sturm-Liouville operator on a finite interval.
In the method of spectral mappings we begin from Cauchy’s integral formula for analytic functions. We apply this theorem in the complex plane of the spectral parameter for specially constructed analytic functions having Coulomb singularity connected with the given spectral characteristics. This permits us to reduce the inverse problem to the so-called main equation which is a linear equation in a corresponding Banach space of sequences.
In this paper, first, integral representation for solution which satisfies certain initial conditions of differential equation generated by Sturm-Liouville operator with Coulomb potential is handled. Second, properties of spectral characteristics and uniqueness theorems for solution of inverse problem are discussed. After that we give a derivation of the main equation and prove its unique solvability. Finally, we obtain an algorithm for the solution of the inverse problem by using the solution of the main equation.
and let us write the expression of the left-hand side of (1) as follows:
Here, has a limit as ; that is, .
Theorem 1 (see ). Solution of the problem which satisfies the initial conditions and the discontinuity conditions (3) has the following form.
If , if , where
In this part, we consider the properties of the spectrum of the problem . In the case of and , the problem is denoted by . Let us verify that a solution of (1) with the initial conditions and and discontinuity conditions (3). When and , it is easily shown that solution with the same conditions has the form
We denote the characteristic function of the problem by : The roots of equation for are the eigenvalues of the problem .
On the other hand, normalizing constants of the problem are defined as follows:
Lemma 2 (see ). The eigenvalues of the problem satisfy the asymptotic equality where , , , , and
Lemma 3 (see ). The normalizing constants of the problem satisfy the asymptotic equality where .
On the other hand, eigenfunctions of problem have the following asymptotic behaviour.
We consider the inverse problem of the reconstruction of problem according to Weyl function. For this reason, we can also consider a boundary value problem of the same form together with .
Assume that is a solution of (1) which satisfies the conditions and the conditions (3).
Let and denote solutions of (1) that satisfy the initial conditions and , respectively, and jump conditions (3).
We set . The functions and are called the Weyl solution and Weyl function of the boundary value problem , respectively.
It is obvious that
Therefore, the Weyl function is meromorphic with simple poles in the points .
Lemma 4. If , then . Therefore, the boundary value problem is uniquely determined by the Weyl function.
Proof. We define the matrix by the formula
Using (19) and (20), we get
On the other hand, from the hypothesis , by using (17) and (21), we get
So, from (23), when , are entire on .
Since relations (17), (21), and (24) yield Let where is sufficiently small number. Since in and from the representations of the solutions we see that are bounded with respect to . Hence it is clear from Liouville’s theorem that do not depend on . Therefore, from the asymptotic formulae of the solutions , ,
for all and for . Finally,
uniformly with respect to . Therefore,
If we substitute these results in (22), we get , for all and . Therefore .
3. The Main Equation of the Inverse Problem
In this section, we solve the inverse problem of recovering from the given spectral data by using ideas of the contour integral method.
We denote that if and are solutions of the equations and separately, then
Let us select a model boundary value problem with real such that . Let be the spectral data of .
It follows from (12) and (14) and analogous formulas for and that
Then, according to (31),
It is clear that
Lemma 5 (Schwarz’s lemma). Let be an analytic function for such that and . Then for .
Lemma 6. The following estimates are valid for , , :
The similar estimates are also current for .
Proof. It follows from (12), (14), (15), and (16) that , when and .
Besides, for a fixed ,
Applying Schwarz’s lemma in the -plane to the function with fixed , and , we obtain
Here is denoted by for .
As a result, and (37) is proved.
Let us show that
For definiteness, let . Get a fixed . For , we have by the help of (31), (15), and (12),
Since , we obtain
Thus, this yields (42) for . For ,
That is, (42) is also valid for . Likewise it can be shown that
Using Schwarz’s lemma, we get
Particularly, this yields
If we apply Schwarz’s lemma to the function for fixed and , we obtain
These approximates together with (34), (36), (12), and (14) suggest (38).
Lemma 7. The following relations hold:
Proof. (1) Denote and take a fixed . In the -plane we regard closed contours , (with counterclockwise circuit) (see Figure 1) where
Denote (with clockwise circuit) (see Figure 1).
Let be the matrix determined by (20). It follows from (21) and (17) that, for each fixed , the functions are meromorphic in with simple poles and . By Cauchy’s integral formula, where and is the Kronecker delta. Therefore, where is used with counterclockwise circuit. Substituting into (22), we get where By the help of (29), uniformly with respect to and on compact sets.
If we take (25) into account, we calculate In view of (17), this supplies where , since the terms with vanish by Cauchy’s theorem.
It follows from that Calculating the integral in (61) by the residue theorem and using (59) we get (52).
Since , we have by Cauchy’s integral formula Considering in the same way as above and using (29), we get where . From (21) and (19), it follows that for any .
If we take (65) and (67) into account, we calculate According to (20), .
Moreover, using (66) we obtain So, As a result, (68) for supplies By virtue of (17), (62), and residue theorem, we obtain (53).
Similarly, the following relation can be obtained:
It follows from the definition of , and (52), (53) that Denote
Lemma 8. The series in (75) converges absolutely and uniformly on . The function is absolutely continuous and .
Proof. We can write in the form where It follows from (12), (14), (33), and (37) that the series in (77) converge absolutely and uniformly on and Moreover, using the asymptotic formulae (15), (12), (14), and (33) we calculate where and for . Therefore, . Analogously, we obtain and consequently .
Lemma 9. The following relation holds: where is defined by (75).
Proof. If we differentiate (52) twice with respect to and use (30) and (75), we obtain
If we replace and by using (1) and then replace using (52), then
Remark 10. For each fixed , (73) can be considered as a system of linear equations with respect to , . However, the series in (73) converges only “with brackets.” Hence, it is not appropriate to use (73) as the main equation of the inverse problem. Below we will convert (73) to a linear equation in a corresponding Banach space of sequences.
Let be a set of indices , , . For each fixed , we define the vector by the formulae We also define the block matrix
by the formulae
Analogously we define by replacing in the former descriptions by and by . It follows from (37) and (38) that Similarly, and also where does not depend on , , , , , and .
Let us consider the Banach space of bounded sequences with the norm . It follows from (87) and (88) that, for each fixed , the operators and , such that is the identity operator mapping from to , are linear bounded operators and
Theorem 11. In the Banach space , the vector satisfies the equation