On the Differential Operators with Periodic Matrix Coefficients

Abstract

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.

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Let 𝐿(𝑃₂,𝑃₃,…,𝑃_𝑛) be the differential operator generated in the space 𝐿^𝑚₂(−∞,∞) of vector-valued functions by the differential expression (−𝑖)^𝑛𝑦^(𝑛)(𝑥)+(−𝑖)^𝑛−2𝑃₂(𝑥)𝑦^(𝑛−2)(𝑥)+^𝑛_𝑣=3𝑃_𝑣(𝑥)𝑦^{(𝑛−𝑣)}(𝑥),(1) where 𝑛 is an integer greater than 1 and 𝑃_𝑘(𝑥), for 𝑘=2,3,…,𝑛, is the 𝑚×𝑚 matrix with the complex-valued summable entries 𝑝_{𝑘,𝑖,𝑗}(𝑥) satisfying 𝑝_{𝑘,𝑖,𝑗}(𝑥+1)=𝑝_{𝑘,𝑖,𝑗}(𝑥) for all 𝑖=1,2,…,𝑚 and 𝑗=1,2,…,𝑚. It is well known that (see [1–4]) the spectrum of the operator 𝐿(𝑃₂,𝑃₃,…,𝑃_𝑛) is the union of the spectra of the operators 𝐿_𝑡(𝑃₂,𝑃₃,…,𝑃_𝑛) for 𝑡∈[0,2𝜋) generated in 𝐿^𝑚₂(0,1) by expression (1) and the quasiperiodic conditions 𝑈_𝜈(𝑦)≡𝑦^(𝜈)(1)−𝑒^𝑖𝑡𝑦^(𝜈)(0)=0,𝜈=0,1,…,(𝑛−1).(2) Note that 𝐿^𝑚₂(𝑎,𝑏) is the set of vector-valued functions 𝑓=(𝑓₁,𝑓₂,…,𝑓_𝑚) with 𝑓_𝑘∈𝐿₂(𝑎,𝑏) for 𝑘=1,2,…,𝑚. The norm ‖⋅‖ and inner product (⋅,⋅) in 𝐿^𝑚₂(𝑎,𝑏) are defined by ‖𝑓‖²=^𝑏_𝑎|𝑓(𝑥)|²𝑑𝑥,(𝑓,𝑔)=^𝑏_𝑎⟨𝑓(𝑥),𝑔(𝑥)⟩𝑑𝑥,(3) where |⋅| and ⟨⋅,⋅⟩ are the norm and inner product in ℂ^𝑚.

The first works concerned with the differential operator 𝐿_𝑡(𝑃₂,𝑃₃,…,𝑃_𝑛) were by Birkhoff [5], Tamarkin [6] in the beginning of 20th century. There exist enormously many papers concerning with the operators 𝐿_𝑡(𝑃₂,𝑃₃,…,𝑃_𝑛) and 𝐿(𝑃₂,𝑃₃,…,𝑃_𝑛). 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 (−𝑖)^𝑛𝑌^(𝑛)+(−𝑖)^𝑛−2𝑃₂𝑌^(𝑛−2)+^𝑛_𝑣=3𝑃_𝑣𝑌^{(𝑛−𝑣)}=𝜆𝑌,(4) 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 𝐿_𝑡(𝑃) 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 𝐿_𝑡(𝑃₂,𝑃₃,…,𝑃_𝑛) 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 𝐿(𝑃₂,𝑃₃,…,𝑃_𝑛).

Let us introduce some preliminary results and describe the results of this paper. Clearly, 𝜑_𝑘,1,𝑡=(𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥},0,…,0),𝜑_𝑘,2,𝑡=(0,𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥},0,…,0),…,𝜑_{𝑘,𝑚,𝑡}=(0,0,…,0,𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥})(5) are the eigenfunctions of the operator 𝐿_𝑡(0) corresponding to the eigenvalue (2𝜋𝑘+𝑡)^𝑛, where 𝑘∈ℤ, and the operator 𝐿_𝑡(𝑃₂,…,𝑃_𝑛) is denoted by 𝐿_𝑡(0) when 𝑃₂(𝑥)=0,…,𝑃_𝑛(𝑥)=0. Furthermore, for brevity of notation, the operators 𝐿_𝑡(𝑃₂,…,𝑃_𝑛) and 𝐿(𝑃₂,…,𝑃_𝑛) are denoted by 𝐿_𝑡(𝑃) and 𝐿(𝑃), respectively. It easily follows from the classical investigations [7, Chapter 3, Theorem  2] that the large eigenvalues of the operator 𝐿_𝑡(𝑃) consist of 𝑚 sequences {𝜆_𝑘,1(𝑡)∶|𝑘|≥𝑁},{𝜆_𝑘,2(𝑡)∶|𝑘|≥𝑁},…,{𝜆_𝑘,𝑚(𝑡)∶|𝑘|≥𝑁},(6) satisfying the following, uniform with respect to 𝑡 in [0,2𝜋), asymptotic formulas: 𝜆_𝑘,𝑗(𝑡)=(2𝜋𝑘+𝑡)^𝑛+𝑂(𝑘^{𝑛−1−1/2𝑚})(7) for 𝑗=1,2,…,𝑚, where 𝑁 is a sufficiently large positive number, that is, 𝑁≫1. We say that the formula 𝑓(𝑘,𝑡)=𝑂(ℎ(𝑘)) is uniform with respect to 𝑡 in a set 𝑆 if there exists a positive constant 𝑐₁, independent of 𝑡, such that |𝑓(𝑘,𝑡))|<𝑐₁|ℎ(𝑘)| for all 𝑡∈𝑆 and 𝑘∈ℤ. Thus formula (7) means that there exist positive numbers 𝑁 and 𝑐₁, independent of 𝑡, such that |𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑘+𝑡)^𝑛|<𝑐₁|𝑘|^{𝑛−1−1/2𝑚},∀|𝑘|≥𝑁,∀𝑡∈[0,2𝜋).(8) In this paper, by the suggested method, we obtain the uniform asymptotic formulas of high accuracy for the eigenvalues 𝜆_𝑘,𝑗(𝑡) and for the corresponding normalized eigenfunctions Ψ_{𝑘,𝑗,𝑡}(𝑥) of 𝐿_𝑡(𝑃) when the entries 𝑝_2,𝑖,𝑗(𝑥),𝑝_3,𝑖,𝑗(𝑥),…,𝑝_{𝑛,𝑖,𝑗}(𝑥) of 𝑃₂(𝑥),𝑃₃(𝑥),…,𝑃_𝑛(𝑥) belong to 𝐿₁[0,1], that is, when there is not any condition about smoothness of the coefficients. Then using these formulas, we find the conditions on the coefficient 𝑃₂(𝑥) for which the number of the gaps in the spectrum of the self-adjoint differential operator 𝐿(𝑃) is finite.

Now let us describe the scheme of the paper. Inequality (8) shows that the eigenvalue 𝜆_𝑘,𝑗(𝑡) of 𝐿_𝑡(𝑃) is close to the eigenvalue (2𝑘𝜋+𝑡)^𝑛 of 𝐿_𝑡(0). To analyze the distance of the eigenvalue 𝜆_𝑘,𝑗(𝑡) of 𝐿_𝑡(𝑃) from the other eigenvalues (2𝑝𝜋+𝑡)^𝑛 of 𝐿_𝑡(0), which is important in perturbation theory, we take into account the following situations. If the order 𝑛 of the differential expression (1) is odd number, 𝑛=2𝑟−1, and |𝑘|≫1, then the eigenvalue (2𝜋𝑘+𝑡)^𝑛 of 𝐿_𝑡(0) lies far from the other eigenvalues (2𝑝𝜋+𝑡)^𝑛 of 𝐿_𝑡(0) for all values of 𝑡∈[0,2𝜋). We have the same situation if 𝑛=2𝑟 and 𝑡 does not lie in the small neighborhoods of 0 and 𝜋. However, if 𝑛 is even number and 𝑡 lies in the neighborhoods of 0 and 𝜋, then the eigenvalue (2𝜋𝑘+𝑡)^𝑛 is close to the eigenvalues (2𝜋(−𝑘)+𝑡)^𝑛 and (2𝜋(−𝑘−1)+𝑡)^𝑛, respectively. For this reason instead of [0,2𝜋) we consider 𝑡∈[−𝜋/2,3𝜋/2) and use the following notation.

Notation 1.
Case 1. (a) 𝑛=2𝑟−1 and 𝑡∈[−𝜋/2,3𝜋/2), (b) 𝑛=2𝑟 and 𝑡∈𝑇(𝑘), where 𝜋𝑇(𝑘)=[−2,3𝜋21)⧵((−,1ln|𝑘|1ln|𝑘|)∪(𝜋−1ln|𝑘|,𝜋+ln|𝑘|)).(9)Case 2. 𝑛=2𝑟 and 𝑡∈(−(ln|𝑘|)⁻¹,(ln|𝑘|)⁻¹).Case 3. 𝑛=2𝑟 and 𝑡∈(𝜋−(ln|𝑘|)⁻¹,𝜋+(ln|𝑘|)⁻¹).
Denote by 𝐴(𝑘,𝑛,𝑡) the sets {𝑘}, {𝑘,−𝑘}, {𝑘,−𝑘−1} for Cases 1, 2, and 3, respectively.

