Uniform Convergence of the Spectral Expansion for a Differential Operator with Periodic Matrix Coefficients

In this paper, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and the quasiperiodic boundary conditions. Using these asymptotic formulas, we find conditions on the coefficients for which the root functions of this operator form a Riesz basis. Then we obtain the uniformly convergent spectral expansion of the differential operators with the periodic matrix coefficients

It is well-known that ( see [2,10] ) the spectrum σ(L) of L is the union of the spectra σ(L t ) of L t for t ∈ [0, 2π). To construct the uniformly convergent spectral expansion for L we first obtain the uniform, with respect to t ∈ Q ε (n), asymptotic formula for the eigenvalues and eigenfunctions of L t , where Q ε (2µ) = {t ∈ Q : |t − πk| > ε, ∀k ∈ Z}, Q ε (2µ + 1) = Q, Q is compact connected subset of C containing a neighborhood of the interval [−a, 2π − a], a ∈ (0, π 2 ), ε ∈ (0, a 2 ) and µ = 1, 2, .... Then we prove that the root functions of L t for t ∈ C(n) form a Riesz basis in L m 2 (0, 1), where C(2µ) = C\{πk : k ∈ Z}, C(2µ + 1) = C. Let us introduce some preliminary results and describe the scheme of the paper. Clearly where N ≫ 1, satisfying the following, uniform with respect to t ∈ Q, asymptotic formulas for j = 1, 2, ..., m. We say that the formula f (k, t) = O(h(k)) is uniform with respect to t ∈ Q if there exists a positive constant c, independent of t, such that | f (k, t)) |< c | h(k) | for all t ∈ Q and | k |≫ 1. The method proposed here allows us to obtain the asymptotic formulas of high accuracy for the eigenvalues λ k,j (t) and the corresponding normalized eigenfunctions Ψ k,j,t (x) of L t when p ν,i,j ∈ L 1 [0, 1] for all ν, i, j . Note that to obtain the asymptotic formulas of high accuracy by the classical methods it is required that P 2 , P 3 , ..., P n be differentiable (see [12]). To obtain the asymptotic formulas for L t we take the operator L t (C), where L t (P 2 , ..., P n ) is denoted by L t (C) when P 2 (x) = C, P 3 (x) = 0, ..., P n (x) = 0, for an unperturbed operator and L t −L t (C) for a perturbation. One can easily verify that the eigenvalues and normalized eigenfunctions of L t (C) are µ k,j (t) = (2πki + ti) n + µ j (2πki + ti) n−2 , Φ k,j,t (x) = v j e itx e i(2πk+t)x (5) for k ∈ Z, j = 1, 2, ..., m, where v 1 , v 2 , ..., v m are the normalized eigenvectors of the matrix C corresponding to the eigenvalues µ 1 , µ 2 , ..., µ m respectively. In section 2 we investigate the operator L t and prove the following 2 theorems.
(b) If λ k,j (t) ∈ U (µ k,p(j) (t), c 1 |k| n−3 ln |k|), then there exists unique eigenfunction Ψ k,j,t (x) corresponding to λ k,j (t) and this eigenfunction satisfies where c 2 is a constant independent of t and j.
Note that here and in forthcoming relations we denote by c i for i = 1, 2, ..., the positive constants, independent of t, whose exact values are inessential. Using Theorem 1 and investigating associated functions of L t we prove: Theorem 2 (a) The large eigenvalues of L t consist of m sequences (3) satisfying the following, uniform with respect to t ∈ Q ε (n), formula is a simple eigenvalue of L t and the corresponding normalized eigenfunction Ψ k,j,t (x) satisfies where v * j is the eigenvector of C * corresponding to µ j and (v * j , v j ) = 1. Note that A. A. Shkalikov [13,14] proved that the root functions of the operators generated by a ordinary differential expression, in the scalar case, with summable coefficients and more complicated boundary conditions form a Riesz basis with brackets. L. M. Luzhina [8] generalized these results for the matrix case. In [22] we prove that if n = 2 and the eigenvalues of the matrix C are simple then the root functions of L t for t ∈ (0, π) ∪ (π, 2π) form a ordinary Riesz basis without brackets. The case n > 2 is more complicated and the most part of the method of the paper [22] does not work here, since in the case n > 2 the adjoint operator of the operator generated by l(y) with arbitrary summable coefficients can not be defined by the Lagrange's formula.
