ON THE SPECTRALITY AND SPECTRAL EXPANSION OF THE NON-SELF-ADJOINT MATHIEU-HILL OPERATOR IN L 2 ( −∞ , ∞ )

. In this paper we investigate the non-self-adjoint operator H generated in L 2 ( −∞ , ∞ ) by the Mathieu-Hill equation with a complex-valued potential. We ﬁnd a necessary and suﬃcient conditions on the potential for which H has no spectral singularity at inﬁnity and it is an asymptotically spectral operator. Moreover, we give a detailed classiﬁcation, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.


1.
Introduction. Let L(q) be the Hill operator generated in L 2 (−∞, ∞) by the expression l(y) = −y + qy, (1) where q is a complex-valued summable function on [0, 1] and q(x + 1) = q(x) a.e.. It is well-known that (see [3,8,9]) the spectrum S(L(q)) of the operator L(q) is the union of the spectra S(L t (q)) of the operators L t (q) for t ∈ (−π, π], where L t (q) is the operator generated in L 2 [0, 1] by (1) and the boundary conditions y(1) = e it y(0), y (1) = e it y (0). ( The spectrum of L t (q) for t ∈ C consist of the eigenvalues that are the roots of where F (λ) = ϕ (1, λ) + θ(1, λ), ϕ and θ are the solutions of the equation l(y) = λy satisfying the initial conditions θ(0, λ) = ϕ (0, λ) = 1 and θ (0, λ) = ϕ(0, λ) = 0. The operators L t (q) and L(q) are denoted by H t (a, b) and H(a, b) respectively when where a and b are the complex numbers. In this paper we consider the spectrality and spectral expansion of the non-self-adjoint Mathieu-Hill operator H(a, b) defined in L 2 (−∞, ∞). For this aim, first, in Section 3 we obtain the uniform with respect to t in some neighborhood of 0 and π asymptotic formulas for the eigenvalues of the operators L t (q) and H t (a, b). These formulas are the preliminary investigations and have an auxiliary nature. Then, in Section 4 using these asymptotic formulas, we find a necessary and sufficient condition, stated in term of potential (4), for the asymptotic spectrality of the operator H(a, b). Finally, in Section 5 we classify in detail the form of the spectral expansion of H(a, b) in term of a and b.

O. A. VELIEV
Gesztesy and Tkachenko [6] proved two versions of a criterion for the Hill operator L(q) with q ∈ L 2 [0, 1] to be a spectral operator of scalar type, in sense of Danford, one analytic and one geometric. The analytic version was stated in term of the solutions of Hill's equation. The geometric version of the criterion uses algebraic and geometric properties of the spectra of periodic/antiperiodic and Dirichlet boundary value problems.
The problem of describing explicitly, for which potentials q the Hill operators L(q) are spectral operators appeared to have been open for about 60 years. In paper [13] we found the explicit conditions on the potential q such that L(q) is an asymptotically spectral operator. In this paper we find a criterion for asymptotic spectrality of H(a, b) stated in term of a and b. Note that these investigations show that the set of potentials q for which L(q) is spectral is a small subset of the periodic functions and it is very hard to describe explicitly the required subset. Moreover, the papers [17,18] and this paper show that the investigation of the spectrality is ineffective for the construction of the spectral expansion for L(q). For this in [17,18] we introduced a new notions essential spectral singularity (ESS) and ESS at infinity and proved that they determine the form of the spectral expansion for L(q). In this paper investigating the ESS and ESS at infinity for H(a, b) we classify the form of its spectral expansion in term of a and b.
