On the Riesz Basisness of Systems Composed of Root Functions of Periodic Boundary Value Problems

In this paper, we consider the nonselfadjoint 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 q.

where q is a complex-valued summable function on [0, 1]. We will consider only the periodic problem. The anti-periodic problem is completely similar. The operator L is regular, but not strongly regular. It is well known [6,14] that the system of root functions of an ordinary differential operator with strongly regular boundary conditions forms a Riesz basis in L 2 [0, 1]. 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 [16,17] showed that the system of root functions of an ordinary differential operator with regular boundary conditions forms a basis with parentheses. In [9], they proved that under the conditions the system of root functions of L forms a Riesz basis of L 2 [0, 1]. A new approach in terms of the Fourier coefficients of q is due to Dernek and Veliev [1]. They proved that if the following conditions hold, then the root functions of L form a Riesz basis in L 2 [0, 1],where q m =: (q , e i2mπx ) =: is the Fourier coefficient of q and without loss of generality we always suppose that q 0 = 0 and the notation a m ∼ b m means that there exist constants c 1 , c 2 such that 0 < c 1 < c 2 and c 1 < |a m /b m | < c 2 for all large m. Makin [12] extended this result as follows: Let the first condition (5) hold. But the second condition (5) is replaced by a less restrictive one: holds and |q 2m | > cm −s−1 with some c > 0 for large m, where s is a nonnegative integer. Then the root functions of the operator L form a Riesz basis in L 2 [0, 1]. In addition, some conditions which imply that the system of root functions does not form a Riesz basis of L 2 [0, 1] were established in [12] (see also [2,3,4]). In [11], we proved that the Riesz basis property is valid if the first condition (4) holds, but the second is replaced by q ∈ W 1 1 [0, 1]. The results of Shkalilov and Veliev [18] are more general and inclusive. The assertions in various forms concerning the Riesz basis property were proved. One of the basic results in the paper [18] is the following statement: Let p ≥ 0 be an arbitrary integer, q ∈ W p 1 [0, 1] and (6) holds with some s ≤ p, and let one of the following conditions hold: with some ε > 0. Then a normal system of root functions of the operator L forms a Riesz basis if and only if q 2m ∼ q −2m .
Here, for large m, denote by Ψ m,j (x) for j = 1, 2 the normalized eigenfunctions corresponding to the simple eigenvalues λ m,j . If the multiplicities of these eigenvalues equal to 2, 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 λ m,1 = λ m,2 has geometric multiplicity 2, we take the normalized eigenfunctions Ψ m,1 (x), Ψ m,2 (x). Secondly, if there is one eigenfunction Ψ m,1 (x) corresponding to the multiple eigenvalue λ m,1 = λ m,2 , then we take the Jordan chain consisting of a normalized eigenfunction Ψ m,1 (x) and corresponding associated function denoted again by Ψ m,2 (x) and orthogonal to Ψ m,1 (x). 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 anti-periodic problems, we refer in particular to [5,7,10,13] and [19,20].
In this paper, we prove the following main result: Theorem 1 Let q ∈ L 1 [0, 1] be arbitrary complex-valued function and suppose that at least one of the conditions is satisfied, where ρ(m), defined in (30), is a common order of the Fourier coefficients q 2m and q −2m of q. Then a normal system of root functions of the operator L forms a Riesz basis if and only if q 2m ∼ q −2m .
This form of Theorem 1 is not novel (see, for example, [18]). The novelty is in the term ρ(m) defined in (30) (see also Lemma 2). Indeed, if we take p = 0 in the Sobolev space W p 1 [0, 1] given above in [18], that is, if q ∈ L 1 [0, 1] then the nonnegative integer s in the conditions (7) must be zero and the assertion on the Riesz basis property remains valid with a less restrictive condition (8) instead of (7). For example, let ρ(m) = o(m −1/2 ). If instead of (8) we suppose that at least one of the following conditions holds |q 2m | > εm −3/2 or |q −2m | > εm −3/2 for all large m with some ε, then the assertion of Theorem 1 is obvious.
It is well known (see, e.g., [15], Theorem 2 in page 64) that the periodic eigenvalues λ m,1 , λ m,2 are located in pairs, satisfying the following asymptotic formula for m ≥ N . Here, by N ≫ 1, we denote large enough positive integer. From this formula, the pair of the eigenvalues {λ m,1 , λ m,2 } is close to the number (2mπ) 2 and isolated from the remaining eigenvalues of L by a distance m.
That is, we have, for j = 1, 2, for all k = 0, 2m and k ∈ Z, where m ≥ N and, here and in subsequent relations, C is some positive constant whose exact value is not essential. For the potential q = 0 and m ≥ 1, clearly, the system {e −i2mπx , e i2mπx } is a basis of the eigenspace corresponding to the eigenvalue (2mπ) 2 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 [18]) The following assertions are equivalent: i) a normal system of root functions of the operator L forms a Riesz basis in the space L 2 [0, 1]; ii) the number of Jordan chains is finite and the relation holds for all indices m and j corresponding only to the simple eigenvalues λ m,j for j = 1, 2, where u m,j , v m,j are the Fourier coefficients defined in (18); iii) the number of Jordan chains is finite and the relation (11) for either j = 1, or j = 2 holds.