In section 3 using Theorem 2 we obtain spectral expansion for the operator L. The spectral expansion for the Hill operator with real-valued potential q(x) was constructed by Gelfand in [4] and Titchmarsh in [15]. Tkachenko proved in [16] that the Hill operator, namely the operator L in the case m = 1, n = 2 can be reduced to triangular form if all eigenvalues of the corresponding operators L t for t ∈ [0, 2π) are simple. McGarvey in [10,11] proved that L, in the case m = 1, is spectral operator if the projections of the operator L are uniformly bounded. Gesztesy and Tkachenko in the recent paper [5] proved that the Hill operator is a spectral operator of scalar type if and only if for all t ∈ [0, 2π) the operators L t have not associated function, the multiple point of either the periodic or anti-periodic spectrum is a point of its Dirichlet spectrum and some other condition hold. However, in general, the eigenvalues are not simple, projections are not uniformly bounded, and L t has associated function, since the Hill operator with simple potential q(x) = e i2πx has infinitely many spectral singularities ( see [3], where Gasymov investigated the Hill operator with special potential, analytically continuable onto the upper half plane). Note that the spectral singularity of L is the point of S(T ) in neighborhood on which the projections of the operator L are not uniformly bounded and we proved in [18] that a number λ ∈ S(L t ) ⊂ S(L) is a spectral singularity if and only if L t has an associated function corresponding to the eigenvalue λ. The existence of the spectral singularities and the absence of the Parseval's equality for the nonself-adjoint operator L t do not allow us to apply the elegant method of Gelfand ( see [4]) for construction of the spectral expansion for the nonself-adjoin operator L. These situation essentially complicate the construction of the spectral expansion for the nonself-adjoint case. In [17] and [20] we constructed the spectral expansion for the Hill operator with continuous complex-valued potential q(x) and with locally summable complex-valued potential q(x) respectively. Then in [19] and [21] we constructed the spectral expansion for the nonself-adjoint operator L, in the case m = 1, with coefficients p k ∈ C (k−1) [0, 1] and with p k ∈ L 1 [0, 1] for k = 2, 3, ..., n respectively. In the paper [9] we constructed the spectral expansion of L when p k,i,j ∈ C (k−1) [0, 1]. In this paper we do it when p k,i,j (x) are arbitrary Lebesgue integrable on (0, 1) functions. Besides, in [9] the expansion is obtained for compactly supported continuous vector functions, while in this paper for each function when n = 2µ. Moreover, using Theorem 2, we prove that the spectral expansion of L converges uniformly in every bounded subset of (−∞, ∞) if f is absolutely continuous compactly supported function and f ′ ∈ L m 2 (−∞, ∞). Note that the spectral expansion obtained in [9], when p k,i,j ∈ C (k−1) [0, 1], converges in the norm of L m 2 (a, b), where a and b are arbitrary real number. Some parts of the proofs of the spectral expansions for L is just writing in vector form of the corresponding proofs obtained in [19] for the case m = 1. These parts are given in appendices, in order to give a possibility to reed this paper independently.

On the eigenvalues and root functions of L t
The formula (4) shows that the eigenvalue λ k,j (t) of L t is close to the eigenvalue (2kπi + ti) n of L t (0). If t ∈ Q ε (n), | k |≫ 1 then the eigenvalue (2πki + ti) n of L t (0) lies far from the other eigenvalues (2pπi + ti) n . It follows from (4) that where | k |≫ 1, ν ≥ 2, and (12), (13) are uniform with respect to t ∈ Q ε (n).
The boundary conditions adjoint to (2) is U ν,t (y) = 0 for ν = 0, 1, ..., (n − 1). Therefore the eigenfunction ϕ * k,s,t (x) and Φ * k,s,t (x) of the operators L * t (0) and L * t (C) corresponding to the eigenvalues (2πpi + ti) n and µ k,j (t) respectively and satisfying (ϕ k,j,t , ϕ * k,s,t ) = 1, where v * s is defined in Theorem 2(c). To prove the asymptotic formulas for the eigenvalues λ k,j (t) and the corresponding normalized eigenfunctions Ψ k,j,t (x) of L t we use the formula which can be obtained from by multiplying scalarly by Φ * k,s,t (x). To estimate the right-hand side of (15) we use (12), (13), the following lemma, and the formula which can be obtained from (16) by multiplying scalarly by ϕ * p,s,t (x).