To describe more precisely the main results of this paper let us introduce some notations and definitions of the needed notions. The spectrum of L t (q) consist of the eigenvalues. In [16] we proved that the eigenvalues λ n (t) of L t can be numbered (counting the multiplicity) by elements of Z such that, for each n the function λ n (·) is continuous on (−π, π] and |λ ±n (t)| → ∞ as n → ∞. The spectrum of L(q) is the union of the continuous curves Γ n = {λ n (t) : t ∈ (−π, π]} for n ∈ Z. Let Ψ n,t be the normalized eigenfunction corresponding to the simple eigenvalue λ n (t) and Ψ * n,t be the normalized eigenfunction of (L t (q)) * corresponding to λ n (t). It is wellknown that (see p. 39 of [10]) if λ n (t) is a simple eigenvalue of L t , then the spectral projection e(λ n (t)) defined by contour integration of the resolvent of L t (q) over the closed contour containing only the eigenvalue λ n (t), has the form e(λ n (t))f = 1 d n (t) (f, Ψ * n,t )Ψ n,t , where d n (t) = Ψ n,t , Ψ * n,t , e (λ n (t)) = |d n (t)| −1 , and (·, ·) is the inner product in L 2 [0, 1]. Note that in this paper the number d n (t) is defined only for the simple eigenvalues λ n (t). If λ n (t) is a simple eigenvalue then the normalized eigenfunctions Ψ n,t and Ψ * n,t are determined uniquely up to constant of modulus 1. Therefore | d n (t) | is uniquely defined and it is the norm of the projection e (λ n (t)). Note also that λ n (t) is a simple eigenvalue if F (λ n (t)) = 0 and the roots of the equation F (λ) = 0 is a discrete set, since F is an entire function. Thus λ n (t) is a simple eigenvalue for t ∈ ((−π, π]\A n ), where A n is at most a finite set. Moreover | d n | is continuous at t if λ n (t) is a simple eigenvalue. Therefore in this paper we prefer the following definitions stated in term of d n (t).
McGarvey [8] proved that L(q) is a spectral operator if and only if there exists c 1 > 0 such that e (λ n (t)) < c 1 for n ∈ Z and for almost all t ∈ (−π, π]. It can be stated in terms of d n (t) and A n as follows. Definition 1. We say that L(q) is a spectral operator if there exists c 1 > 0 such that |d n (t)| −1 < c 1 (7) for all n ∈ Z and t ∈ ((−π, π]\A n ).
Note that here and in subsequent relations we denote by c i for i = 1, 2, ... the positive constants whose exact values are inessential. Similarly, we use the following definition.
Definition 2. We say that L(q) is an asymptotically spectral operator if there exists N > 0 such that (7) holds for all |n| > N and t ∈ ((−π, π]\A n ).
As was noted in the paper [16], the spectral singularity of the operator L(q) are the points λ ∈ S(L(q)) for which the projections e (λ n (t)) corresponding to the simple eigenvalues λ n (t) lying in some neighborhood of λ are not uniformly bounded. Therefore we have the following definitions for the spectral singularities in term of d n .
Definition 3. A point λ ∈ σ(L(q)) is said to be a spectral singularity of L(q) if there exist n ∈ Z and sequence {t k } ⊂ ((−π, π]\A n ) such that λ n (t k ) → λ and |d n (t k )| → 0 as k → ∞. We say that the operator L(q) has a spectral singularity at infinity if there exist sequences {n k } ⊂ Z and {t k } ⊂ ((−π, π]\A n k ) such that |n k | → ∞ and |d n k (t k )| → 0 as k → ∞.
It is clear that the operator L(q) has no the spectral singularity at infinity if and only if it is asymptotically spectral operator. Now let us list the main results.
This main result of Section 4 implies the following Corollary 1. Let ab ∈ R. Then H(a, b) is a spectral operator if and only it is self adjoint.
These results show that the theory of spectral operator is ineffective for the study of the spectral expansion for the non-self-adjoint operator H(a, b) too. It was proven in [5] that in the self-adjoint case the spectral expansion of L(q) has the following elegant form where a n (t) = 1 In the non-self-adjoint case to obtain the spectral expansion, we need to consider the integrability of a n (t)Ψ n,t with respect to t over (−π, π] which is connected with the integrability of 1 dn . Therefore in [17] we introduced the following notions. Definition 4. A number λ 0 ∈ σ(L) is said to be an essential spectral singularity (ESS) of L if there exist t 0 ∈ (−π, π] and n ∈ Z such that λ 0 = λ n (t 0 ) and 1 dn is not integrable over (t 0 − δ, t 0 + δ) for all δ > 0.
It is clear that λ 0 = λ n (t 0 ) is ESS if and only is there exists sequence of closed intervals I(s) approaching t 0 such that λ n (t) for t ∈ I(s) are the simple eigenvalue and lim It the similar way in [17] we defined ESS at infinity.
