#### Abstract

We consider the nonself-adjoint Sturm-Liouville operator with and either periodic or antiperiodic boundary conditions. We obtain necessary and sufficient conditions for systems of root functions of these operators to be a Riesz basis in in terms of the Fourier coefficients of .

#### 1. Introduction

Let be Sturm-Liouville operator generated in by the expression either with the periodic boundary conditions or with the antiperiodic boundary conditions where is a complex-valued summable function on . We will consider only the periodic problem. The antiperiodic problem is completely similar. The operator is regular, but not strongly regular. It is well known [1, 2] that the system of root functions of an ordinary differential operator with strongly regular boundary conditions forms a Riesz basis in . Generally, the normalized eigenfunctions and associated functions, that is, the root functions of the operator with only regular boundary conditions, do not form a Riesz basis. Nevertheless, Shkalikov [3, 4] showed that the system of root functions of an ordinary differential operator with regular boundary conditions forms a basis with parentheses. In [5], they proved that under the conditions the system of root functions of forms a Riesz basis in . A new approach in terms of the Fourier coefficients of is due to Dernek and Veliev [6]. They proved that if the following conditions hold, then the root functions of form a Riesz basis in , where is the Fourier coefficient of and without loss of generality we always suppose that and the notation means that there exist constants and such that and for all large . Makin [7] extended this result as follows.

*Let the first condition in (5) hold. But the second condition in (5) is replaced by a less restrictive one: **,**holds and ** with some ** for large **, where ** is a nonnegative integer. Then the root functions of the operator ** form a Riesz basis in *.

In addition, some conditions which imply that the system of root functions does not form a Riesz basis of were established in [7] (see also [8–10]). In [11], we proved that the Riesz basis property is valid if the first condition in (4) holds, but the second is replaced by . The results of Veliev and Shkalikov [12] are more general and inclusive. The assertions in various forms concerning the Riesz basis property were proved. One of the basic results in [12] is the following statement.

*Let ** be an arbitrary integer, ** and (7) holds with some **, and let one of the following conditions hold:**with some **. Then a normal system of root functions of the operator ** forms a Riesz basis if and only if **. *

Here, for large , by for denote the normalized eigenfunctions corresponding to the simple eigenvalues . If the multiplicities of these eigenvalues are equal to , then the root subspace consists either of two eigenfunctions or of Jordan chains comprising one eigenfunction and one associated function. First, if the multiple eigenvalue has geometric multiplicity 2, we take the normalized eigenfunctions , . Secondly, if there is one eigenfunction corresponding to the multiple eigenvalue , then we take the Jordan chain consisting of a normalized eigenfunction and corresponding associated function denoted again by and orthogonal to . Thus the system of root functions obtained in this way will be called a* normal system*.

Moreover, for the other interesting results about the Riesz basis property of root functions of the periodic and antiperiodic problems, we refer in particular to [13–18].

In this paper, we prove the following main result.

Theorem 1. *Let be arbitrary complex-valued function and suppose that at least one of the conditions
**
is satisfied, where , defined in (40), is a common order of the Fourier coefficients and of .**Then a normal system of root functions of the operator forms a Riesz basis if and only if
*

This form of Theorem 1 is not novel (see, e.g., [12]). The novelty is in the term defined in (40) (see also Lemma 4). Indeed, if we take in the Sobolev space given above in [12], that is, if , then the nonnegative integer in the conditions in (8) must be zero and the assertion on the Riesz basis property remains valid with a less restrictive condition in (9) instead of (8). For example, let . If instead of (9) we suppose that at least one of the following conditions holds with some , then the assertion of Theorem 1 is obvious.

It is well known (see, e.g., [19], Theorem 2 in page 64) that the periodic eigenvalues are located in pairs satisfying the following asymptotic formula: for . Here, by we denote large enough positive integer. From this formula, the pair of the eigenvalues is close to the number and is isolated from the remaining eigenvalues of by a distance . That is, we have, for , for all and , where and, here and in subsequent relations, is some positive constant whose exact value is not essential. For the potential and , clearly, the system is a basis of the eigenspace corresponding to the eigenvalue of the periodic boundary value problems.

Finally, let us state the following relevant theorem which will be used in the proof of Theorem 1.

Theorem 2 (see [12]). *The following assertions are equivalent.**(i) A normal system of root functions of the operator forms a Riesz basis in the space .**(ii) The number of Jordan chains is finite and the relation
**
holds for all indices and corresponding only to the simple eigenvalues for , where , are the Fourier coefficients defined in (21).**(iii) The number of Jordan chains is finite and the relation (14) for either or holds.*

#### 2. Preliminaries

The following well-known relation will be used to obtain, for large , the asymptotic formulas for periodic eigenvalues corresponding to the normalized eigenfunctions : where , . From Lemma 1 in [20], we iterate (15) by using the following relations: where for all , and , where .