Therefore there exist a positive constant M (k, j) and indices p 0 , s 0 satisfying max p∈Z, s=1,2,...,m n ν=2 Then using (17) and (12), we get where d > 2|k|. This implies that the decomposition of Ψ k,j,t (x) by basis where sup Now using the integration by parts, (1), and the inequality (21), we obtain Therefore arguing as in the proof of (22) and using (12) we get where ν = 2, 3, . . . , n, and k,j,t , ϕ * p,s,t ) and tending q to ∞, we obtain (18). Let us we prove (19). It follows from (20) and (18) that By (21) and (13) we have On the other hand Therefore using (24) we get Now using this we prove the following lemma.

Lemma 2
The following equalities Proof. Using (18) for ν = 2, p = k and the obvious relation we see that This with (25) and (13) for ν = 2 implies that Similarly, using (18), (25), (13) we obtain Since (13) is uniform with respect to t ∈ Q ε (n) and the constant c 5 in (25) does not depend on t ( recall that we denote by c k the constant independent of t) these formulas are uniform with respect to t ∈ Q ε (n). Therefore recalling the definitions of Φ * k,s,t and ϕ * k,q,t ( see (14)) we get the proof of (26) and (27) Proof. It follows from (25) and (13) that and this formula is uniform with respect to t ∈ Q ε (n). Then the decomposition of Ψ k,j,t (x) by the basis {ϕ p,s,t (x) : s = 1, 2, ..., m, p ∈ Z} has the form Since Ψ k,j,t = ϕ k,j,t = 1 and (30) is uniform with respect to t ∈ Q ε (n), there exists a positive constant N 1 , independent of t, such that for all | k |≥ N 1 , t ∈ Q ε (n) and j = 1, 2, ..., m. Therefore using (14) and taking into account that the vectors v * 1 , v * 2 , ..., v * m form a basis in C m , that is, e s is a linear combination of these vectors we get the proof of (28) THE PROOF OF THEOREM 1(a). It follows from Lemma 2 that there exists a positive constant N 2 , independent of t, such that if | k |≥ N 2 , t ∈ Q ε (n) then the right-hand side of (15) is less than c 10 |k| n−3 ln |k|. Therefore (15) and Lemma 3 give the proof of the Theorem 1(a).
THE PROOF OF THEOREM 1(b). Let λ k,j be an eigenvalue of L t lying in U (µ k,p(j) (t), c 1 |k| n−3 ln |k|) and Ψ k,j,t be any normalized eigenfunction corresponding to λ k,j . Then using (5) and taking into account that the eigenvalues of C are simple we get (15), (26), (27) gives On the other hand by (14) and (29) Since (26), (27), (29) are uniform with respect to t ∈ Q ε (n) the formulas (31) and (32) are also uniform. Therefore decomposing Ψ k,j,t (x) by basis {Φ p,s,t (x) : s = 1, 2, ..., m, p ∈ Z} we see that any normalized eigenfunction corresponding to λ k,j satisfies (6). If there are two linearly independent eigenfunctions corresponding to λ k,j , then one can find two orthogonal eigenfunctions satisfying (6), which is impossible. Theorem 1 is proved.
To proof of the main results for L t (Theorem 2) we need to investigate the normalized associated function Ψ k,j,1,t (x) of L t corresponding to the eigenvalue λ k,j (t). By definition of the associated function we have where Ψ k,j,0,t (x) is an eigenfunction of L t . Note that, in general, the eigenfunction Ψ k,j,0,t (x) is not normalized. For investigation of the associated function we use the following formulas.
By Proposition 1 the eigenvalue λ k,j (t) of L t for |k| ≥ N 0 is simple and by Theorem 1 the corresponding eigenfunction satisfy (6), where p(j) = j (see the definition of p(j) in Theorem 1), that is, (8), (7) and Theorem 2(a) is proved.