By (8) there exists a positive constant 𝑐₂, independent of 𝑡, such that the inequalities |(2𝑘𝜋+𝑡)^𝑛−(2𝜋𝑝+𝑡)^𝑛|>𝑐₂(ln|𝑘|)⁻¹(||𝑘|−|𝑝||+1)(|𝑘|+|𝑝|)^𝑛−1,|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|>𝑐₂(ln|𝑘|)⁻¹(||𝑘|−|𝑝||+1)(|𝑘|+|𝑝|)^𝑛−1,(10) where |𝑘|>𝑁, hold in Cases 1, 2, and 3 for 𝑝≠𝑘, for 𝑝≠𝑘,−𝑘, and for 𝑝≠𝑘,−(𝑘+1), respectively. To avoid the listing of these cases, using Notation 1, we see that the inequalities in (10) hold for 𝑝∉𝐴(𝑘,𝑛,𝑡). To obtain the asymptotic formulas we essentially use the following lemma that easily follows from (8) and (10).

Lemma 1. The equalities _{𝑝∶|𝑝|>𝑑}𝑝^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|1=𝑂(𝑑),(11)_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}𝑝^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|=𝑂(ln|𝑘|𝑘),(12)_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}𝑘^2𝑛−4|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|²=𝑂((ln|𝑘|)²𝑘²),(13)_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}𝑝^2𝑛−4|𝜆_𝑝,𝑗(𝑡)−(2𝜋𝑘+𝑡)^𝑛|²=𝑂((ln|𝑘|)²𝑘²)(14) hold uniformly with respect to 𝑡 in [−𝜋/2,3𝜋/2), where 𝑑≥2|𝑘|, |𝑘|≥𝑁≫1.

Proof. The proof of (11). It follows from (8) that if |𝑝|>𝑑≥2|𝑘|, then |𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|>|𝑝|^𝑛𝜋,∀𝑡∈[−2,3𝜋2).(15) Therefore the left-hand side of (11) is less than _{𝑝∶|𝑝|>𝑑}1𝑝²𝜋,∀𝑡∈[−2,3𝜋2)(16) which is 𝑂(1/𝑑). Thus (11) holds uniformly with respect to 𝑡∈[−𝜋/2,3𝜋/2).
The proof of (12). The summation in the left-hand side of (12) is taking over all 𝑝∈ℤ⧵𝐴(𝑘,𝑛,𝑡). Since ℤ⧵𝐴(𝑘,𝑛,𝑡)=𝑆(1)∪𝑆(2)∪𝑆(3),(17) where 𝑆(1)={𝑝∶|𝑝|>2|𝑘|}, 𝑆(2)={𝑝∶|𝑝|≤2|𝑘|,𝑝∉𝐴(𝑘)}, 𝑆(3)={𝑝∶𝑝∈𝐴(𝑘)⧵𝐴(𝑘,𝑛,𝑡)}, and 𝐴(𝑘)={±𝑘,±(𝑘+1),±(𝑘+2)}, the left-hand side of (12) can be written as 𝑆(9,1)+𝑆(9,2)+𝑆(9,3), where 𝑆(9,1)=_{𝑝∶𝑝∈𝑆(1)}𝑝^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|,𝑆(9,2)=_{𝑝∶𝑝∈𝑆(2)}𝑝^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|,𝑆(9,3)=_{𝑝∶𝑝∈𝑆(3)}𝑝^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|.(18) Taking 𝑑=2|𝑘|, from (11) we obtain that 𝑆(9,1)=𝑂(𝑘⁻¹). If 𝑝∉𝐴(𝑘), then using (8) one can readily see that |𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|>(||𝑘|−|𝑝||)(|𝑘|+|𝑝|)^𝑛−1,|𝑝^𝑛−2||𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|<1(||𝑘|−|𝑝||)|𝑘|(19) for all 𝑡∈[−𝜋/2,3𝜋/2). Let 𝑠=||𝑘|−|𝑝||. Clearly, if |𝑝|≤2|𝑘|, 𝑝∉𝐴(𝑘), then 2<𝑠≤|𝑘| and the number 𝑠 attains the same value at most 4 times. Therefore 1|𝑆(9,2)|≤4(|𝑘|^𝑘_𝑠=31𝑠)=𝑂(ln|𝑘|𝑘).(20) Since the set 𝑆(3) has at most 6 elements, and for 𝑝∉𝐴(𝑘,𝑛,𝑡) the inequalities in (10) hold, we have 𝑆(9,3)=𝑂(ln|𝑘|𝑘).(21) Now, estimations for 𝑆(9,1), 𝑆(9,2), 𝑆(9,3) 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 𝑆(10,1)+𝑆(10,2)+𝑆(10,3) and 𝑆(11,1)+𝑆(11,2)+𝑆(11,3), respectively, where the summations in 𝑆(10,𝑖) and in 𝑆(11,𝑖) are taking over 𝑝∈𝑆(𝑖)(𝑖=1,2,3). Then, repeating the arguments by which we estimated 𝑆(9,1), 𝑆(9,2), and 𝑆(9,3), we get 𝑆(10,1)=𝑂(𝑘⁻³),𝑆(11,1)=𝑂(𝑘⁻³),𝑆(10,2)=𝑂(𝑘⁻²),𝑆(11,1)=𝑂(𝑘⁻²),𝑆(10,3)=𝑂((ln|𝑘|)²𝑘²),𝑆(11,3)=𝑂((ln|𝑘|)²𝑘²).(22) These equalities imply the proof of (13) and (14).

To obtain the asymptotic formulas we use (11)–(14) and consider the operator 𝐿_𝑡(𝑃) as perturbation of 𝐿_𝑡(𝐶) by 𝐿_𝑡(𝑃)−𝐿_𝑡(𝐶), where 𝐶=∫¹₀𝑃₂(𝑥)𝑑𝑥, 𝐿_𝑡(𝐶) is the operator generated by (2) and by the expression (−𝑖)^𝑛𝑦^(𝑛)(𝑥)+(−𝑖)^𝑛−2𝐶𝑦^(𝑛−2)(𝑥).(23) Therefore, first of all, we analyze the eigenvalues and eigenfunction of the operator 𝐿_𝑡(𝐶). We assume that 𝐶 is the Hermitian matrix. Then the expression in (23) is the self-adjoint expression. Since the boundary conditions (2) are self-adjoint, the operator 𝐿_𝑡(𝐶) is also self-adjoint. The eigenvalues of 𝐶, counted with multiplicity, and the corresponding orthonormal eigenvectors are denoted by 𝜇₁≤𝜇₂≤⋯≤𝜇_𝑚 and 𝑣₁,𝑣₂,…,𝑣_𝑚. Thus 𝐶𝑣_𝑗=𝜇_𝑗𝑣_𝑗,⟨𝑣_𝑖,𝑣_𝑗⟩=𝛿_𝑖,𝑗.(24) One can easily verify that the eigenvalues and eigenfunctions of 𝐿_𝑡(𝐶) are 𝜇_𝑘,𝑗(𝑡)=(2𝜋𝑘+𝑡)^𝑛+𝜇_𝑗(2𝜋𝑘+𝑡)^𝑛−2, Φ_{𝑘,𝑗,𝑡}(𝑥)=𝑣_𝑗𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥}, that is, (𝐿(𝐶)−𝜇_𝑘,𝑗(𝑡))Φ_{𝑘,𝑗,𝑡}(𝑥)=0.(25)

To prove the asymptotic formulas for the eigenvalues 𝜆_𝑘,𝑗(𝑡) and for the corresponding normalized eigenfunctions Ψ_{𝑘,𝑗,𝑡} of 𝐿_𝑡(𝑃) we use the formula (𝜆_𝑘,𝑗(𝑡)−𝜇_𝑝,𝑠(𝑡))(Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})=(−𝑖)^𝑛−2((𝑃₂−𝐶)Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})+^𝑛_𝜈=3(𝑃_𝜈Ψ^𝑛−𝜈_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})(26) which can be obtained from 𝐿(𝑃)Ψ_{𝑘,𝑗,𝑡}(𝑥)=𝜆_𝑘,𝑗(𝑡)Ψ_{𝑘,𝑗,𝑡}(𝑥)(27) by multiplying both sides by Φ_{𝑝,𝑠,𝑡}(𝑥) and using (25). Then we estimate the right-hand side of (26) (see Lemma 3) by using Lemma 2. At last, estimating (Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡}) (see Lemma 4) and using these estimations in (26), we find the asymptotic formulas for the eigenvalues and eigenfunctions of 𝐿_𝑡(𝑃) (see Theorems 5 and 6). Then using these formulas, we find the conditions on the eigenvalues of the matrix 𝐶 for which the number of the gaps in the spectrum of the operator 𝐿(𝑃) is finite (see Theorem 7). Some of these results for differentiable 𝑃₂(𝑥) are obtained in [3, 11] by using the classical asymptotic expansions for the solutions of (4). The case 𝑛=2 is investigated in [12]. In this case, an interesting spectral estimates were done in the paper [13], 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 𝑛>2.

To estimate the right-hand side of (26) we use (11), (12), the following lemma, and the formula (𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛)(Ψ_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡})=(−𝑖)^𝑛−2(𝑃₂Ψ^𝑛−2_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡})+^𝑛_𝜈=3(𝑃_𝜈Ψ^𝑛−𝜈_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡})(28) which can be obtained from (27) by multiplying both sides by 𝜑_{𝑝,𝑠,𝑡} and using 𝐿_𝑡(0)𝜑_{𝑝,𝑠,𝑡}=(2𝜋𝑝+𝑡)^𝑛𝜑_{𝑝,𝑠,𝑡}.

Lemma 2. Let Ψ_{𝑘,𝑗,𝑡}(𝑥) be normalized eigenfunction of 𝐿_𝑡(𝑃). Then sup_𝑥∈0,1]|Ψ^(𝜈)_{𝑘,𝑗,𝑡}(𝑥)|=𝑂(𝑘^𝜈)(29) for 𝜈=0,1,…,𝑛−2. Equality (29) is uniform with respect to 𝑡 in [−𝜋/2,3𝜋/2).

