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On the basis property of the root functions of some class of non-self-adjoint Sturm-Liouville operators
Boundary Value Problems volume 2014, Article number: 57 (2014)
Abstract
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis.
MSC:34L05, 34L20.
1 Introduction and preliminary facts
Let , , , and be the operators generated in by the differential expression
and the following boundary conditions:
and
respectively, where is a complex-valued summable function on , and .
In conditions (2), (3), (4), and (5) if , , , and , respectively, then any is an eigenvalue of infinite multiplicity. In (2) and (4) if and then they are periodic boundary conditions; in (3) and (5) if and then they are antiperiodic boundary conditions.
These boundary conditions are regular but not strongly regular. Note that, if the boundary conditions are strongly regular, then the root functions form a Riesz basis (this result was proved independently in [1, 2] and [3]). In the case when an operator is associated with the regular but not strongly regular boundary conditions, the root functions generally do not form even a usual basis. However, Shkalikov [4, 5] proved that they can be combined in pairs, so that the corresponding 2-dimensional subspaces form a Riesz basis of subspaces.
In the regular but not strongly regular boundary conditions, periodic and antiperiodic boundary conditions are the ones more commonly studied. Therefore, let us briefly describe some historical developments related to the Riesz basis property of the root functions of the periodic and antiperiodic boundary value problems. First results were obtained by Kerimov and Mamedov [6]. They established that, if
then the root functions of the operator form a Riesz basis in , where denotes the operator generated by (1) and the periodic boundary conditions.
The first result in terms of the Fourier coefficients of the potential q was obtained by Dernek and Veliev [7]. They proved that if the conditions
hold, then the root functions of form a Riesz basis in , where is the Fourier coefficient of q and everywhere, without loss of generality, it is assumed that . Here denotes the inner product in and means that and as . Makin [8] improved this result. Using another method he proved that the assertion on the Riesz basis property remains valid if condition (7) holds, but condition (6) is replaced by a less restrictive one: ,
holds and with some for sufficiently large n, where s is a nonnegative integer. Besides, some conditions which imply the absence of the Riesz basis property were presented in [8]. Shkalilov and Veliev obtained in [9] more general results, which cover all results discussed above.
The other interesting results as regards periodic and antiperiodic boundary conditions were obtained in [10–19].
The basis properties of other some operators with regular but not strongly regular boundary conditions are studied in [20–23]. It was proved in [22] that the system of the root functions of the operator generated by (1) and the boundary conditions
forms an unconditional basis of the space , where is an arbitrary complex-valued function from the class , γ is an arbitrary nonzero complex number and . Kerimov and Kaya [20, 21] investigated the basis properties of fourth order differential operators with some regular boundary conditions.
In this paper we prove that if
where , then the large eigenvalues of the operators and are simple. Moreover, if there exists a sequence such that (8) holds when n is replaced by , then the root functions of these operators do not form a Riesz basis.
Similarly, if
then the large eigenvalues of the operators and are simple and if there exists a sequence such that (9) holds when n is replaced by , then the root functions of these operators do not form a Riesz basis.
Moreover, we obtain asymptotic formulas of arbitrary precision for the eigenvalues and eigenfunctions of the operators , , , and .
2 Main results
We will focus only on the operator . The investigations of the operators , , and are similar. It is well known that (see (47a) and (47b) on page 65 of [24]) the eigenvalues of the operators consist of the sequences , satisfying
for . From this formula one can easily obtain the following inequality:
for ; ; , and , where N denotes a sufficiently large positive integer, that is, .
Let us denote by the operator when . The eigenvalues of the operator are for . The eigenvalue 0 is simple and the corresponding eigenfunction is 1. The eigenvalues for are double and the corresponding eigenfunctions and associated functions are
respectively. Note that for any constant c, is also an associated function corresponding to , since one can easily verify that it satisfies the equation and boundary conditions for the associated functions. It can be shown that the adjoint operator is associated with the boundary conditions
It is easy to see that 0 is a simple eigenvalue of and the corresponding eigenfunction is . The other eigenvalues for , are double and the corresponding eigenfunctions and associated functions are
respectively.
Let
and
(see (12) and (13)). The system of the root functions of can be written as , where
One can easily verify that it forms a basis in and the biorthogonal system is the system of the root functions of , where
since .