THE PROOF OF THEOREM 2(b). It follows from (8) that the root functions of L t quadratically close to the system {v j e itx −1 e i(2πk+t)x : k ∈ Z, l = 1, 2, ..., m} which form a Riesz in L m 2 (0, 1). On the other hand the system of the root functions of L t is complete and minimal in L m 2 (0, 1) ( see [8]). Therefore, by Bari theorem ( see [1,6]), the system of the root functions of L t forms a Riesz basis in L m 2 (0, 1). THE PROOF OF THEOREM 2(c). To prove the asymptotic formulas for normalized eigenfunction Ψ * k,j,t (x) of L * t corresponding to the eigenvalue λ k,j (t) we use the formula obtained from L * t Ψ * k,j,t = λ k,j (t)Ψ * k,j,t by multiplying by ϕ p,s,t and using (L * t Ψ * k,j,t , ϕ p,s,t ) = (Ψ * k,j,t , L t ϕ p,s,t ). Instead of (17) using these formula and arguing as in the proof of (25) we obtain This with (5) and (13) implies the following relations On the other hand (8) and equality Ψ * k,j,t , Ψ k,s,t = 0 for j = s give Since (8), (13) hold uniformly the formulas (46)-(48) are uniform with respect to t ∈ Q ε (n) and they yield where v * j is defined in Theorem 2(c). Now (8) and (49) imply (9), since

THE PROOF OF THEOREM 2(d).
To investigate the convergence of the expansion series of L t we consider the series k:|k|≥N , j=1,2,...,m where N ≥ N 0 and N 0 is defined in Theorem 1, f (x) is absolutely continuous function satisfying (1) and f ′ (x) ∈ L m 2 (0, 1). Without loss of generality instead of the series (51) we consider the series k:|k|≥N , j=1,2,...,m where f t (x) is defined by Gelfand transform ( see [4]) f is absolutely continuous compactly supported function and f ′ ∈ L m 2 (−∞, ∞), since we use (52) in next section for spectral expansion of L. It follows from (53) that To prove the uniform convergence of (52) we consider the series To estimate the terms of this series we decompose X k,j,t by basis {Φ * p,s,t : p ∈ Z, s = 1, 2, ..., m} and then use the inequality Using the integration by parts and then Schwarz inequality we get |k|≥N , s=1,2,...,m Again using the integration by parts, Schwarz inequality and (46), (50) we obtain that the expression in the in the second row of (56) is less than It is not hard to see that this expression is less than c 17 k −2 , that is, the expression in the second row of (56) is less than c 17 k −2 . Therefore the relations (56), (57) imply that the expressions in (55) and (52) tend to zero uniformly with respect to t ∈ Q ε (n) and t ∈ Q ε (n), x ∈ [0, 1] respectively as N → ∞. Since in the proof of the uniform convergence of (52) we used only the properties (54) of f t the series (51) converges uniformly with respect to x ∈ [0, 1], that is, Theorem 2(d) is proved.
Note that in the proof of Theorem 2(d) we proved the following theorem, which will be used in next section.
Theorem 3 If f is absolutely continuous, compactly supported function and f ′ ∈ L m 2 (−∞, ∞) then the series (52), where f t is defined by (53), N ≥ N 0 , N 0 is defined in Theorem 1(a), converges uniformly with respect to t ∈ Q ε (n), x ∈ D for any bounded subset D of (−∞, ∞).

Spectral Expansion for L
Let Y 1 (x, λ), Y 2 (x, λ), . . . , Y n (x, λ) be the solutions of the matrix equation satisfying Y which is a polynomial of e it with entire coefficients f 1 (λ), f 2 (λ), .... Therefore the multiple eigenvalues of the operators L t are the zeros of the resultant R(λ) ≡ R(∆, ∆ ′ ) of the polynomials ∆(λ, t) and ∆ ′ (λ, t) ≡ ∂ ∂λ ∆(λ, t). Since R(λ) is entire function and the large eigenvalues of L t for t = 0, π are simple ( see Theorem 2 (a)), For each a k there are nm values t k,1 , t k,2 , ..., t k,nm of t satisfying ∆(a k , t) = 0. Hence the set is countable and for t / ∈ A all eigenvalues of L t are simple eigenvalues. By Theorem 2(a) the possible accumulation point of the set A are πk, where k ∈ Z.

Lemma 6
The eigenvalues of L t can be numbered as λ 1 (t), λ 2 (t), ..., such that for each p the function λ p (t) is continuous in Q and is analytic in Q\A(p), where A(p) is a subset of A consisting of finite numbers t p 1 , t p 2 , ..., t p sp . Moreover the followings hold: where |k| ≥ N 0 , p(k, j) = 2|k|m + j if k > 0, p(k, j) = (2|k| − 1)m + j if k < 0, the sets Q ε (n), Q and number N 0 are defined in (2) and in Theorem 1(a).