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 𝑡. The eigenfunction Ψ_{𝑘,𝑗,𝑡} corresponding to the eigenvalue 𝜆_𝑘,𝑗(𝑡) has the form Ψ_{𝑘,𝑗,𝑡}(𝑥)=𝑌₁(𝑥,𝜌_𝑘,𝑗)𝑎₁+𝑌₂(𝑥,𝜌_𝑘,𝑗)𝑎₂+⋯+𝑌_𝑛(𝑥,𝜌_𝑘,𝑗)𝑎_𝑛,(30) where 𝑎_𝑘∈ℂ^𝑚, 𝜌_𝑘,𝑗(𝑡)=𝑖(𝜆_𝑘,𝑗(𝑡))^1/𝑛, 𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)) for 𝑠=1,2,…,𝑛 are linearly independent 𝑚×𝑚 matrix solutions of (4) for 𝜆=𝜆_𝑘,𝑗(𝑡) satisfying 𝑑^𝜈𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡))𝑑𝑥^𝜈=(𝜌_𝑘,𝑗(𝑡))^𝜈𝑒^{𝜌𝑘,𝑗(𝑡)𝜔𝑠𝑥}[𝜔^𝜈_𝑠1𝐼+𝑂(𝑘)](31) for 𝜈=0,1,…,(𝑛−1). Here 𝐼 is unit matrix, 𝜔₁,𝜔₂,…,𝜔_𝑛 are the 𝑛th root of 1, and 𝑂(1/𝑘) is an 𝑚×𝑚 matrix satisfying the following conditions: 1𝑂(𝑘)=𝐴(𝑥,𝑡,𝑘)𝑘𝑐,|𝐴(𝑥,𝑡,𝑘)|<³𝜋|𝑘|,∀𝑥∈[0,1],∀𝑡∈[−2,3𝜋2),(32) where 𝑘>𝑁 and 𝑐₃ is a positive constant, independent of 𝑡. To consider the uniformity, with respect to 𝑡, of (29) we use (32).
The proof of (29) in the case 𝑛=2𝑟−1, 𝑟>1. Denote by (𝜆_𝑘,𝑗(𝑡))^1/𝑛 the root of 𝜆_𝑘,𝑗(𝑡) lying in 𝑂(𝑘^−1/2𝑚) neighborhood of (2𝑘𝜋+𝑡) and put 𝜌_𝑘,𝑗(𝑡)=𝑖(𝜆_𝑘,𝑗(𝑡))^1/𝑛. Then we have 𝜌_𝑘,𝑗(𝑡)=(2𝑘𝜋+𝑡)𝑖+𝑂(𝑘^−1/2𝑚).(33) Suppose 𝜔₁,𝜔₂,…,𝜔_𝑛 are ordered in such a way that 𝜔_𝑟=1,ℛ(𝜌_𝑘,𝑗(𝑡)𝜔_𝑠)<0,∀𝑠<𝑟,ℛ(𝜌_𝑘,𝑗(𝑡)𝜔_𝑠)>0,∀𝑠>𝑟,(34) where ℛ(𝑧) is the real part of 𝑧. Using (31), (34), (2), and (33), we get 𝑌_𝑠^(𝜈−1)(1,𝜌_𝑘,𝑗(𝑡))=(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑒^{𝜌𝑘,𝑗(𝑡)𝜔𝑠}[𝜔_𝑠^𝜈−1𝐼],𝑌_𝑠^(𝜈−1)(0,𝜌_𝑘,𝑗(𝑡))=(𝜌_𝑘,𝑗(𝑡))^𝜈−1[𝜔_𝑠^𝜈−1𝑈𝐼],(35)_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))=−(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑒^𝑖𝑡[𝜔_𝑠^𝜈−1𝑈𝐼],∀𝑠<𝑟,_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))=(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑒^{𝜌𝑘,𝑗(𝑡)𝜔𝑠}[𝜔_𝑠^𝜈−1𝑈𝐼],∀𝑠>𝑟,(36)_𝜈(𝑌_𝑟(𝑥,𝜌_𝑘,𝑗(𝑡)))=(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑂(𝑘^−1/2𝑚),(37) where [𝜔_𝑠^𝜈−1𝐼]=𝜔_𝑠^𝜈−1𝐼+𝑂(1/𝑘) and 𝑂(1/𝑘) satisfies the relation (32). Now using these relations and the notations of (30), we prove that 𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡))𝑎_𝑠=𝑂((|𝑎_𝑟|)𝑘^−1/2𝑚),∀𝑠≠𝑟.(38) Since Ψ_{𝑘,𝑗,𝑡}(𝑥) satisfies (2) and (30), we have the system of equations _𝑠≠𝑟𝑈_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))𝑎_𝑠=−𝑈_𝜈(𝑌_𝑟(𝑥,𝜌_𝑘,𝑗(𝑡)))𝑎_𝑟,𝜈=0,1,…,(𝑛−2)(39) with respect to 𝑎_𝑠,𝑞 for 𝑠≠𝑟 and 𝑞=1,2,…,𝑚, where 𝑎_𝑠,𝑞 are coordinates of the vector 𝑎_𝑠. Using (36) and (37) in (39) and then dividing both parts of (𝜈+1)th equation of (39), for 𝜈=0,1,…,(𝑛−2), by (𝜌_𝑘,𝑗(𝑡))^𝜈, we get the system of equations whose coefficient matrix 𝐴 is −𝑒^𝑖𝑡[𝐼]⋯−𝑒^𝑖𝑡[𝐼]𝑒^{𝜌𝑘,𝑗𝜔𝑟+1}[𝐼]⋯𝑒^{𝜌𝑘,𝑗𝜔𝑛}[𝐼]−𝑒^𝑖𝑡[𝜔₁𝐼]⋯−𝑒^𝑖𝑡[𝜔_𝑟−1𝐼]𝑒^{𝜌𝑘,𝑗𝜔𝑟+1}[𝜔_𝑟+1𝐼]⋯𝑒^{𝜌𝑘,𝑗𝜔𝑛}[𝜔_𝑛𝐼]⋯⋯⋯⋯⋯⋯−𝑒^𝑖𝑡[𝜔₁^𝑛−2𝐼]⋯−𝑒^𝑖𝑡[𝜔^𝑛−2_𝑟−1𝐼]𝑒^{𝜌𝑘,𝑗𝜔𝑟+1}[𝜔^𝑛−2_𝑟+1𝐼]⋯𝑒^{𝜌𝑘,𝑗𝜔𝑛}[𝜔_𝑛^𝑛−2𝐼],(40) and the right-hand side is 𝑂((|𝑎_𝑟|)𝑘^−1/2𝑚). To estimate det𝐴 let us denote by 𝐴(𝑚) the matrix obtained from 𝐴 by replacing [𝜔^𝑗_𝑠𝐼] with 𝜔^𝑗_𝑠𝐼 and by dividing the 𝑠th column (note that the entries of the 𝑠th column are the 𝑚×𝑚 matrices) for 𝑠<𝑟 and for 𝑠>𝑟 by −𝑒^𝑖𝑡 and by 𝑒^{𝜌𝑘,𝑗𝜔𝑠}, respectively. Clearly, 𝜔𝐴(1)=1⋯11⋯1₁⋯𝜔_𝑟−1𝜔_𝑟+1⋯𝜔_𝑛𝜔⋯⋯⋯⋯⋯⋯_1𝑛−2⋯𝜔^𝑛−2_𝑟−1𝜔^𝑛−2_𝑟+1⋯𝜔_𝑛^𝑛−2(41) and det𝐴(1)≠0. Besides, interchanging the rows and then interchanging the columns of 𝐴(𝑚), we obtain det𝐴(𝑚)=(𝐴(1))^𝑚. Using this and solving (39) by Cramer's rule, we get 𝑎_𝑠,𝑞=det𝐴_𝑠,𝑞det𝐴=𝑂((|𝑎_𝑟|)𝑒^{−𝜌𝑘,𝑗𝜔𝑠}𝑘^−1/2𝑚),∀𝑠>𝑟,(42) since 𝐴_𝑠,𝑞 is obtained from 𝐴 by replacing the ((𝑠−1)𝑚+𝑞)th column of 𝐴, which is the 𝑞th column of 𝑒^{𝜌𝑘,𝑗𝜔𝑠}𝑒[𝐼]_{𝜌𝑘,𝑗𝜔𝑠}[𝜔_𝑘⋮𝑒𝐼]_{𝜌𝑘,𝑗𝜔𝑠}[𝜔_𝑘^𝑛−2],with𝑂(|𝑎_𝑟|𝑘^−1/2𝑚)𝑂(|𝑎_𝑟|𝑘^−1/2𝑚)⋮𝑂(|𝑎_𝑠|𝑘^−1/2𝑚).(43) In the same way, we obtain 𝑎_𝑠,𝑞=𝑂((|𝑎_𝑟|)𝑘^−1/2𝑚),∀𝑠<𝑟.(44) Now (38) follows from (44), (42), and (34). Therefore, the normalization condition ‖Ψ_{𝑘,𝑗,𝑡}‖=1, and (38), (30), (31), (33), and (34) imply that Ψ_{𝑘,𝑗,𝑡}(𝑥)=(𝑌_𝑟(𝑥,𝜌_𝑘,𝑗(𝑡)))𝑎_𝑟+𝑂(𝑘^−1/2𝑚)=𝑒^{𝑖(2𝑘𝜋+𝑡)𝑥}𝑎_𝑟+𝑂(𝑘^−1/2𝑚),(45) where |𝑎_𝑟|²=1+𝑂(𝑘^−1/2𝑚), from which we get the proof of (29) for 𝜈=0. Differentiating both sides of (30) and using (42) and (44), we get the proof of (29) for arbitrary 𝜈 in the case 𝑛=2𝑟−1.
The proof of (29) in the case 𝑛=2𝑟. In this case the 𝑛th roots 𝜔₁,𝜔₂,…,𝜔_𝑛 of 1 are ordered in such a way that 𝜔_𝑟=1,𝜔_𝑟+1=−1,ℛ(𝜌_𝑘,𝑗𝜔_𝑠)<0,∀𝑠<𝑟;ℛ(𝜌_𝑘,𝑗𝜔_𝑠)>0,∀𝑠>𝑟+1.(46) Hence we have 𝑈_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))=−(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑒^𝑖𝑡[𝜔_𝑠^𝑣−1𝑈𝐼],∀𝑠<𝑟,_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))=(𝜌_𝑘,𝑗(𝑡))^𝜈−1𝑒^{𝜌𝑘,𝑗(𝑡)𝜔𝑠}[𝜔_𝑠^𝑣−1𝐼],∀𝑠>𝑟+1.(47) Now using these equalities, we prove that 𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡))𝑎_𝑠=𝑂((|𝑎_𝑟|+|𝑎_𝑟+1|)𝑘^−1/2𝑚),∀𝑠≠𝑟,𝑟+1.(48) Using (47) and arguing as in the case 𝑛=2𝑟−1, we get the system of equations _{𝑠≠𝑟,𝑟+1}𝑈_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))𝑎_𝑠=−_{𝑠=𝑟,𝑟+1}𝑈_𝜈(𝑌_𝑠(𝑥,𝜌_𝑘,𝑗(𝑡)))𝑎_𝑠(49) for 𝜈=0,1,2,…,(𝑛−3). Arguing as in the proof of (42)–(45) and using (46), we get 𝑎_𝑠,𝑞=𝑂((|𝑎_𝑟|+|𝑎_𝑟+1|)𝑒^{−𝜌𝑘,𝑗𝜔𝑠}𝑘^−1/2𝑚𝑎),∀𝑠>𝑟+1,_𝑠,𝑞=𝑂((|𝑎_𝑟|+|𝑎_𝑟+1|)𝑘^−1/2𝑚Ψ),∀𝑠<𝑟,_{𝑘,𝑗,𝑡}(𝑥)=𝑒^{𝑖(2𝑘𝜋+𝑡)𝑥}𝑎_𝑟+𝑒^{−𝑖(2𝑘𝜋+𝑡)𝑥}𝑎_𝑟+1+𝑂(𝑘^−1/2𝑚),(50) where |𝑎_𝑟|²+|𝑎_𝑟+1|²=1+𝑂(𝑘^−1/2𝑚), which implies the proof of (29) in the case 𝑛=2𝑟.