Definition 5. We say that the operator L(q) has ESS at infinity if there exist sequence of integers n s and sequence of closed subsets I(s) of (−π, π] such that λ ns (t) for t ∈ I(s) are the simple eigenvalues and Note that it follows from the above definitions that the boundlessness of |d n (·)| −1 is the characterization of the spectral singularities and the considerations of the spectral singularities play only the crucial rule for the investigations of the spectrality of L(q). On the other hand, the periodic differential operators, in general, is not a spectral operator. Therefore to construct the spectral expansion for the operator L, in the general case, in [17,18] we introduced the new concepts ESS which connected with the nonintegrability of |d n (·)| −1 and proved that the spectral expansion has the elegant form (9) if and only if L(q) has no ESS and ESS at infinity. In Section 5 investigating the ESS and ESS at infinity for H(a, b) we obtained the following main results for its spectral expansion.
Theorem 2. If 0 < |ab| < 16/9, then H(a, b) has no ESS and ESS at infinity and its spectral expansion has the elegant form (9).
For the largest subclass of the potentials (4) we prove the following criterion Theorem 3. The non-self-adjoint operator H(a, b) has no ESS at infinity, has at most finite number of ESS and its spectral expansion has the asymptotically elegant form   (−π,π] n∈S a n (t)Ψ n,t (x)dt + n∈Z\S (−π,π] a n (t)Ψ n,t (x)dt   (13) if and only if ab = 0, where S is at most a finite set and is the set of the indices n for which Γ n contains at least one ESS.
For the remaining potentials we prove the following criterion where (a −n (t)Ψ −n,t (x) + a n (t)Ψ n,t (x)) dt, . Note that if the conditions requested for H(a, b) in Theorem 3 do not hold then either a = 0 or b = 0, that is, the conditions requested for H(a, b) in Theorem 4 hold. It means that all cases of the potential (4) are investigated in Theorem 3 and Theorem 4. In Theorem 2 some subcase of Theorem 3 is studied.
2. Preliminary facts. In this section we present some results of [12,13,2] which are used in this paper.
Note that, the formula f (n, t) = O(h(n)) is said to be uniform with respect to t in a set I if there exist positive constants M and N, independent of t, such that | f (n, t)) |< M | h(n) | for all t ∈ I and | n |≥ N. We use Remark 2.1 and lot of formulas of [13] that are listed in Remark 1 and as formulas (20)-(36). Remark 1. In Remark 2.1 of [13] we proved that here exists a positive integer N (0) such that the disk U (n, t, ρ) =: {λ ∈ C : λ − (2πn + t) 2 ≤ 15πnρ} for t ∈ [0, ρ], where 15πρ < 1, and n > N (0) contains two eigenvalues (counting with multiplicities) denoted by λ n,1 (t) and λ n,2 (t) and these eigenvalues can be chosen as a continuous function of t on the interval [0, ρ]. Similarly, there exists a positive integer N (π) such that the disk U (n, t, ρ) for t ∈ [π − ρ, π] and n > N (π) contains two eigenvalues (counting with multiplicities) denoted again by λ n,1 (t) and λ n,2 (t) that are continuous function of t on the interval [π − ρ, π].
Theorem 6. A number λ ∈ U (n, t, ρ) is an eigenvalue of L t (q) for t ∈ [0, ρ] and n > N, where U (n, t, ρ) and N are defined in Remark 1, if and only if Proof. If u n,j (t) = 0, then by (36) we have v n,j (t) = 0. Therefore, (22) and (26) imply that that is, the right-hand side and the left-hand side of (40) vanish when λ is replaced by λ n,j (t). Hence λ n,j (t) satisfies (40). In the same way we prove that if v n,j (t) = 0 then λ n,j (t) is a root of (40). It remains to consider the case u n,j (t)v n,j (t) = 0. In this case multiplying (22) and (26) side by side and canceling u n,j (t)v n,j (t) we get an equality obtained from (40) by replacing λ with λ n,j (t). Thus, in any case λ n,j (t) is a root of (40). Now we prove that the