Proof. Let t ∈ Q. It easily follows from the classical investigations [12, chapter 3, theorem 2] ( see (3), (4)) that there exist a large numbers r and c, independent of t, such that the all eigenvalues of the operators L t,z for z ∈ [0, 1], where L t,z is defined by (45), lie in the set where U (µ, c) = {λ ∈ C :| λ − µ |< c}. Clearly there exist a closed curve Γ such that: (a) The curve Γ lies in the resolvent set of the operators L t,z for all z ∈ [0, 1]. (b) All eigenvalues of L t,z for all z ∈ [0, 1] that do not lie in U ((2πki + ti) n , ck n−1− 1 2m ) for |k| ≥ N 0 belong to the set enclosed by Γ.
(ii) U (t, δ) ∩ A(U 0 ) = ∅ and d s,t (z) ∈ U 0 for z ∈ U (t, δ), s = 1, 2, ..., 2m. Now take any point t 0 from U (0, ε)\A(U 0 ). Let γ be line segment in U (0, ε)\A(U 0 ) joining t 0 and a point of the circle S(0, ε) = {t : |t| = ε}. For any t from γ there exist U (t, δ) satisfying (i) and (ii). Since γ is a compact set the cover {U (t, δ) : t ∈ γ} of γ contains a finite cover U (t 0 , δ), U (t 1 , δ), ..., U (t v , δ), where t v ∈ S(0, ε). Now we are ready to continue analytically the function λ p(k,j) (t) into the set U (0, ε). For any z ∈ U (t v , δ) ∩ Q ε (n) the eigenvalue λ p(k,j) (z) coincides with one of the eigenvalues d 1,tv (z), d 2,tv (z), ..., d 2m,tv (z), since there exists 2m eigenvalue of L z lying in U 0 . Denote by B s the subset of the set U (t v , δ) ∩ Q ε (n) for which the function λ p(k,j) (z) coincides with d s,tv (z). Since d s,t (z) = d i,t (z) for s = i the sets B 1 , B 2 , ..., B 2m are pairwise disjoint and the union of these sets is U (t v , δ) ∩ Q ε (n). Therefore there exists index s for which the set B s contains accumulation point and hence λ p(k,j) (z) = d s,tv (z) for all z ∈ U (t v , δ) ∩ Q ε (n). Thus d s,tv (z) is analytic continuation of λ p(k,j) (z) to U (t v , δ). In the same way we get the analytic continuation of λ p(k,j) (z) to U (t v−1 , δ), U (t v−2 , δ), ..., U (t 0 , δ). Since t 0 is arbitrary point of U (0, ε)\A(U 0 ) we obtain the analytic continuation of λ p(k,j) (z) to U (0, ε)\A(U 0 ). The analytic continuation of λ p(k,j) (z) to U (π, ε)\A(U π ) can be obtained in the same way, where A(U π ) can be defined as A(U 0 ). Thus the function λ p(k,j) (t) is analytic in Q\A(p), where A(p) consist of finite numbers t p 1 , t p 2 , ..., t p sp . Since ∆(λ, t) is continuos with respect (λ, t), the function λ p(k,j) (t) can be extended continuously to the set Q. Now let us define the eigenvalues λ p (t) for p ≤ (2N 1 − 1)m, t ∈ Q which are apart from the eigenvalues defined by (63). These eigenvalues lies in a bounded set B and by (61) the set B ∩ ker R and the subset A(B) of A corresponding to B are finite. Take a point a from the set Q\A. Denote the eigenvalues of L a in increasing ( of absolute value) order | λ 1 (a) |≤| λ 2 (a) |≤ ... ≤| λ (2N1−1)m (a) | . If | λ p (a) |=| λ p+1 (a) | then by λ p (a) we denote the eigenvalue that has a smaller argument, where argument is taken in [0, 2π). Since a / ∈ A the eigenvalues λ 1 (a), λ 2 (a), ..., λ (2N1−1)m (a) are simple zeros of ∆(λ, a) = 0. Therefore using the implicit function theorem we obtain the analytic functions λ 1 (t), λ 2 (t), ..., λ (2N1−1)m (t) on a neighborhood U (a, δ) of a which are eigenvalues of L t for t ∈ U (a, δ). These functions can be analytically continued to Q ε (n)\A, being the eigenvalues of L t , where, as we noted above, A∩Q ε (n) consist of a finite number of points. Taking into account that A(B) is finite, arguing as we have done in the proof of analytic continuation and continuous extension of λ p (t) for p > (2N 1 − 1)m, we obtain the analytic continuations of these functions to the set Q except finite points and continuous extension to Q By Gelfand's Lemma ( see [4]) every compactly supported vector function f (x) can be represented in the form where f t (x) is defined by (53). This representation can be extended to all function of L m 2 (−∞, ∞), and where {X k,t : k = 1, 2, ...