It follows from this lemma that the equalities (𝑃_𝜈Ψ^𝑛−𝜈_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡})=𝑂(𝑘^𝑛−𝜈),(𝑃_𝜈Ψ^𝑛−𝜈_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})=𝑂(𝑘^𝑛−𝜈)(51) for 𝜈=2,3,…,𝑛 and for 𝑗=1,2,…,𝑚 hold uniformly with respect to 𝑡 in [−𝜋/2,3𝜋/2). Now (51) together with (28) implies that |(Ψ_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡}𝑐)|≤⁴|𝑘|^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑝+𝑡)^𝑛|(52) for 𝑝∉𝐴(𝑘,𝑛,𝑡), |𝑘|≥𝑁, and 𝑠,𝑗=1,2,…,𝑚, where 𝑐₄ is a positive constant, independent of 𝑡. Using this we prove the following lemma.

Lemma 3. Let 𝑏_𝑠,𝑞(𝑥) be the entries of 𝑃₂(𝑥) and 𝑏_{𝑠,𝑞,𝑝}=∫¹₀𝑏_𝑠,𝑞(𝑥)𝑒^{−2𝜋𝑖𝑝𝑥}𝑑𝑥. Then (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝑃₂𝜑_{𝑝,𝑠,𝑡})=_{𝑞=1,2,…𝑚;𝑙∈ℤ}𝑏_{𝑠,𝑞,𝑝−𝑙}(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡}),(53)(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},(𝑃₂−𝐶)Φ_{𝑝,𝑠,𝑡})=𝑂(𝑘^𝑛−3ln|𝑘|)+𝑂(𝑘^𝑛−2𝑏_𝑘)(54) for 𝑝∈𝐴(𝑘,𝑛,𝑡) and 𝑠=1,2,…,𝑚, where 𝑏_𝑘=max{|𝑏_{𝑖,𝑗,𝑝}|∶𝑖,𝑗=1,2,…,𝑚;𝑝=2𝑘,−2𝑘,2𝑘+1,−2𝑘−1},(55) and 𝐶 is the Hermitian matrix defined in (23). Formula (54) is uniform with respect to 𝑡 in [−𝜋/2,3𝜋/2). Moreover, in Case 1 of Notation  1 the formula (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},(𝑃₂−𝐶)Φ_{𝑘,𝑠,𝑡})=𝑂(𝑘^𝑛−3ln|𝑘|)(56) holds. If 𝑛=2𝑟−1, then (56) is uniform with respect to 𝑡 in [−𝜋/2,3𝜋/2).

Proof. Note that if the entries 𝑏_𝑠,𝑞 of 𝑃₂ belong to 𝐿₂[0,1], then (53) is obvious, since {𝜑_{𝑙,𝑞,𝑡}∶𝑙∈ℤ,𝑞=1,2,…,𝑚} is an orthonormal basis in 𝐿^𝑚₂[0,1]. Now we prove (53) in case 𝑏_𝑠,𝑞∈𝐿₁[0,1]. Using (2), (52), and the integration by parts, we see that there exists a constant 𝑐₅, independent of 𝑡, such that |(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡})|=|(2𝜋𝑙+𝑡)^𝑛−2(Ψ_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡}𝑐)|≤⁵|𝑘|^𝑛−2|𝑙|^𝑛−2|𝜆_𝑘,𝑗(𝑡)−(2𝜋𝑙+𝑡)^𝑛|,(57) for 𝑙∉𝐴(𝑘,𝑛,𝑡), |𝑘|≥𝑁. This and (11) imply that there exists a constant 𝑐₆, independent of 𝑡, such that _{𝑙∶|𝑙|>𝑑}|(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡}𝑐)|<⁶|𝑘|^𝑛−2𝑑,(58) where 𝑑≥2|𝑘|, 𝑡∈[−𝜋/2,3𝜋/2). Hence the decomposition of Ψ^(𝑛−2)_{𝑘,𝑗,𝑡} by the basis {𝜑_{𝑙,𝑞,𝑡}∶𝑙∈ℤ,𝑞=1,2,…,𝑚} has the form
Ψ^(𝑛−2)_{𝑘,𝑗,𝑡}(𝑥)=_{|𝑙|≤𝑑;𝑞=1,2,…,𝑚}(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡})𝜑_{𝑙,𝑞,𝑡}(𝑥)+𝑔_𝑑(𝑥),wheresup_𝑥∈0,1]|𝑔_𝑑𝑐(𝑥)|<⁶|𝑘|^𝑛−2𝑑.(59) Using (59) in (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝑃₂𝜑_{𝑝,𝑠,𝑡}) and letting 𝑑 tend to ∞, we obtain (53).
Since Φ_{𝑝,𝑠,𝑡}(𝑥)≡𝑣_𝑠𝑒^{𝑖(2𝜋𝑝+𝑡)𝑥}, to prove (54), it is enough to show that (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},(𝑃₂−𝐶)𝜑_{𝑝,𝑠,𝑡})=𝑂(𝑘^𝑛−3ln|𝑘|)+𝑂(𝑘^𝑛−2𝑏_𝑘)(60) for 𝑠=1,2,…,𝑚 and 𝑝∈𝐴(𝑘,𝑛,𝑡). Using the obvious relation (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝐶𝜑_{𝑝,𝑠,𝑡})=_{𝑞=1,2,…,𝑚}𝑏_𝑠,𝑞,0(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑞,𝑡})(61) and (53), we see that (Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},(𝑃₂−𝐶)𝜑_{𝑝,𝑠,𝑡})=_{𝑙∶𝑙∈𝐴(𝑘,𝑛,𝑡)⧵𝑝;𝑞=1,2,…,𝑚}𝑏_{𝑠,𝑞,𝑝−𝑙}(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡})+_{𝑙∶𝑙∉𝐴(𝑘,𝑛,𝑡);𝑞=1,2,…,𝑚}𝑏_{𝑠,𝑞,𝑝−𝑙}(Ψ^(𝑛−2)_{𝑘,𝑗,𝑡},𝜑_{𝑙,𝑞,𝑡}).(62) Since |𝑏_{𝑗,𝑖,𝑠}|≤max_{𝑝,𝑞=1,2,…,𝑚}¹₀|𝑏_𝑝,𝑞(𝑥)|𝑑𝑥=𝑂(1)(63) for all 𝑗, 𝑖, 𝑠, using (57) and (12), we see that the second summation of the right-hand side of (62) is 𝑂((𝑘^𝑛−3ln|𝑘|). Besides, it follows from (29) and (55) that the first summation of the right-hand side of (62) is 𝑂(𝑘^𝑛−2𝑏_𝑘), since for 𝑝∈𝐴(𝑘,𝑛,𝑡) and 𝑙∈𝐴(𝑘,𝑛,𝑡)⧵𝑝, we have 𝑝−𝑙∈{2𝑘,−2𝑘,2𝑘+1,−2𝑘−1}. Hence (54) is proved. In Case 1 of Notation 1 the first summation of the right-hand side of (62) is absent, since in this case 𝐴(𝑘,𝑛,𝑡)={𝑘} and 𝐴(𝑘,𝑛,𝑡)⧵𝑝=∅ for 𝑝∈𝐴(𝑘,𝑛,𝑡). 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 𝑁₀, independent of 𝑡, such that for |𝑘|>𝑁₀ and for 𝑝∈𝐴(𝑘,𝑛,𝑡) the following assertions hold.
(a)If 𝐶 is Hermitian matrix, then for each eigenfunction Ψ_{𝑘,𝑗,𝑡} of 𝐿_𝑡(𝑃) there exists an eigenfunction Φ_{𝑝,𝑠,𝑡} of 𝐿_𝑡(𝐶) satisfying |(Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡}1)|>.3𝑚(64)(b)If 𝐿_𝑡(𝑃) is self-adjoint operator, then for each eigenfunction Φ_{𝑘,𝑗,𝑡} of 𝐿_𝑡(𝐶) there exists an eigenfunction Ψ_{𝑝,𝑠,𝑡} of 𝐿_𝑡(𝑃) satisfying |(Φ_{𝑘,𝑗,𝑡},Ψ_{𝑝,𝑠,𝑡}1)|>.3𝑚(65)