roots of (40) lying in U (n, t, ρ) are the eigenvalues of L t (q). Let F (λ, t) be the left-hand side minus the right-hand side of (40). Using (31) one can easily verify that the inequality where , holds for all λ from the boundary of U (n, t, ρ). Since the function (λ − (2πn + t) 2 )(λ − (2πn − t) 2 ) has two roots in the set U (n, t, ρ), by the Rouche's theorem from (41) we obtain that F (λ, t) has two roots in the same set. Thus L t (q) has two eigenvalue (counting with multiplicities) lying in U (n, t, ρ) (see Remark 1) that are the roots of (40). On the other hand, (40) has preciously two roots (counting with multiplicities) in U (n, t, ρ). Therefore λ ∈ U (n, t, ρ) is an eigenvalue of L t (q) if and only if (40) holds. If λ ∈ U (n, t, ρ) is a double eigenvalue of L t (q), then by Remark 1 L t (q) has no other eigenvalues in U (n, t, ρ) and hence (40) has no other roots. This implies that λ is a double root of (40). By the same argument one can prove that if λ is a double root of (40) then it is a double eigenvalue of L t (q) One can readily verify that equation (40) can be written in the form where and, for brevity, we denote C(λ, t), B(λ, t), A(λ, t) etc. by C, B, A etc. It is clear that λ is a root of (42) if and only if it satisfies at least one of the equations and where Remark 2. It is clear from the construction of D(λ, t) that this function is continuous with respect to (λ, t) for t ∈ [0, ρ] and λ ∈ U (n, t, ρ). Moreover, by Remark 1 the eigenvalues λ n,1 (t) and λ n,2 (t) continuously depend on t ∈ [0, ρ]. Therefore D(λ n,j (t), t) for n > N and j = 1, 2 is a continuous functions of t ∈ [0, ρ]. By (43), (34), (23), (27) and (30) we have as n → ∞. Therefore by (18) and Theorem 2 of [12] the eigenvalues λ n,1 (ρ) and λ n,2 (ρ) are simple, λ n,1 (ρ), satisfies (44) and λ n,2 (ρ) satisfies (45). If λ n,1 (t) and λ n,2 (t) are simple for t ∈ [t 0 , ρ], where 0 ≤ t 0 ≤ ρ, then these functions are analytic function on [t 0 , ρ] and λ n, Theorem 7. Suppose that D(λ n,j (t), t)) continuously depends on t at [t 0 , ρ] and for n > N and j = 1, 2, where ρ and N are defined in Remark 1 and √ D is defined in (46) and 0 ≤ t 0 ≤ ρ. Then for t ∈ [t 0 , ρ] the eigenvalues λ n,1 (t) and λ n,2 (t) defined in Remark 1 are simple, λ n,1 (t) satisfies (44) and λ n,2 (t), satisfies (45). That is for t ∈ [t 0 , ρ], n > N and j = 1, 2.
More detail estimations of B and B are given in the following lemma, where we use the following notation. We say that a n is of order of b n and write a n ∼ b n if a n = O(b n ) and b n = O(a n ) as n → ∞.
From Lemma 1 it easily follows the following statement.
for all t ∈ [0, n −3 ]. Henceforward, for brevity of notation, 1 + O(n −2 ) is denoted by [1]. Now we are ready to prove the main result of this section by using Theorems 6 and 7.
Theorem 9. Let S be the set of integer n > N such that − π + 3cn −2 ≤ arg(β n α n ) ≤ π − 3cn −2 (95) and {t n : n > N } be a sequence defined as follows: t n = 0 if n ∈ S and if n / ∈ S, where c is defined in (94). Then the eigenvalues λ n,1 (t) and λ n,2 (t) defined in Remark 1 are simple and satisfy (48) for t ∈ [t n , ρ].
Proof. By (100), there exists c 2 ∈ (0, 1) such that −π + c 2 < arg((ab) 2n ) < π − c 2 for all n ∈ N. Hence by the definition of β n and α n (see Lemma 1) (101) and hence (102) holds. Moreover, (101) implies that (95) holds. Therefore the proof follows from Theorem 8 Remark 3. Let A, B, A , B and C be the functions obtained from A, B, A , B and C by replacing a k , a k , b k , b k with a k , a k , b k , b k , where a k , a k , b k , b k differ from a k , a k , b k , b k respectively, in the following sense. The sums in the expressions for a k , a k , b k , b k are taken under condition n 1 + n 2 + ... + n s = 0, ±(2n + 1) instead of the condition n 1 + n 2 + ... + n s = 0, ±2n for s = 1, 2, ..., k.