} is the biorthogonal system of {Ψ k,t : k = 1, 2, ...}, Ψ k,t (x) is a normalized eigenfunction corresponding to λ k (t), the eigenvalue λ k (t) is defined in Lemma 6, Ψ k,t (x) and X k,t (x) are extended to (−∞, ∞) by (58) and by X k,t (x + 1) = e it X k,t (x). Let a ∈ (0, π 2 )\A, ε ∈ (0, a 2 ) and let l(ε) be a smooth curve joining the points −a and 2π − a and satisfying where Π(a, ε) = {x + iy : x ∈ [−a, 2π − a], y ∈ [0, 2ε)}, l(−ε) = {t :t ∈ l(ε)}, the sets Q, Q ε (n) and A are defined in (2) Since l(ε) ∈ C(n) ( see (65) and the definition of C(n) in the introduction), it follows from Theorem 2(b) and Lemma 6 that for each t ∈ l(ε) we have a decomposition where a k (t) = (f t , X k,t ). Using (67) in (66) we get Remark 1 If λ ∈ σ(L) then there exists points t 1 , t 2 , ..., t k of [0, 2π) such that λ is an eigenvalue λ(t j ) of L tj of multiplicity s j for j = 1, 2, ..., k. Let S(λ, b) = {z :| z − λ |= b} be a circle containing only the eigenvalue λ(t j ) of L tj for j = 1, 2, ..., k. Using Lemma 6 we see that there exists a neighborhood U (t j , δ) = {t :| t − t j |≤ δ} of t j such that: (a) The circle S(λ, b) lies in the resolvent set of L t for all t ∈ U (t j , δ) and j = 1, 2, ..., k.
Thus the spectrum of L t for t ∈ U (t j , δ), j = 1, 2, ..., k separated by S(λ, b) into two parts in since of [7] ( see §6.4 of chapter 3 of [7]). Since {L t : t ∈ U (t j , δ)} is a holomorphic family of operators in since [7] (see §1 of chapter 7 of [7]), the theory of holomorphic family of finite dimensional operators can be applied to the part of L t for t ∈ U (t j , δ) corresponding to the inside of S(λ, b). Therefore ( see §1 of the chapter 2 of [7] ) the eigenvalue Λ j,1 (t), Λ j,2 (t), ..., Λ j,sj (t) and corresponding eigenprojections P (Λ j,1 (t)), P (Λ j,2 (t)), ..., P (Λ j,sj (t)) are branches of an analytic function. These eigenprojections is represented by a Laurent series in t 1 ν , where ν ≤ s j , with finite principal parts. One can easily see that if λ p (t) is a simple eigenvalue of L t then and P (λ p (t)) is analytic function in some neighborhood of t, where α p (t) = (Ψ p,t , Ψ * p,t ). This and Lemma 6 show that for each p the function a p (t)Ψ p,t is analytic on D(ε) ∪ D(−ε) except finite points.

Theorem 4 (a) If f (x) is absolutely continuous, compactly supported function and
and where Proof. The proof of (70) in case (a) follows from (68), Theorem 3, and Lemma 6. In Appendix A by writing the proof of the Theorem 2 of [19] in the vector form we get the proof of (70) in the case (b). In Appendix B the formula (71) is obtained from (70) by writing the proof of the Theorem 3 of [19] in the vector form Definition 1 Let λ be a point of the spectrum σ(L) of L and t 1 , t 2 , ..., t k be the points of [0, 2π) such that λ is a eigenvalue of L tj of multiplicity s j for j = 1, 2, ..., k. The point λ is called a spectral singularity of L if where supremum is taken over all t ∈ (U (t j , δ)\{t j }), j = 1, 2, ..., k; i = 1, 2, ..., s j , the set U (t j , δ) and the eigenvalues Λ j,1 (t), Λ j,2 (t), ..., Λ j,sj (t) are defined in Remark 1. In other words λ is called a spectral singularity of L if there exists indices j, i such that the point t j is a pole of P (Λ j,i (t)). Briefly speaking a point λ ∈ σ(L) is called a spectral singularity of L if the projections of L t corresponding to the simple eigenvalues lying in the small neighborhood of λ are not uniformly bounded. We denote the set of spectral singularities by S(L).