Proof. It follows from (52) and (13) that _{𝑠=1,2,…,𝑚}(_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}|(Ψ_{𝑘,𝑗,𝑡},𝜑_{𝑝,𝑠,𝑡})|²)=𝑂((ln|𝑘|)²𝑘²).(66) Hence using the equality Φ_{𝑝,𝑠,𝑡}(𝑥)=𝑣_𝑠𝑒^{𝑖(2𝜋𝑝+𝑡)𝑥}, where 𝑣_𝑠 is the normalized eigenvectors of 𝐶, and the Parseval equality, we get _{𝑠=1,2,…,𝑚}(_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}|(Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})|²)=𝑂((ln|𝑘|)²𝑘²),(67)_{𝑠=1,2,…,𝑚;𝑝∈𝐴(𝑘,𝑛,𝑡)}|(Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})|²=1+𝑂((ln|𝑘|)²𝑘²).(68) Since the number of the eigenfunctions Φ_{𝑝,𝑠,𝑡}(𝑥) for 𝑝∈𝐴(𝑘,𝑛,𝑡), 𝑠=1,2,…,𝑚 is less than 2𝑚 (see Notation 1), (64) follows from (68).
Using (52) and (14), we get _{𝑠=1,2,…,𝑚}(_{𝑝∶𝑝∉𝐴(𝑘,𝑛,𝑡)}|(𝜑_{𝑘,𝑗,𝑡},Ψ_{𝑝,𝑠,𝑡})|²)=𝑂((ln|𝑘|)²𝑘²).(69) Therefore, arguing as in the proof of (64) and taking into account that the eigenfunctions of the self-adjoint operator 𝐿_𝑡(𝑃) form an orthonormal basis in 𝐿^𝑚₂(0,1), we get the proof of (65).

Theorem 5. Let 𝐿_𝑡(𝑃) be a self-adjoint operator, and let 𝐶 be a Hermitian matrix. If 𝑛=2𝑟−1, then for arbitrary 𝑡, if 𝑛=2𝑟, then for 𝑡≠0,𝜋 the large eigenvalues of 𝐿_𝑡(𝑃) consist of 𝑚 sequences (6) satisfying 𝜆_𝑘,𝑗(𝑡)=(2𝜋𝑘+𝑡)^𝑛+𝜇_𝑗(2𝜋𝑘+𝑡)^𝑛−2+𝑂(𝑘^𝑛−3ln|𝑘|),(70) and the normalized eigenfunction Ψ_{𝑘,𝑗,𝑡} corresponding to 𝜆_𝑘,𝑗(𝑡) satisfies ‖Ψ_{𝑘,𝑗,𝑡}−𝐸Ψ_{𝑘,𝑗,𝑡}‖=𝑂((ln|𝑘|)𝑘)(71) for 𝑗=1,2,…,𝑚, where 𝜇₁≤𝜇₂≤⋯≤𝜇_𝑚 are the eigenvalues of 𝐶 and 𝐸 is the orthogonal projection onto the eigenspace of 𝐿_𝑡(𝐶) corresponding to 𝜇_𝑘,𝑗(𝑡). If 𝜇_𝑗 is a simple eigenvalue of 𝐶, then the eigenvalue 𝜆_𝑘,𝑗(𝑡) satisfying (70) is a simple eigenvalue, and the corresponding eigenfunction satisfies Ψ_{𝑘,𝑗,𝑡}(𝑥)=𝑣_𝑗𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥}+𝑂((ln|𝑘|)𝑘),(72) where 𝑣_𝑗 is the eigenvector of 𝐶 corresponding to the eigenvalue 𝜇_𝑗. In the case 𝑛=2𝑟−1 the formulas (70)–(72) are uniform with respect to 𝑡 in [−𝜋/2,3𝜋/2).

Proof. By (51) and (56) the right-hand side of (26) is 𝑂(𝑘^𝑛−3ln|𝑘|). On the other hand by Notation 1 if 𝑡≠0,𝜋, then there exists 𝑁 such that 𝑡∈𝑇(𝑘), and hence 𝐴(𝑘,𝑛,𝑡)={𝑘}, for |𝑘|≥𝑁. Thus dividing (26) by (Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡}), where 𝑝∈𝐴(𝑘,𝑛,𝑡), and hence 𝑝=𝑘, and using (64), we get {𝜆_𝑘,1(𝑡),𝜆_𝑘,2(𝑡),…,𝜆_𝑘,𝑚(𝑡)}⊂^𝑚_𝑗=1(𝑈(𝜇_𝑘,𝑗(𝑡),𝛿_𝑘)),(73) where 𝑈(𝜇,𝛿)={𝑧∈ℝ∶|𝜇−𝑧|<𝛿}, |𝑘|≥max{𝑁,𝑁₀}, 𝛿_𝑘=𝑂(|𝑘|^𝑛−3ln|𝑘|). Instead of (64) using (65), in the same way, we obtain 𝑈(𝜇_𝑘,𝑠(𝑡),𝛿_𝑘)∩{𝜆_𝑘,1(𝑡),𝜆_𝑘,2(𝑡),…,𝜆_𝑘,𝑚(𝑡)}≠∅(74) for |𝑘|≥max{𝑁,𝑁₀} and 𝑠=1,2,…,𝑚. Hence to prove (70) we need to show that if the multiplicity of the eigenvalue 𝜇_𝑗 is 𝑞 then there exist precisely 𝑞 eigenvalues of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝛿_𝑘) for |𝑘|≥max{𝑁,𝑁₀}. The eigenvalues of 𝐿_𝑡(𝑃) and 𝐿_𝑡(𝐶) can be numbered in the following way: 𝜆_𝑘,1(𝑡)≤𝜆_𝑘,2(𝑡)≤⋯≤𝜆_𝑘,𝑚(𝑡) and 𝜇_𝑘,1(𝑡)≤𝜇_𝑘,2(𝑡)≤⋯≤𝜇_𝑘,𝑚(𝑡). If 𝐶 has 𝜈 different eigenvalues 𝜇_𝑗1,𝜇_𝑗2,…,𝜇_𝑗𝜈 with multiplicities 𝑗₁,𝑗₂−𝑗₁,…,𝑗_𝜈−𝑗_𝜈−1, then we have 𝑗₁<𝑗₂<⋯<𝑗_𝜈=𝑚,𝜇_𝑗1<𝜇_𝑗2<⋯<𝜇_𝑗𝜈,𝜇₁=𝜇₂=⋯=𝜇_𝑗1,𝜇_𝑗1+1=𝜇_𝑗1+2=⋯=𝜇_𝑗2,…,𝜇_{𝑗𝜈−1+1}=𝜇_{𝑗𝜈−2+2}=⋯=𝜇_𝑗𝜈.(75) Suppose there exist precisely 𝑠₁,𝑠₂,…,𝑠_𝜈 eigenvalues of 𝐿_𝑡(𝑃) lying in the intervals 𝑈(𝜇_𝑘,𝑗1(𝑡),𝛿_𝑘),𝑈(𝜇_𝑘,𝑗2(𝑡),𝛿_𝑘),…,𝑈(𝜇_{𝑘,𝑗𝜈}(𝑡),𝛿_𝑘),(76) respectively. Since 𝛿_𝑘≪(min_{𝑝=1,2,…,𝜈−1}|(𝜇_𝑗𝑝+1−𝜇_𝑗𝑝)(2𝜋𝑘+𝑡)^𝑛−2|)for|𝑘|≫1,(77) these intervals are pairwise disjoints. Therefore using (6) and (7), we get 𝑠₁+𝑠₂+⋯+𝑠_𝜈=𝑚.(78) Now let us prove that 𝑠₁=𝑗₁,𝑠₂=𝑗₂−𝑗₁,…,𝑠_𝜈=𝑗_𝜈−𝑗_𝜈−1. Due to the notations the eigenvalues 𝜆_𝑘,1(𝑡),𝜆_𝑘,2(𝑡),…,𝜆_𝑘,𝑠1(𝑡) of the operator 𝐿_𝑡(𝑃) lie in 𝑈(𝜇_𝑘,1(𝑡),𝛿_𝑘) and by the definition of 𝛿_𝑘 we have |𝜆_𝑘,𝑗(𝑡)−𝜇_𝑘,𝑠1(𝑡)|>2(min_{𝑝∶𝑝>𝑗1}|(𝜇₁−𝜇_𝑝)(2𝜋𝑘+𝑡)^𝑛−2|)(79) for 𝑗≤𝑠₁ and 𝑠>𝑗₁. Hence using (26) for 𝑝=𝑘 and (56), (51), we get _{𝑠∶𝑠>𝑗1}|(Ψ_{𝑘,𝑗,𝑡},Φ_{𝑘,𝑠,𝑡})|²=𝑂((ln|𝑘|)²𝑘²),∀𝑗≤𝑠₁.(80) Using this, (67), and taking into account that 𝐴(𝑘,𝑛,𝑡)={𝑘} for |𝑘|≥𝑁, we conclude that there exists normalized eigenfunction, denoted by Φ_{𝑘,𝑗,𝑡}(𝑥), of 𝐿_𝑡(𝑃) corresponding to 𝜇_𝑘,1(𝑡)=𝜇_𝑘,2(𝑡)=⋯=𝜇_𝑘,𝑗1(𝑡) such that Ψ_{𝑘,𝑗,𝑡}(𝑥)=Φ_{𝑘,𝑗,𝑡}(𝑥)+𝑂(𝑘⁻¹ln|𝑘|)(81) for 𝑗≤𝑠₁. Since Ψ_𝑘,1,𝑡,Ψ_𝑘,2,𝑡,…,Ψ_{𝑘,𝑠1,𝑡} are orthonormal system we have (Φ_{𝑘,𝑗,𝑡},Φ_{𝑘,𝑠,𝑡})=𝛿_𝑠,𝑗+𝑂(𝑘⁻¹ln|𝑘|),∀𝑠,𝑗=1,2,…,𝑠₁.(82) This formula implies that the dimension 𝑗₁ of the eigenspace of 𝐿_𝑡(𝐶) corresponding to the eigenvalue 𝜇_𝑘,1(𝑡) is not less than 𝑠₁. Thus 𝑠₁≤𝑗₁. In the same way we prove that 𝑠₂≤𝑗₂−𝑗₁,…,𝑠_𝜈≤𝑗_𝜈−𝑗_𝜈−1. Now (78) and the equality 𝑗_𝜈=𝑚 (see (75)) imply that 𝑠₁=𝑗₁,𝑠₂=𝑗₂−𝑗₁,…,𝑠_𝜈=𝑗_𝜈−𝑗_𝜈−1. Therefore, taking into account that, the eigenvalues of 𝐿_𝑡(𝑃) consist of 𝑚 sequences satisfying (7), we get (70). The proof of (71) follows from (81).
Now suppose that 𝜇_𝑗 is a simple eigenvalue of 𝐶. Then 𝜇_𝑘,𝑗(𝑡) is a simple eigenvalues of 𝐿_𝑡(𝐶) and, as it was proved above, there exists unique eigenvalues 𝜆_𝑘,𝑗(𝑡) of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑠(𝑡),𝛿_𝑘), where |𝑘|≥max{𝑁,𝑁₀}, and the eigenvalues 𝜆_𝑘,𝑗(𝑡) for |𝑘|≥max{𝑁,𝑁₀} are the simple eigenvalues. Hence (72) is the consequence of (71), since there exists unique eigenfunction Φ_{𝑘,𝑗,𝑡}(𝑥)=𝑣_𝑗𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥} corresponding to the eigenvalue 𝜇_𝑘,𝑗(𝑡). The uniformity of the formulas (70)–(72) follows from the uniformity of (56), (51), (64), and (65).

