Research Article | Open Access
On the Differential Operators with Periodic Matrix Coefficients
We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.
Let be the differential operator generated in the space of vector-valued functions by the differential expression where is an integer greater than and , for , is the matrix with the complex-valued summable entries satisfying for all and . It is well known that (see [1–4]) the spectrum of the operator is the union of the spectra of the operators for generated in by expression (1) and the quasiperiodic conditions Note that is the set of vector-valued functions with for . The norm and inner product in are defined by where and are the norm and inner product in .
The first works concerned with the differential operator were by Birkhoff , Tamarkin  in the beginning of 20th century. There exist enormously many papers concerning with the operators and . For the list of these papers one can look to the monographs [1, 7–10]. Here we only note that in these classical investigations in order to obtain the asymptotic formulas of high accuracy, by using the classical asymptotic expansions for solutions of the matrix equation it is required that the coefficients must be differentiable. Thus, these classical methods never permit us to obtain the asymptotic formulas of high accuracy for the operator with nondifferentiable coefficients. However, the method suggested in this paper is independent of smootness of the coefficients. Using this method we obtain an asymptotic formulas of high accuracy for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with only summable coefficients and then by using these formulas we consider the spectrum of the operator .
Let us introduce some preliminary results and describe the results of this paper. Clearly, are the eigenfunctions of the operator corresponding to the eigenvalue , where , and the operator is denoted by when . Furthermore, for brevity of notation, the operators and are denoted by and , respectively. It easily follows from the classical investigations [7, Chapter 3, Theorem 2] that the large eigenvalues of the operator consist of sequences satisfying the following, uniform with respect to in , asymptotic formulas: for , where is a sufficiently large positive number, that is, . We say that the formula is uniform with respect to in a set if there exists a positive constant , independent of , such that for all and . Thus formula (7) means that there exist positive numbers and , independent of , such that In this paper, by the suggested method, we obtain the uniform asymptotic formulas of high accuracy for the eigenvalues and for the corresponding normalized eigenfunctions of when the entries of belong to , that is, when there is not any condition about smoothness of the coefficients. Then using these formulas, we find the conditions on the coefficient for which the number of the gaps in the spectrum of the self-adjoint differential operator is finite.
Now let us describe the scheme of the paper. Inequality (8) shows that the eigenvalue of is close to the eigenvalue of . To analyze the distance of the eigenvalue of from the other eigenvalues of , which is important in perturbation theory, we take into account the following situations. If the order of the differential expression (1) is odd number, , and , then the eigenvalue of lies far from the other eigenvalues of for all values of . We have the same situation if and does not lie in the small neighborhoods of and . However, if is even number and lies in the neighborhoods of and , then the eigenvalue is close to the eigenvalues and respectively. For this reason instead of we consider and use the following notation.
By (8) there exists a positive constant , independent of , such that the inequalities where , hold in Cases 1, 2, and 3 for , for , and for respectively. To avoid the listing of these cases, using Notation , we see that the inequalities in (10) hold for . To obtain the asymptotic formulas we essentially use the following lemma that easily follows from (8) and (10).
Lemma 1. The equalities hold uniformly with respect to in , where , .
Proof. The proof of (11). It follows from (8) that if , then
Therefore the left-hand side of (11) is less than
which is . Thus (11) holds uniformly with respect to .
The proof of (12). The summation in the left-hand side of (12) is taking over all . Since where , , , and , the left-hand side of (12) can be written as , where Taking , from (11) we obtain that . If , then using (8) one can readily see that for all . Let . Clearly, if , , then and the number attains the same value at most times. Therefore Since the set has at most elements, and for the inequalities in (10) hold, we have Now, estimations for , , imply (12).
The proofs of (13) and (14). The proofs of (13) and (14) are similar to the proof of (12). Namely, again we consider the left-hand sides of (13) and (14) as and respectively, where the summations in and in are taking over . Then, repeating the arguments by which we estimated , and , we get These equalities imply the proof of (13) and (14).