As we noted in Section 2 (see Theorem 2 of [12] and Remark 1 ) the large eigenvalues of H t for t ∈ [ρ, π − ρ] consist of the simple eigenvalues λ n (t) for |n| > N satisfying the, uniform with respect to t in [ρ, π −ρ], asymptotic formula (17). Thus by Theorem 8 and by the just noted similar investigation, the eigenvalues λ n,j (t) for n > N, j = 1, 2 and t ∈ [t n , ρ] ∪ [π − ρ, t n ] and the eigenvalues λ n (t) for t ∈ [ρ, π − ρ] and |n| > N are simple.. These eigenvalues satisfy (48), (17) and (103) for t ∈ [t n , ρ], t ∈ [ρ, π − ρ] and t ∈ [π − ρ, t n ] respectively. Finally, note that (100) and (104) hold if and only if (8) holds. H(a, b). In this section we find necessary and sufficient condition on a and b for the asymptotic spectrality of the operator H(a, b), that is, we prove Theorem 1 formulated in the introduction. To prove this main result of this section we first prove the following two statements which easily follows from the results of Section 3.

On the spectrality of
Theorem 10. If (8) holds, then there exists N such that for |n| > N the component Γ n of the spectrum S(H) of the operator H is a separated simple analytic arc with the end points λ n (0) and λ n (π). These components do not contain spectral singularities. In other words, the number of the spectral singularities of H is finite.
Proof. As we noted in the end of Remark 3 if (8) holds, then (100) and (104) hold too. Therefore by Corollary 2, Theorem 2 of [13], and Remark 3 the eigenvalues λ n (t) for |n| > N and t ∈ [0, π] are simple. Therefore for |n| > N the component Γ n of the spectrum of the operator H is a separated simple analytic arc with the end points λ n (0) and λ n (π). It is well-known that the spectral singularities of H are contained in the set of multiple eigenvalues of H t (see Proposition 2 of [17]). Hence, Γ n for |n| > N does not contain the spectral singularities. On the other hand, the multiple eigenvalues are the zeros of the entire function dF (λ) dλ , where F (λ) is defined in (3). Since the entire function has a finite number of roots on the bounded sets the number of the spectral singularities of H(a, b) is finite It was noted in [2] that (see page 539 of [2]) if |a| = |b|, then the results of [6] and [2] show that H(a, b) is not a spectral operator. Since our aim is to prove the necessary and sufficient condition for asymptotic spectrality and the fact that  H(a, b) is not a spectral operator does not imply that it is not asymptotic spectral operator, here we prove the following fact which easily follows from the formulas of Section 3. then the operator H(a, b) has the spectral singularity at infinity and hence is not an asymptotically spectral operator.
Using formula (48) for j = 2 in (50) and (51) and taking into account the notations and arguments of Remark 4 we obtain 4πnt [1] = C(λ n (t)) + 4πnt, s(t) is −1 or 1 if λ n (t) satisfies (44) or (45) respectively. Since the boundary condition (2) is self-adjoint we have (H t (q)) * = H t (q). Therefore, all formulas and theorems obtained for H t are true for H * t if we replace a and b by b and a respectively. For instance, (35) and (36) hold for the operator H * t and hence we have Similarly the formulas (109) and (110) for the operator H * t have the form For | n |> N it follows from (35), (36), (112) and (113) that By (6) and Definition 2 to study the asymptotic spectrality we need to consider the expression (Ψ n,t , Ψ * n,t ). First let us note the following simple statement. Proposition 2. The equality (Ψ n,t , Ψ * n,t ) = 1 + O(n −1 ) holds uniformly for t ∈ [n −3 , ρ].
Therefore in the following lemma we investigate F + (λ n (t)) and u n (t)u * n (t).

Theorem 11. (a)
The operator H has no the spectral singularity at infinity and is an asymptotically spectral operator if and only if | a |=| b | and (8) holds.
If α is a rational number, that is, α = m q where m and q are irreducible integers and α is defined in (8), then the operator H has no the spectral singularity at infinity and is an asymptotically spectral operator if and only if m is an even integer. If α is an irrational number, then H has the spectral singularity at infinity and is not an asymptotically spectral operator if and only if there exists a sequence of pairs {(q k , p k )} ⊂ N 2 such that where 2p k − 1 and q k are irreducible integers.
By Theorem 8, for the sequence {t n k } defined by (96) and now, for simplicity, redenoted by {t k } the eigenvalues λ n k ,j (t k ) are simple and the following relations . It with (124) implies that F + (λ n k (t k )) = o(1) as k → ∞. Thus |d n k (t k )| → 0 as k → ∞, due to (115) and (118). In the same way we prove it when {q k } contains infinite number of odd number. It contradicts to the assumption that H(a, b) is an asymptotically spectral operator, due to Definition 2 Now using Theorem 1, that is, Theorem 10 (a) we prove Corollary 1 (see introduction).