Theorem 6. Let 𝐿_𝑡(𝑃) be a self-adjoint operator, let 𝐶 be a Hermitian matrix, let 𝑛=2𝑟, 𝜇_𝑗 be a simple eigenvalue of 𝐶, let 𝛼_𝑗 be a positive constant satisfying 𝛼_𝑗<min_{𝑞∶𝑞≠𝑗}|𝜇_𝑗−𝜇_𝑞|, and let 𝐵(𝛼_𝑗,𝑘,𝜇_𝑗) be a set defined by 𝐵(𝛼_𝑗,𝑘,𝜇_𝑗)=𝐵(0,𝛼_𝑗,𝑘,𝜇_𝑗)∪𝐵(𝜋,𝛼_𝑗,𝑘,𝜇_𝑗), where 𝐵(0,𝛼_𝑗,𝑘,𝜇_𝑗)=_{𝑠=1,2…,𝑚}(𝜇_𝑠−𝜇_𝑗−𝛼_𝑗,𝜇4𝑛𝜋𝑘^𝑠−𝜇_𝑗+𝛼_𝑗4𝑛𝜋𝑘),𝐵(𝜋,𝛼_𝑗,𝑘,𝜇_𝑗)=_{𝑠=1,2…,𝑚}𝜇(𝜋+^𝑠−𝜇_𝑗−𝛼_𝑗𝜇2𝑛𝜋(2𝑘+𝑛−1),𝜋+^𝑠−𝜇_𝑗+𝛼_𝑗2𝑛𝜋(2𝑘+𝑛−1)).(83) There exist a positive number 𝑁₁ such that if |𝑘|≥𝑁₁ and 𝑡∉𝐵(𝛼_𝑗,𝑘,𝜇_𝑗), then there exists a unique eigenvalue, denoted by 𝜆_𝑘,𝑗(𝑡), of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗,𝜀_𝑘), where 𝜀_𝑘=𝑐₇(|𝑘|^𝑛−3ln|𝑘|)+|𝑘|^𝑛−2𝑏_𝑘, 𝑏_𝑘 is defined by (55), and 𝑐₇ is a positive constant, independent of 𝑡. The eigenvalue 𝜆_𝑘,𝑗(𝑡) is a simple eigenvalue of 𝐿_𝑡(𝑃) and the corresponding normalized eigenfunction Ψ_{𝑘,𝑗,𝑡}(𝑥) satisfies Ψ_{𝑘,𝑗,𝑡}(𝑥)=𝑣_𝑗𝑒^{𝑖(2𝜋𝑘+𝑡)𝑥}+𝑂(𝑘⁻¹ln|𝑘|)+𝑂(𝑏_𝑘).(84)

Proof. To consider the simplicity of 𝜇_𝑘,𝑗(𝑡) and 𝜆_𝑘,𝑗(𝑡) we introduce the set 𝜋𝑆(𝑘,𝑗,𝑝,𝑠)={𝑡∈[−2,3𝜋2)∶|𝜇_𝑘,𝑗(𝑡)−𝜇_𝑝,𝑠(𝑡)|<𝛼_𝑗|𝑘|^𝑛−2}(85) for (𝑝,𝑠)≠(𝑘,𝑗). It follows from (10) that 𝑆(𝑘,𝑗,𝑝,𝑠)=∅ for 𝑝≠𝑘,−𝑘,−𝑘−1. Moreover, if 𝜇_𝑗 is a simple eigenvalue, then 𝑆(𝑘,𝑗,𝑘,𝑠)=∅ for 𝑠≠𝑗, since |𝜇_𝑘,𝑗(𝑡)−𝜇_𝑘,𝑠(𝑡)|=|(𝜇_𝑗−𝜇_𝑠)(2𝜋𝑘+𝑡)^𝑛−2|>𝛼_𝑗|𝑘|^𝑛−2.(86) It remains to consider the sets 𝑆(𝑘,𝑗,−𝑘,𝑠), 𝑆(𝑘,𝑗,−𝑘−1,𝑠). Using the equality 𝜇_𝑘,𝑗(𝑡)−𝜇_−𝑘,𝑠(𝑡)=(2𝜋𝑘)^𝑛−2(4𝑛𝑘𝜋𝑡+𝜇_𝑗−𝜇_𝑠)+𝑂(𝑘^𝑛−3), we see that 𝜇𝑆(𝑘,𝑗,−𝑘,𝑠)⊂(^𝑠−𝜇_𝑗−𝛼_𝑗,𝜇4𝑛𝜋𝑘^𝑠−𝜇_𝑗+𝛼_𝑗4𝑛𝜋𝑘).(87) Similarly, by using the obvious equality 𝜇_𝑘,𝑗(𝑡)−𝜇_{−𝑘−1,𝑠}(𝑡)=(2𝜋𝑘+𝑡)^𝑛+𝜇_𝑗(2𝜋𝑘+𝑡)^𝑛−2−(2𝜋𝑘+2𝜋−𝑡)^𝑛−𝜇_𝑗(2𝜋𝑘+2𝜋−𝑡)^𝑛−2=(2𝜋𝑘)^𝑛−21(𝑛2𝜋𝑘𝑡−𝑛2𝜋𝑘(2𝜋−𝑡)+2𝑛(𝑛−1)(𝑡²−(2𝜋−𝑡)²)+𝜇_𝑗−𝜇_𝑠)+𝑂(𝑘^𝑛−3)=(2𝜋𝑘)^𝑛−2[(𝑡−𝜋)(2𝑘+(𝑛−1))2𝜋𝑛+𝜇_𝑗−𝜇_𝑠]+𝑂(𝑘^𝑛−3),(88) we get 𝜇𝑆(𝑘,𝑗,−𝑘−1,𝑠)⊂(𝜋+^𝑠−𝜇_𝑗−𝛼_𝑗𝜇2𝑛𝜋(2𝑘+𝑛−1),𝜋+^𝑠−𝜇_𝑗+𝛼_𝑗2𝑛𝜋(2𝑘+𝑛−1)).(89) Using these relations and the definition of 𝐵(𝛼_𝑗,𝑘,𝜇_𝑗), we obtain _{𝑝∈ℤ,𝑠=1,2,…,𝑚,(𝑝,𝑠)≠(𝑘,𝑗)}𝑆(𝑘,𝑗,𝑝,𝑠)=_{𝑝=−𝑘,−𝑘−1,𝑠=1,2,…,𝑚}𝑆(𝑘,𝑗,𝑝,𝑠)⊂𝐵(𝛼_𝑗,𝑘,𝜇_𝑗).(90) Therefore it follows from (85) that if 𝑡∉𝐵(𝛼_𝑗,𝑘,𝜇_𝑗), then |𝜇_𝑘,𝑗(𝑡)−𝜇_𝑝,𝑠(𝑡)|≥𝛼_𝑗|𝑘|^𝑛−2(91) for all (𝑝,𝑠)≠(𝑘,𝑗). Hence 𝜇_𝑘,𝑗(𝑡) is a simple eigenvalue of 𝐿_𝑡(𝐶) for 𝑡∉𝐵(𝛼_𝑗,𝑘,𝜇_𝑗). Instead of (56) using (54) and arguing as in the proof of (74), we obtain that there exists 𝑁₁ such that if |𝑘|≥𝑁₁, then there exists an eigenvalue, denoted by 𝜆_𝑘,𝑗(𝑡), of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘). Now using the definition of 𝜀_𝑘 and then (91), we see that |𝜆_𝑘,𝑗(𝑡)−𝜇_𝑘,𝑗(𝑡)|<𝜀_𝑘=𝑜(𝑘^𝑛−2),|𝜆_𝑘,𝑗(𝑡)−𝜇_𝑝,𝑠1(𝑡)|>2𝛼_𝑗|𝑘|^𝑛−2(92) for |𝑘|≥𝑁₁, 𝑠=1,2,…,𝑚, (𝑝,𝑠)≠(𝑘,𝑗) and for any eigenvalue 𝜆_𝑘,𝑗(𝑡) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘). Let Ψ_{𝑘,𝑗,𝑡}(𝑥) be any normalized eigenfunction corresponding to 𝜆_𝑘,𝑗(𝑡). Dividing both sides of (26) by 𝜆_𝑘,𝑗(𝑡)−𝜇_𝑝,𝑠(𝑡) and using (54), (51), and (92), we get (Ψ_{𝑘,𝑗,𝑡},Φ_{𝑝,𝑠,𝑡})=𝑂(𝑘⁻¹ln|𝑘|)+𝑂(𝑏_𝑘)(93) for (𝑝,𝑠)≠(𝑘,𝑗) and 𝑝∈𝐴(𝑘,𝑛,𝑡). This, (67) and (68) imply that Ψ_{𝑘,𝑗,𝑡}(𝑥) satisfies (84). Thus we have proved that (84) holds for any normalized eigenfunction of 𝐿_𝑡(𝑃) corresponding to any eigenvalue lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘). If there exist two different eigenvalues of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘) or if there exists a multiple eigenvalue of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘), then we obtain that there exist two orthonormal eigenfunctions satisfying (84) which is impossible. Therefore there exists unique eigenvalue 𝜆_𝑘,𝑗(𝑡) of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝜀_𝑘) and 𝜆_𝑘,𝑗(𝑡) is a simple eigenvalue of 𝐿_𝑡(𝑃).