Hence, substituting (16) in (15) for and then isolating the terms with indices , we deduce, in view of , that First, we use (15) for in the right-hand side of (18). Then, considering (16) with the indices and isolating the terms with indices , we get by repeating this procedure once again, and where , Using (13), (17), and the relation one can prove the estimates In the same way, by using the eigenfunction of the operator for , we can obtain the relations where Here the similar estimates as in (28) are valid for .

In addition, by using (13), (15), and (17), we get Thus, we obtain that the normalized eigenfunction by the basis on has the following expansion: where Now, let us consider the following form of the Riemann-Lebesgue lemma. By this we set and clearly as . As the proof of lemma is similar to that of Lemma 6 in [21], we pass to the proof.

Lemma 3. *If then as uniformly in .*

#### 3. Main Results

To prove the main results of the paper we need the following lemmas.

Lemma 4. *The eigenvalues of the operator for and satisfy
**
where is defined in (40).*

*Proof. *For the proof we have to estimate the terms of (19) and (29). It is easily seen that
where . Thus, we get
From the argument in Lemma 2(a) of [18] we deduce, with our notations,
where
for and
Thus, from the equalities
(see (40) and (45)) and since , integration by parts gives for the integral in (44) the estimate
for large . It is easily seen by substituting into the relation for (see (29)) that
In a similar way, by (42), and so forth, we get
where ,
Thus, by using and (40), integration by parts again gives for the integral in (51) the following estimate:
Similarly
To estimate (see (24)), let us show that
for and some . Since is summable function on , there exists such that
and the integral (56) is bounded for all . Hence, multiplying the integrand of (56) by and then using integration by parts, we get
which implies (55).

Thus by (13), (17), and relation (27), we deduce that
Also .

From relation (39), for large , it follows that either or . We first consider the case when . Hence, by using (19), (49), and (53) with we obtain
This with definition (40) gives . Similarly, for the other case , by using (29), (49), (54), and , we get (41). The lemma is proved.

Lemma 5. *For all large , we have the following estimates (see, resp., (23), (32) and (25), (34)):
*

*Proof. *Let us estimate the sum . By using estimate (28) and inequality (55) for large , we deduce that
In the same way .

Arguing as in [12] (see the proof of Lemma 6), let us now estimate the sum . Taking into account (42) and Lemma 4, we have
where
By using the identity
and the substitutions , in the formula , we obtain with the indices in the following form:
where
From (46)–(48), (52), and and using integration by parts only in , we obtain the following estimates:
Then, in view of (65) and (67), . This with equality (62) implies that . In the same way satisfies the same estimate. The lemma is proved.

Thus by using Lemmas 4 and 5, Theorem 2, and an argument similar to that of Theorem 2 in [12] under the conditions in (9), let us prove the following main result.

* Proof of Theorem 1. *In view of Lemma 4, substituting the values of
given by (53), (54), and (60) in relations (20) and (30), we get the following reversion of the relations
for .

It is easily seen again by substituting , , in the sum (see (29)) and using (50) that for . Hence, multiplying (69) by and (70) by and subtracting we obtain the following equality:

Suppose, for example, that satisfies the condition in (9). Then using this equality we get
for . In addition, for large , the condition in (9) for implies that the geometric multiplicity of the eigenvalue is 1. Arguing as in Lemma 4 of [12], if there exist mutually orthogonal two eigenfunctions corresponding to , then one can choose an eigenfunction such that . Thus combining this with (39) and (71), we get which contradicts (9).

Let the normal system of root functions form a Riesz basis. To prove , from (72) it is enough to show that all the large periodic eigenvalues are simple, since in this case we have, by Theorem 2,
for . For large , again by Theorem 2 and the condition in (9) for , respectively, the number of Jordan chains and the eigenvalues of geometric multiplicity are finite; that is, all large eigenvalues are simple.

Now let . From (72), we obtain the relation (73) for which implies that the number of Jordan chains is finite. In fact, if there exists a Jordan chain consisting of an eigenfunction and an associated function corresponding to the eigenvalue , then, for example, for , using the eigenfunction of the adjoint operator and the relation
we obtain that . Thus, from expansion (37) for , we get which contradicts (73) for . Thus, using Theorem 2, we prove that a normal system of root functions of the operator forms a Riesz basis.

Arguing as in the proof of Theorem 1, we obtain a similar result established below for the antiperiodic problems.

Theorem 6. *Let be arbitrary complex-valued function and suppose that at least one of the conditions
**
is satisfied, where is obtained from (40) by replacing with and a common order of both Fourier coefficients and of .**Then a normal system of root functions of the operator with antiperiodic boundary conditions forms a Riesz basis if and only if .*

*Remark 7. *Clearly if instead of (9) we assume that at least one of the conditions
holds, then the assertion of Theorem 1 is satisfied. In this way one can easily write a similar result for the antiperiodic problem.

In addition to all the above results, we note that if either the first condition of (9) and (10) or the second condition of (9) and (10) holds then all the periodic eigenvalues are asymptotically simple. We can write a similar result for the antiperiodic problem.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.