Preliminaries
The following well-known relation will be used to obtain, for large m, the asymptotic formulas for periodic eigenvalues λ m,j corresponding to the normalized eigenfunctions Ψ m,j (x): where , we iterate (12) by using the following relations where for all m ≥ N , m 1 ∈ Z and j = 1, 2, where M = sup m∈Z |q m |. Hence, substituting (13) in (12) for k = 0 and then isolating the terms with indices m 1 = 0, 2m, we deduce, in view of q 0 = 0, that (15) First, we use (12) for k = m 1 in the right-hand side of (15). Then, considering (13) with the indices m 2 and isolating the terms with indices m 1 +m 2 = 0, 2m, we get by repeating this procedure once again, and Using (10), (14) and the relation m1 =0,2m one can prove the estimates In the same way, by using the eigenfunction e −i2mπx of the operator L for q = 0, we can obtain the relations Here the similar estimates as in (23) are valid for R ′ i (m), i = 1, 2. In addition, by using (10), (12) and (14), we get Thus, we obtain that the normalized eigenfunctions Ψ m,j (x) by the basis {e i2kπx : k ∈ Z} on [0, 1] has the following expansion where Now, let us consider the following form of the Riemann-Lebesgue lemma. By this we set and clearly ρ(m) → 0 as m → ∞. As the proof of lemma is similar to that of Lemma 6 in [8], we pass to the proof.

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

Lemma 2
The eigenvalues λ m,j of the operator L for m ≥ N and j = 1, 2, satisfy where ρ(m) is defined in (30).
Proof For the proof we have to estimate the terms of (16) and (24). It is easily seen that where Λ 0 m∓m1 = (2mπ) 2 − (2(m ∓ m 1 )π) 2 . Thus, we get From the argument in Lemma 2(a) of [20] we deduce, with our notations, where for m 1 = 0 and Thus, from the equalities for large m. It is easily seen by substituting m 1 = −k into the relation for a ′ 1 (λ m,j ) (see (24)) that a 1 (λ m,j ) = a ′ 1 (λ m,j ).
By using the identity and the substitutions k 1 = m 1 , k 2 = 2m − m 1 − m 2 in the formula I(m), we obtain I(m) with the indices m 1 , m 2 in the following form where q m1 q m2 q 2m−m1−m2 m 2 (2m − m 1 ) , .
From (35)-(36), (40), 2I 2 (m) = I 1 (m) and using integration by parts only in I 1 , we obtain the following estimates Then, in view of (47) and (48), This with the equality (46) implies that b 2 (λ m,j ) = O ρ(m)m −2 . In the same way b ′ 2 (λ m,j ) satisfies the same estimate. The lemma is proved. ⊓ ⊔ Thus by using Lemma 2-3, Theorem 2 and an argument similar to that of Theorem 2 in [18] under the conditions (8), let us prove the following main result.

Proof of Theorem 1
In view of Lemma 2, substituting the values of  (17) and (25), we get the following reversion of the relations It is easily seen again by substituting m 1 + m 2 = −k 1 , m 2 = k 2 in the sum a ′ 2 (λ m,j ) (see (24)) and using (38) that a i (λ m,j ) = a ′ i (λ m,j ) for i = 1, 2. Hence, multiplying (49) by v m,j and (50) by u m,j and subtracting we obtain the following equality Suppose, for example, that q 2m satisfies the condition in (8). Then using this equality we get for j = 1, 2. In addition, for large m, the condition (8) for q 2m implies that the geometric multiplicity of the eigenvalue λ m,j is 1. Arguing as in Lemma 4 of [18], if there exist mutually orthogonal two eigenfunctions Ψ m,j (x) corresponding to λ m,1 = λ m,2 , then one can choose an eigenfunction Ψ m,j (x) such that u m,j = 0. Thus combining this with (29) and (51), we get q 2m = O(ρ(m)m −1 ) which contradicts (8). Let the normal system of root functions form a Riesz basis. To prove κ m ∼ 1, from (52) it is enough to show that all the large periodic eigenvalues λ m,j are simple, since in this case we have, by Theorem 2, for j = 1, 2. For large m, again by Theorem 2 and the condition (8) for q 2m , respectively, the number of Jordan chains and the eigenvalues of geometric multiplicity 2 is finite, that is, all large eigenvalues are simple. Now let q 2m ∼ q −2m . From the second formula of (52), we obtain that κ m ∼ 1 and then, from the first, that (53) for the eigenfunction Ψ m,1 (x), that is, j = 1 which implies that the number of Jordan chains is finite. In fact, if there exists a Jordan chain consists of an eigenfunction Ψ m,1 (x) and an associated function Ψ m,2 (x) corresponding to the eigenvalue λ m,1 = λ m,2 , then, for example for λ m,1 , using the eigenfunction Ψ m,1 (x) of the adjoint operator L * and the relation (L − λ m,1 )Ψ m,2 (x) = Ψ m,1 (x), we obtain that (Ψ m,1 , Ψ m,1 ) = 0. Thus, from the expansion (28) for j = 1, we get u m,1 v m,1 = O(m −2 ) which contradicts (53) for j = 1. Thus, using Theorem 2, we prove that a normal system of root functions of the operator L forms a Riesz basis.
⊓ ⊔ Arguing as in the proof of Theorem 1, we obtain a similar result established below for the anti-periodic problems. Remark 1 Clearly if instead of (8) we assume that at least one of the conditions ρ(m) ∼ q 2m , ρ(m) ∼ q −2m holds, then the assertion of Theorem 1 is satisfied. In this way one can easily write a similar result for the anti-periodic problem. In addition to all the above results, we note that if either the first condition of (8) and (9), or the second condition of (8) and (9) hold then all the periodic eigenvalues are asymptotically simple. We can write a similar result for the anti-periodic problem.