Theorem 7. Let 𝐿(𝑃) be self-adjoint operator generated in 𝐿^𝑚₂(−∞,∞) by the differential expression (1), and let 𝐶 be Hermitian matrix.
(a)If 𝑛 and 𝑚 are odd numbers then the spectrum 𝜎(𝐿(𝑃)) of 𝐿(𝑃) coincides with (−∞,∞).(b)If 𝑛 is odd number, 𝑛>1, and the matrix 𝐶 has at least one simple eigenvalue, then the number of the gaps in 𝜎(𝐿(𝑃)) is finite.(c)Suppose that 𝑛 is even number, and the matrix 𝐶 has at least three simple eigenvalues 𝜇_𝑗1<𝜇_𝑗2<𝜇_𝑗3 such that diam({𝜇_𝑗1+𝜇_𝑖1,𝜇_𝑗2+𝜇_𝑖2,𝜇_𝑗3+𝜇_𝑖3})≠0 for each triple (𝑖₁,𝑖₂,𝑖₃), where 𝑖_𝑝=1,2,…,𝑚 for 𝑝=1,2,3 and diam(𝐴) is the diameter sup_{𝑥,𝑦∈𝐴}|𝑥−𝑦| of the set 𝐴. Then the number of the gaps in the spectrum of 𝐿(𝑃) is finite.