To obtain the asymptotic formulas we use (11)–(14) and consider the operator as perturbation of by , where , is the operator generated by (2) and by the expression Therefore, first of all, we analyze the eigenvalues and eigenfunction of the operator . We assume that is the Hermitian matrix. Then the expression in (23) is the self-adjoint expression. Since the boundary conditions (2) are self-adjoint, the operator is also self-adjoint. The eigenvalues of , counted with multiplicity, and the corresponding orthonormal eigenvectors are denoted by and . Thus One can easily verify that the eigenvalues and eigenfunctions of are , , that is,
To prove the asymptotic formulas for the eigenvalues and for the corresponding normalized eigenfunctions of we use the formula which can be obtained from by multiplying both sides by and using (25). Then we estimate the right-hand side of (26) (see Lemma 3) by using Lemma 2. At last, estimating (see Lemma 4) and using these estimations in (26), we find the asymptotic formulas for the eigenvalues and eigenfunctions of (see Theorems 5 and 6). Then using these formulas, we find the conditions on the eigenvalues of the matrix for which the number of the gaps in the spectrum of the operator is finite (see Theorem 7). Some of these results for differentiable are obtained in [3, 11] by using the classical asymptotic expansions for the solutions of (4). The case is investigated in . In this case, an interesting spectral estimates were done in the paper , whose main goal was to reformulate some spectral problems for the differential operator with periodic matrix coefficients as problems of conformal mapping theory. In this paper we consider the more complicated case .
Lemma 2. Let be normalized eigenfunction of . Then for . Equality (29) is uniform with respect to in .
Proof. To prove (29) we use the arguments of the proof of the asymptotic formulas (6) and take into consideration the uniformity with respect to . The eigenfunction corresponding to the eigenvalue has the form
where , , for are linearly independent matrix solutions of (4) for satisfying
for . Here is unit matrix, are the th root of , and is an matrix satisfying the following conditions:
where and is a positive constant, independent of . To consider the uniformity, with respect to , of (29) we use (32).
The proof of (29) in the case , . Denote by the root of lying in neighborhood of and put . Then we have Suppose are ordered in such a way that where is the real part of . Using (31), (34), (2), and (33), we get where and satisfies the relation (32). Now using these relations and the notations of (30), we prove that Since satisfies (2) and (30), we have the system of equations with respect to for and , where are coordinates of the vector . Using (36) and (37) in (39) and then dividing both parts of th equation of (39), for , by , we get the system of equations whose coefficient matrix is and the right-hand side is . To estimate let us denote by the matrix obtained from by replacing with and by dividing the th column (note that the entries of the th column are the matrices) for and for by and by respectively. Clearly, and . Besides, interchanging the rows and then interchanging the columns of , we obtain . Using this and solving (39) by Cramer's rule, we get since is obtained from by replacing the th column of , which is the th column of In the same way, we obtain Now (38) follows from (44), (42), and (34). Therefore, the normalization condition , and (38), (30), (31), (33), and (34) imply that where , from which we get the proof of (29) for . Differentiating both sides of (30) and using (42) and (44), we get the proof of (29) for arbitrary in the case .
The proof of (29) in the case . In this case the th roots of are ordered in such a way that Hence we have Now using these equalities, we prove that Using (47) and arguing as in the case , we get the system of equations for . Arguing as in the proof of (42)–(45) and using (46), we get where , which implies the proof of (29) in the case .
It follows from this lemma that the equalities for and for hold uniformly with respect to in . Now (51) together with (28) implies that for , , and , where is a positive constant, independent of . Using this we prove the following lemma.
Lemma 3. Let be the entries of and . Then for and , where and is the Hermitian matrix defined in (23). Formula (54) is uniform with respect to in . Moreover, in Case 1 of Notation 1 the formula holds. If , then (56) is uniform with respect to in .