The proof of Corollary 1. Since any self-adjoint operator is spectral, we need to prove that if H(a, b) is a spectral operator and ab is real, then (4) is a real potential. By Definitions 1 and 2 the spectral operator is also asymptotically spectral operator. Thus ab is real and by Theorem 1, (8) holds. If ab < 0 then α − 1 = 0,where α = π −1 arg(ab), which contradicts (8). Hence we have ab > 0. On the other hand, Proposition 1 implies that | a |=| b | . From the last two relations we obtain b = a . It means that (4) is a real potential and H(a, b) is a self-adjoint operator. H(a, b). Now we consider the forms of the spectral expansion of H(a, b). For this as is noted in the introduction we need to investigate in detail the ESS and ESS at infinity for H(a, b). Besides, we use the following results of the papers [14,17,18] formulated as summary. Summary 1. (a) The spectral expansion has the elegant form (9) if and only if L(q) has no ESS and ESS at infinity (see page 7 of [18]).

On the spectral expansion of
(b) If L(q) has no ESS at infinity, then the number of ESS is at most finite and the spectral expansion has the asymptotically elegant form (13) (see Theorem 3.13 of [18]).
(c) ESS of L(q) is a multiple 2-periodic eigenvalue. Note that the eigenvalues of L 0 (q) and L π (q) is called as 2-periodic eigenvalues. If the geometric multiplicity of the multiple 2-periodic eigenvalue is 1, then it is ESS (see Proposition 4 of [17]).
(e) If Λ = λ n (t 0 ) is a multiple eigenvalue, then the sum of the expressions a k (t)Ψ k,t (x) for k ∈ {s ∈ Z : λ s (t 0 ) = Λ} is integrable in some neighborhood of t 0 . If Λ is an ESS then at least two of these expressions are nonintegrable (see Remark 2 of [17]).
As we noted at the end of introduction, in this section we consider the spectral expansion of H(a, b) for all potential of the form (4) by dividing it into two complementary cases: Case 1: ab = 0 and Case 2: either a = 0 or b = 0. First we consider Case 1 and prove that in this case the operator H(a, b) has no ESS at infinity. Therefore by Summary 1(b) the number of ESS is at most finite and the spectral expansion has the asymptotically elegant form (13). For this, due to Definition 5, we need to study the existence and the behavior of (−π,π] |d n (t)| −1 dt for large n. Note that the estimations that was done in Section 4 for |d n (t)| −1 are not enough for the estimations of (125). Here we need more sharp and complicated estimations due to the followings. In Section 4 some estimations for (Ψ n,t , Ψ * n,t ) were done under assumption | a |=| b | (first condition) and the estimations for t ∈ I 2 , where a multiple eigenvalues may appear, were done under condition (8) (second condition), while in this section the estimations are done for all cases of the potential (4). Moreover in Section 4 we considered only boundlessness of |d n (t)| −1 , while here we consider its integrability and investigate the limit of (125) as n → ∞.
In this section to estimate (Ψ n,t , Ψ * n,t ) we also use (118). However, if the first condition does not hold, say if | a |<| b |, then the multiplicand u n (t)u * n (t) and hence the left-hand side of (118) is O (| a | n | b | −n ) . Therefore (115) is ineffective for the estimation of (Ψ n,t , Ψ * n,t ). For this first of all we consider the Bloch functions in detail which was done in Theorem 11. Moreover, in Section 4 we have used essentially the second condition (8) to estimate (Ψ n,t , Ψ * n,t ) for t ∈ I 2 . To do this estimation without condition (8) we develop a new approach at the end in this section. Besides we use the first statements of Lemma 2(a) and Lemma 2(b) which hold without assumption | a |=| b | . Finally, note that in the proof of Lemma 2(b) we proved that if (102) holds then (119) holds uniformly for t ∈ I 2 .
By Summary 1(e) if k = 0, then the sum of two expressions a k (t)Ψ k,t (x) and a −k (t)Ψ −k,t (x) corresponding to the ESS λ k (0) is integrable on [−h, h], while both of them is nonintegrable. Besides a 0 (t)Ψ 0,t is integrable since λ 0 (0) is a simple eigenvalue and hence is not an ESS. Therefore we have (a n (t)Ψ n,t +a n (t)Ψ n,t ) dt.
Using it in (146) we get (15). In the same way from the last equality for f π we obtain (16).