To obtain the asymptotic formulas for the eigenvalues and the corresponding normalized eigenfunctions of we use (11) and the well-known relations
and
where
which can be obtained by multiplying both sides of the equality
by and , respectively. It follows from (18) and (19) that
Moreover, we use the following relations:
for , where . These relations are obvious for , since to obtain (22) and (23) we can use the decomposition of and by the basis (16). For see Lemma 1 of [25].
To obtain the asymptotic formulas for the eigenvalues and eigenfunctions we iterate (18) and (19) by using (22) and (23). First let us prove the following obvious asymptotic formulas, namely (29), for the eigenfunctions . The expansion of by the basis (17) can be written in the form
where
and , are defined in (14) and (15), respectively. Using (20), (21), (24), and (25) one can readily see that there exists a constant C such that
Hence by (26) and (28) we obtain
Since is normalized, we have
that is,
where
Note that , since and by (30) we see that at least one of and is different from zero.
Now let us iterate (18). Using (22) in (18) we get
Isolating the terms in the right-hand side of this equality containing the multiplicands and (i.e., the case ), using (20) and (21) for the terms and , respectively (in the case ), we obtain
where
Using (22) and (23) for the terms and of the last summation we obtain
Now isolating the terms for we get
Here and below the summations are taken under the conditions and for . Introduce the notations
where
Using these notations and repeating this iteration k times we get
where
It follows from (10), (11), (24), and (25) that
for and for all , where .
Therefore letting k tend to infinity, we obtain
where
and by (32) we have
for and for all .
Thus iterating (18) we obtain (31). Now iterating (19) instead of (18), using (23) and (22), and arguing as in the previous iteration, we get
where
Similar to (32) one can verify that
for and for all . Now letting k tend to infinity in (35), we obtain
where
and by (37) we have
for and for all .
To get the main results of this paper we use the following system of equations, obtained above, with respect to and ,
where
(see (33) and (36)). Note that (39), (40) with (34), (38) give
Introduce the notations
Then, by (41)-(46) and (49) we have
Theorem 1 The following statements hold:
-
(a)
Any eigenfunction of corresponding to the eigenvalue defined in (10) satisfies
(54)Moreover, there exists N such that for all the geometric multiplicity of the eigenvalue is 1.
-
(b)
A complex number , where is defined in (32), is an eigenvalue of if and only if it is a root of the equation
(55)Moreover, is a double eigenvalue of if and only if it is a double root of (55).
Proof (a) By (10) the left-hand side of (48) is , which implies that . Therefore from (29) we obtain (54). Now suppose that there are two linearly independent eigenfunctions corresponding to . Then there exists an eigenfunction satisfying
which contradicts (54).
(b) First we prove that the large eigenvalues are the roots of (55). It follows from (54), (27), and (15) that . If then multiplying (39) and (40) side by side and then canceling we obtain (55). If then by (39) and (40) we have and , which means that (55) holds. Thus in any case is a root of (55).
Now we prove that the roots of (55) lying in are the eigenvalues of . Let be the left-hand side of (55), which can be written as
and
Using (34) and (38), one can easily verify that the inequality
holds for all λ from the boundary of . Since the function has two roots in the set , by the Rouche theorem we find that has two roots in the same set. Thus has two eigenvalues (counting with multiplicities) lying in that are the roots of (55). On the other hand, (55) has preciously two roots (counting with multiplicities) in . Therefore is an eigenvalue of if and only if (55) holds.
If is a double eigenvalue of then it has no other eigenvalues in and hence (55) has no other roots. This implies that λ is a double root of (55). By the same way one can prove that if λ is a double root of (55) then it is a double eigenvalue of . □
Let us consider (55) in detail. By (56) we have
If we substitute in (57), then it becomes
The solutions of (58) are
where
which can be written in the form
and, as we shall see below, can be defined as analytic function on . Clearly the eigenvalue is a root either of the equation
or of the equation
Now let us examine and in detail. If (8) holds then one can readily see from (34), (38), (50)-(53), and (59) that
for . By (62) there exists an appropriate choice of branch of (depending on n) which is analytic on . Taking into account (62), (34), (38), (50), and (51), we see that (60) and (61) have the form
Theorem 2 If (8) holds, then the large eigenvalues are simple and satisfy the following asymptotic formulas:
for . Moreover, if there exists a sequence such that (8) holds when n is replaced by , then the root functions of do not form a Riesz basis.