Proof. Note that if the entries of belong to , then (53) is obvious, since is an orthonormal basis in . Now we prove (53) in case . Using (2), (52), and the integration by parts, we see that there exists a constant , independent of , such that
for , . This and (11) imply that there exists a constant , independent of , such that
where , . Hence the decomposition of by the basis has the form
Using (59) in and letting tend to , we obtain (53).
Since , to prove (54), it is enough to show that for and . Using the obvious relation and (53), we see that Since for all , , , using (57) and (12), we see that the second summation of the right-hand side of (62) is . Besides, it follows from (29) and (55) that the first summation of the right-hand side of (62) is , since for and , we have . Hence (54) is proved. In Case 1 of Notation the first summation of the right-hand side of (62) is absent, since in this case and for . Thus (56) is proved. The uniformity of formulas (54) and (56) follows from the uniformity of (29), (11), and (12).
Lemma 4. There exists a positive number , independent of , such that for and for the following assertions hold.
(a)If is Hermitian matrix, then for each eigenfunction of there exists an eigenfunction of satisfying (b)If is self-adjoint operator, then for each eigenfunction of there exists an eigenfunction of satisfying
Proof. It follows from (52) and (13) that
Hence using the equality , where is the normalized eigenvectors of , and the Parseval equality, we get
Since the number of the eigenfunctions for , is less than (see Notation ), (64) follows from (68).
Using (52) and (14), we get Therefore, arguing as in the proof of (64) and taking into account that the eigenfunctions of the self-adjoint operator form an orthonormal basis in , we get the proof of (65).
Theorem 5. Let be a self-adjoint operator, and let be a Hermitian matrix. If , then for arbitrary , if , then for the large eigenvalues of consist of sequences (6) satisfying and the normalized eigenfunction corresponding to satisfies for , where are the eigenvalues of and is the orthogonal projection onto the eigenspace of corresponding to . If is a simple eigenvalue of , then the eigenvalue satisfying (70) is a simple eigenvalue, and the corresponding eigenfunction satisfies where is the eigenvector of corresponding to the eigenvalue . In the case the formulas (70)–(72) are uniform with respect to in .
Proof. By (51) and (56) the right-hand side of (26) is . On the other hand by Notation if , then there exists such that , and hence , for . Thus dividing (26) by , where , and hence , and using (64), we get
where , , . Instead of (64) using (65), in the same way, we obtain
for and . Hence to prove (70) we need to show that if the multiplicity of the eigenvalue is then there exist precisely eigenvalues of lying in for . The eigenvalues of and can be numbered in the following way: and . If has different eigenvalues with multiplicities , then we have
Suppose there exist precisely eigenvalues of lying in the intervals
these intervals are pairwise disjoints. Therefore using (6) and (7), we get
Now let us prove that . Due to the notations the eigenvalues of the operator lie in and by the definition of we have
for and . Hence using (26) for and (56), (51), we get
Using this, (67), and taking into account that for , we conclude that there exists normalized eigenfunction, denoted by , of corresponding to such that
for . Since are orthonormal system we have
This formula implies that the dimension of the eigenspace of corresponding to the eigenvalue is not less than . Thus . In the same way we prove that . Now (78) and the equality (see (75)) imply that . Therefore, taking into account that, the eigenvalues of consist of sequences satisfying (7), we get (70). The proof of (71) follows from (81).
Now suppose that is a simple eigenvalue of . Then is a simple eigenvalues of and, as it was proved above, there exists unique eigenvalues of lying in , where , and the eigenvalues for are the simple eigenvalues. Hence (72) is the consequence of (71), since there exists unique eigenfunction corresponding to the eigenvalue . The uniformity of the formulas (70)–(72) follows from the uniformity of (56), (51), (64), and (65).
Theorem 6. Let be a self-adjoint operator, let be a Hermitian matrix, let , be a simple eigenvalue of , let be a positive constant satisfying , and let be a set defined by , where There exist a positive number such that if and , then there exists a unique eigenvalue, denoted by , of lying in , where , is defined by (55), and is a positive constant, independent of . The eigenvalue is a simple eigenvalue of and the corresponding normalized eigenfunction satisfies