Remark 2 Note that if γ = {λ p (t) : t ∈ (α, β)} is a curve lying in σ(L) and containing no multiple eigenvalues of L t , where t ∈ [0, 2π), then arguing as in papers [18,9] one can prove that for the projection P (γ) of L corresponding to γ the following hold that is, the definition 1 is equivalent to the definition of the spectral singularities given in [18,9], where the spectral singularities is defined as a points in the neighborhoods of which the projections P (γ) are not uniformly bounded. The proof of (74) is long technical. In order to avoid eclipsing the essence by technical detail and taking into account that in the spectral expansion of L the eigenfunctions and eigenprojections of L t for t ∈ [0, 2π) are used ( see (71)), and using that there are the closed relationship between projections (see (74)) of L and L t for t ∈ [0, 2π), in this paper, in the definition of the spectral singularities, without loss of naturalness, instead of the boundlessness of projections P (γ) of L we use the boundlessness of projections P (λ p (t)), of L t , that is, we use the definition 1. In any case the spectral singularity is a point of σ(L) that requires the regularization in order to get the spectral expansion.  (61).
where U is a neighborhood of t 0 such that if t ∈ U then λ p (t) is not a spectral singularity.
(c) If the operator L has not spectral singularities then we have the following spectral expansion in term of the parameter t : If f (x) is absolutely continuous, compactly supported function and f ′ ∈ L m 2 (−∞, ∞) then the series in (76) converges uniformly in any bounded subset of (−∞, ∞). If f (x) ∈ S then the series converges in the norm of L m 2 (a, b) for every a, b ∈ R.
Proof. (a) If λ p (t 0 ) is a simple eigenvalue of L t0 then due to the Remark 1 ( see (69) and the end of Remark 1) the projection P (λ p (t)) and | α p (t) | continuously depend on t in some neighborhood of t 0 . On the other hand α p (t 0 ) = 0, since the system of the root functions of L t0 is complete. Therefore it follows from the Definition 1 that λ is not a spectral singularities of L.
(b) It follows from (61) and Theorem 5(a) that there exists a neighborhood U of t 0 such that if t ∈ U then λ p (t) is not spectral spectral singularities of L. If λ p (t 0 ) ∈ σ(L)\S(L) then by Definition 1 t 0 is not a pole of P (λ p (t)), that is, by Remark 1 the Laurent series in t 1 ν , where ν ≤ s, of P (λ p (t)) at t 0 has not principal part. Therefore (69) implies that 1 |αp(t)| and hence 1 |αp(t)| (f t , Ψ * p,t )Ψ p,t is a bounded continuous functions in some neighborhood of t 0 , which implies the proof of (b).
(c) It follows from Theorem 5(b) that if the operator L has not spectral singularities then where the left-hand side is defined by (72). Thus (76) follows from (77), (71) Now we change the variables to λ by using the characteristic equation ∆(λ, t) = 0 and the implicit-function theorem. By (60) ∆(λ, t) and ∂∆(λ,t) ∂t are polynomials of e it and their resultant is entire function. It is clear that this resultant is not zero function. Let b 1 , b 2 , ..., be zeros of the resultant, i.e., are the common zeros of the polynomials ∆(λ, t) and ∂∆(λ,t) ∂t .
To obtain (70) we must to prove that the last integral in (A2) tends to zero as N → ∞. For this we prove the following Lemma 7 On l ε the functions g N,t , k=1,2,...,N b N k (t))Ψ k,t (A3) tend to zero as N → ∞ uniformly with respect to t.
Proof. First we prove that g N,t tends to zero uniformly. Let P N,t and P ∞,t be projections of L m 2 [0, 1] onto H N,t and H ∞,t respectively, where H ∞,t = ∪ ∞ n=1 H N,t . If follows from (67) that f t ∈ H ∞,t . On the other hand one can readily see that H N,t ⊂ H N +1,t ⊂ H ∞,t , P N,t ⊂ P ∞,t , P N,t → P ∞,t .