Proof To prove that the large eigenvalues are simple let us show that one of the eigenvalues, say satisfies (65) for and the other satisfies (65) for . Let us prove that each of (60) and (61) has a unique root in by proving that
is a contraction mapping. For this we show that there exist positive real numbers , , such that
for , where . The proof of (66) is similar to the proof of (56) of the paper [26].
Now let us prove (67). By (62) and (8) we have
for . On the other hand arguing as in the proof of (56) of the paper [26] we get
Hence for the large values of n we have
for . Thus by the fixed point theorem, each of (60) and (61) has a unique root and in respectively. Clearly by (63) and (64), we have which implies that (55) has two simple roots in . Therefore by Theorem 1(b), and are the eigenvalues of lying in , that is, they are and , which proves the simplicity of the large eigenvalues and the validity of (65).
If there exists a sequence such that (8) holds when n is replaced by , then by Theorem 1(a)
Now it follows from the theorems of [4, 5] (see also Lemma 3 of [18]) that the root functions of do not form a Riesz basis. □
Now let us consider the operators , , and . First we consider the operator .
It is well known that (see (47a) and (47b) on page 65 of [24]) the eigenvalues of the operators consist of the sequences , satisfying (10) when is replaced by . The eigenvalues, eigenfunctions and associated functions of are
respectively. The biorthogonal systems analogous to (16) and (17) are
respectively.
Analogous formulas to (18) and (19) are
respectively, where
Instead of (16)-(19) using (68)-(71) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for .
Theorem 3 If (8) holds, then the large eigenvalues are simple and satisfy the following asymptotic formulas:
for . The eigenfunctions corresponding to obey
Moreover, if there exists a sequence such that (8) holds when n is replaced by , then the root functions of do not form a Riesz basis.
Now let us consider the operator . It is well known that (see (47a) and (47b) on page 65 of [24]) the eigenvalues of the operators consist of the sequences , satisfying
for . The eigenvalues, eigenfunctions, and associated functions of are
for , respectively. The biorthogonal systems analogous to (16) and (17) are
respectively.
Analogous formulas to (18) and (19) are
respectively, where
Instead of (16)-(19) using (73)-(76) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for .
Theorem 4 If (9) holds, then the large eigenvalues are simple and satisfy the following asymptotic formulas:
for . The eigenfunctions corresponding to obey
Moreover, if there exists a sequence such that (9) holds when n is replaced by , then the root functions of do not form a Riesz basis.
Lastly we consider the operator . It is well known that (see (47a) and (47b) on page 65 of [24]) the eigenvalues of the operators consist of the sequences , satisfying (72) when is replaced by . The eigenvalues, eigenfunctions, and associated functions of are
for , respectively. The biorthogonal systems analogous to (16) and (17) are
respectively.
Analogous formulas to (18) and (19) are
respectively, where
Instead of (16)-(19) using (77)-(80) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for .
Theorem 5 If (9) holds, then the large eigenvalues are simple and satisfy the following asymptotic formulas:
for . The eigenfunctions corresponding to obey
Moreover, if there exists a sequence such that (9) holds when n is replaced by , then the root functions of do not form a Riesz basis.
Now suppose that
If
where B is defined by (34), then one can readily see from (59), (34), (38), and (50)-(53) that there exists a positive constant K such that
for and for the large values of n. Therefore arguing as in the proof of Theorem 2, we obtain the following.
Theorem 6 Suppose that (81) holds. If (82) holds, then the large eigenvalues of the operator are simple. Moreover, if there exists a sequence such that (82) holds when n is replaced by , then the root functions of do not form a Riesz basis. Similar results continue to hold for the operators , , and .
Remark 1 Since the eigenvalues and are the fixed points of (60) and (61) respectively, using the fixed point iteration one can determine these eigenvalues with arbitrary precision. Moreover, using these better approximations of the eigenvalues, one can also determine the better approximations for the eigenfunctions of the operator . Similar results can be obtained for the operators , , and .
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Nur, C., Veliev, O.A. On the basis property of the root functions of some class of non-self-adjoint Sturm-Liouville operators. Bound Value Probl 2014, 57 (2014). https://doi.org/10.1186/1687-2770-2014-57
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DOI: https://doi.org/10.1186/1687-2770-2014-57