Proof. (a) In case 𝑚=1 the assertion (a) is proved in [4]. Our proof is carried out analogous fashion. Since 𝐿(𝑃) is self-adjoint, 𝜎(𝐿(𝑃)) is a subset of (−∞,∞). Therefore we need to prove that (−∞,∞)⊂𝜎(𝐿(𝑃)). Suppose to the contrary that there exists a real number 𝜆 such that 𝜆∉𝜎(𝐿(𝑃)). It is not hard to see that the characteristic determinant Δ(𝜆,𝑡)=det(𝑈_𝜈(𝑌_𝑠(𝑥,𝜆))) of 𝐿_𝑡(𝑃) has the form Δ(𝜆,𝑡)=𝑒^{𝑖𝑛𝑚𝑡}+𝑎₁(𝜆)𝑒^{𝑖(𝑛𝑚−1)𝑡}+𝑎₂(𝜆)𝑒^{𝑖(𝑛𝑚−2)𝑡}+⋯+𝑎_𝑛𝑚(𝜆),(94) that is, Δ(𝜆,𝑡) is a polynomial 𝑆_𝜆(𝑢) of 𝑢=𝑒^𝑖𝑡 of order 𝑛𝑚 with entire coefficients 𝑎₁(𝜆),𝑎₂(𝜆),…. It is well known that if 𝜆∉𝜎(𝐿(𝑃)), then the absolute values of all roots 𝑢₁=𝑒^𝑖𝑡1,𝑢₂=𝑒^𝑖𝑡2,…,𝑢_𝑛𝑚=𝑒^{𝑖𝑡𝑛𝑚} of 𝑆_𝜆(𝑢)=0 differ from 1, that is, 𝑡_𝑘≠𝑡_𝑘 and 𝜆 is the eigenvalue of 𝐿_𝑡𝑘(𝑃) for 𝑘=1,2,…,𝑛𝑚. It is not hard to see that 𝐿^∗_𝑡𝑘=𝐿_𝑡𝑘, 𝜆=𝜆∈𝜎(𝐿_𝑡𝑘). Moreover, if 𝜆 is the eigenvalue of 𝐿_𝑡𝑘(𝑃) of multiplicity 𝑚_𝑘 then 𝜆 is the eigenvalue of 𝐿_𝑡𝑘(𝑃) of the same multiplicity 𝑚_𝑘. Now taking into account that 𝑢_𝑘=𝑒^𝑖𝑡𝑘 is the root of 𝑆_𝜆(𝑢)=0 of multiplicity 𝑚_𝑘 if and only if 𝜆 is the eigenvalue of 𝐿_𝑡𝑘(𝑃) of multiplicity 𝑚_𝑘, we obtain that 𝑒^𝑖𝑡𝑘 is also root of 𝑆_𝜆(𝑢)=0 of the same multiplicity 𝑚_𝑘. Since 𝑒^𝑖𝑡𝑘≠𝑒^𝑖𝑡𝑘, we see that the number 𝑛𝑚 of the roots of 𝑆_𝜆(𝑢)=0 (see (94)) is an even number which contradicts the assumption that 𝑛 and 𝑚 are odd numbers.
(b) It follows from the uniform asymptotic formula (70) that there exists a positive numbers 𝑁₂, 𝑐₈, independent of 𝑡, such that if |𝑘|≥𝑁₂ and 𝜇_𝑗 is a simple eigenvalue of the matrix 𝐶 then there exists unique simple eigenvalue 𝜆_𝑘,𝑗(𝑡) of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝛿_𝑘), where 𝛿_𝑘=𝑐₈|𝑘|^𝑛−3ln|𝑘| and 𝑡∈[−𝜋/2,3𝜋/2). Therefore 𝜆_𝑘,𝑗(𝑡₀) for 𝑡₀∈(−𝜋/2,3𝜋/2), |𝑘|≥𝑁₂ is a simple zero of the characteristic determinant Δ(𝜆,𝑡₀). By implicit function theorem there exists a neighborhood 𝑈(𝑡₀)⊂(−𝜋/2,3𝜋/2) of 𝑡₀ and a continuous in 𝑈(𝑡₀) function Λ(𝑡) such that Λ(𝑡₀)=𝜆_𝑘,𝑗(𝑡₀), Λ(𝑡) is an eigenvalue of 𝐿_𝑡(𝑃) for 𝑡∈𝑈(𝑡₀) and |Λ(𝑡)−𝜇_𝑘,𝑗(𝑡)|<𝛿_𝑘, for all 𝑡∈𝑈(𝑡₀), since |Λ(𝑡₀)−𝜇_𝑘,𝑗(𝑡₀)|=|𝜆_𝑘,𝑗(𝑡₀)−𝜇_𝑘,𝑗(𝑡₀)|<𝛿_𝑘 and the functions Λ(𝑡), 𝜇_𝑘,𝑗(𝑡) are continuous. Now taking into account that there exists unique eigenvalue of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝛿_𝑘), we obtain that Λ(𝑡)=𝜆_𝑘,𝑗(𝑡) for 𝑡∈𝑈(𝑡₀), and hence 𝜆_𝑘,𝑗(𝑡) is continuous at 𝑡₀∈(−𝜋/2,3𝜋/2). Therefore the sets Γ_𝑘,𝑗={𝜆_𝑘,𝑗(𝑡)∶𝑡∈(−𝜋/2,3𝜋/2)} for |𝑘|≥𝑁₂ are intervals and Γ_𝑘,𝑗⊂𝜎(𝐿(𝑃)). Similarly there exists a neighborhood 𝜋𝑈(−2𝜋)=(−2𝜋−𝛽,−2+𝛽)(95) of −𝜋/2 and a continuous function 𝐴(𝑡) such that 𝐴(−𝜋/2)=𝜆_𝑘+1,𝑗(−𝜋/2), where 𝑘≥𝑁₂, 𝐴(𝑡) is an eigenvalue of 𝐿_𝑡+2𝜋(𝑃)(𝐿_𝑡(𝑃)=𝐿_𝑡+2𝜋(𝑃)) for 𝑡∈(−𝜋/2−𝛽,−𝜋/2] and 𝐴(𝑡) is an eigenvalue of 𝐿_𝑡(𝑃) for 𝑡∈[−𝜋/2,−𝜋/2+𝛽) and |𝐴(𝑡)−𝜇_𝑘+1,𝑗(𝑡)|<𝛿_𝑘𝜋∀𝑡∈[−2𝜋,−2+𝛽),|𝐴(𝑡)−𝜇_𝑘,𝑗(𝑡+2𝜋)|<𝛿_𝑘𝜋∀𝑡∈(−2𝜋−𝛽,−2),(96) since |𝐴(−𝜋/2)−𝜇_𝑘+1,𝑗(−𝜋/2)|=|𝜆_𝑘+1,𝑗(−𝜋/2)−𝜇_𝑘+1,𝑗(−𝜋/2)|<𝛿_𝑘, 𝜇_𝑘,𝑗(𝑡+2𝜋)=𝜇_𝑘+1,𝑗(𝑡) and the functions Λ(𝑡), 𝜇_𝑘,𝑗(𝑡) are continuous. Again taking into account that there exists unique eigenvalue of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_𝑘+1,𝑗(𝑡),𝛿_𝑘) for 𝑡∈[−𝜋/2,−𝜋/2+𝛽) and lying in 𝑈(𝜇_𝑘,𝑗(𝑡),𝛿_𝑘) for 𝑡∈(3𝜋/2−𝛽,3𝜋/2), we obtain that 𝐴(𝑡)=𝜆_𝑘+1,𝑗𝜋(𝑡),∀𝑡∈[−2𝜋,−2+𝛽),𝐴(𝑡)=𝜆_𝑘,𝑗𝜋(𝑡+2𝜋),∀𝑡∈(−2𝜋−𝛽,−2).(97) Thus one part of the interval {𝐴(𝑡)∶𝑡∈(−𝜋/2−𝛽,−𝜋/2+𝛽)} lies in Γ_𝑘,𝑗 and the other part lies in Γ_𝑘+1,𝑗, that is, the interval Γ_𝑘,𝑗 and Γ_𝑘+1,𝑗 are connected for 𝑘≥𝑁₂. Similarly the interval Γ_𝑘,𝑗 and Γ_{𝑘−1,𝑗} are connected for 𝑘≤−𝑁₂. Therefore the number of the gaps in the spectrum of 𝐿(𝑃) is finite.
(c) In Theorem 6 we proved that if |𝑘|≥𝑁₁ and 𝑡∉𝐵(𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝), where 𝑝=1,2,3, then there exists a unique eigenvalue, denoted by 𝜆_{𝑘,𝑗𝑝}(𝑡), of 𝐿_𝑡(𝑃) lying in 𝑈(𝜇_{𝑘,𝑗𝑝}(𝑡),𝜀_𝑘) and it is a simple eigenvalue. Let us prove that 𝜆_{𝑘,𝑗𝑝}(𝑡) is continuous at 𝑡₀𝜋∈[−2,3𝜋2)⧵𝐵(𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝).(98) Since 𝜆_{𝑘,𝑗𝑝}(𝑡₀) is a simple eigenvalue it is a simple zero of the characteristic determinant Δ(𝜆,𝑡) of the operator 𝐿_𝑡(𝑃). Therefore repeating the argument of the proof of the continuity of 𝜆_𝑘,𝑗(𝑡) in the proof of (b), we obtain that 𝜆_{𝑘,𝑗𝑝}(𝑡) is continuous at 𝑡₀ for |𝑘|≥𝑁₁. Now we prove that there exists 𝐻 such that (𝐻,∞)⊂{𝜆_{𝑘,𝑗𝑝}𝜋(𝑡)∶𝑡∈[−2,3𝜋2)⧵𝐵(𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝),𝑘=𝑁₁,𝑁₁+1,…}.(99) It is clear that (ℎ,∞)⊂{𝜇_{𝑘,𝑗𝑝}𝜋(𝑡)∶𝑡∈[−2,3𝜋2),𝑘=𝑁₁,𝑁₁+1,…},∀𝑝=1,2,3,(100) where ℎ=𝜇_𝑁1,𝑗3(−𝜋/2). Since 𝜇_{𝑘,𝑗𝑝}(𝑡) is increasing function for 𝑘≥𝑁₁, it follows from the obvious equality 𝜇_{𝑘,𝑗𝑝}(𝜇_𝑠−𝜇_𝑗𝑝∓𝛼_𝑗𝑝4𝑛𝜋𝑘)=(2𝜋𝑘)^𝑛+𝑛(2𝜋𝑘)^𝑛−1𝜇_𝑠−𝜇_𝑗𝑝∓𝛼_𝑗𝑝4𝑛𝜋𝑘+𝜇_𝑗𝑝(2𝜋𝑘)^𝑛−2+𝑂(𝑘^𝑛−4)=(2𝜋𝑘)^𝑛+(2𝜋𝑘)^𝑛−2𝜇_𝑠+𝜇_𝑗𝑝∓𝛼_𝑗𝑝2+𝑂(𝑘^𝑛−4)(101) and from the definition of 𝐵(0,𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝) that {𝜇_{𝑘,𝑗𝑝}(𝑡)∶𝑡∈𝐵(0,𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝)}⊂_{𝑠=1,2,…,𝑚}𝐶(0,𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝),(102) where 𝐶(0,𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝)={𝑥∈ℝ∶|𝑥−(2𝜋𝑘)^𝑛+(2𝜋𝑘)^𝑛−2((𝜇_𝑠+𝜇_𝑗𝑝)/2)|<𝛼_𝑗𝑝(2𝜋𝑘)^𝑛−2}. This inclusion with (100) implies that the set (ℎ,∞)⧵_{𝑘∶𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(0,𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝)(103) is a subset of the set {𝜇_{𝑘,𝑗𝑝}(𝑡)∶𝑡∈[−𝜋/2,3𝜋/2)⧵𝐵(0,𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝),𝑘≥𝑁₁}. Similarly, using 𝜇_{𝑘,𝑗𝑝}𝜇(𝜋+^𝑠−𝜇_𝑗𝑝∓𝛼_𝑗𝑝4𝑛𝜋𝑘)=(2𝜋𝑘+𝜋)^𝑛+(2𝜋𝑘)^𝑛−2𝜇_𝑠+𝜇_𝑗𝑝∓𝛼_𝑗𝑝2+𝑂(𝑘^𝑛−3),(104) which can be proved by direct calculations, we obtain that the set (ℎ,∞)⧵_{𝑘∶𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(𝜋,𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝),(105) where 𝐶(𝜋,𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝)={𝑥∈ℝ∶|𝑥−(2𝜋𝑘+𝜋)^𝑛+(2𝜋𝑘)^𝑛−2((𝜇_𝑠+𝜇_𝑗𝑝)/2)|<𝛼_𝑗𝑝(2𝜋𝑘)^𝑛−2}, is a subset of {𝜇_{𝑘,𝑗𝑝}𝜋(𝑡)∶𝑡∈[−2,3𝜋2)⧵𝐵(𝜋,𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝),𝑘≥𝑁₁}.(106) Now using (92) and the continuity of 𝜆_{𝑘,𝑗𝑝}(𝑡) on [−𝜋/2,3𝜋/2)⧵𝐵(𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝), we see that the set (𝐻,∞)⧵(_{𝑘∶𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝)),(107) where 𝐻=ℎ+1, 𝐶(𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝)=𝐶(0,𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝)∪𝐶(𝜋,𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝), is a subset of the set {𝜆_{𝑘,𝑗𝑝}(𝑡)∶𝑡∈[−𝜋/2,3𝜋/2)⧵𝐵(𝛼_𝑗𝑝,𝑘,𝜇_𝑗𝑝),𝑘≥𝑁₁}. Thus we have _𝑝=1,2,3((𝐻,∞)⧵(_{𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝)))⊂𝜎(𝐿(𝑃)).(108) To prove the inclusion (𝐻,∞)⊂𝜎(𝐿(𝑃)) it is enough to show that the set _𝑝=1,2,3(_{𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(𝑘,𝑗_𝑝,𝑠,2𝛼_𝑗𝑝))(109) is empty. If this set contains an element 𝑥, then 𝑥∈_{𝑘≥𝑁1;𝑠=1,2,…,𝑚}𝐶(𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝)(110) for all 𝑝=1,2,3. Using this and the definition of 𝐶(𝑘,𝑗_𝑝,𝑠,𝛼_𝑗𝑝), we obtain that there exist 𝑘≥𝑁₁; 𝜈=0,1 and 𝑠=𝑖_𝑝 such that |𝑥−(𝜋(2𝑘+𝜈))^𝑛−𝜇_𝑗𝑝+𝜇_𝑖𝑝2(2𝜋𝑘)^𝑛−2|<2𝛼_𝑗𝑝(2𝜋𝑘)^𝑛−2(111) for all 𝑝=1,2,3 and hence |𝜇_𝑗𝑞+𝜇_𝑖𝑞2−𝜇_𝑗𝑝+𝜇_𝑖𝑝2|<4𝛼_𝑗𝑝(112) for all 𝑝,𝑞=1,2,3. Clearly, the constant 𝛼_𝑗𝑝 can be chosen so that 8𝛼_𝑗𝑝<min_{𝑖1,𝑖2,𝑖3}(diam({𝜇_𝑗1+𝜇_𝑖1,𝜇_𝑗2+𝜇_𝑖2,𝜇_𝑗3+𝜇_𝑖3})),(113) since, by assumption of the theorem, the right-hand side of (113) is a positive constant. If (113) holds then (112) and hence (110) do not hold which implies that (𝐻,∞)⊂𝜎(𝐿(𝑃)). Hence the number of the gaps in the spectrum of 𝐿(𝑃) is finite.

Acknowledgment

The work was supported by the Scientific and Technological Research Council of Turkey (Tübitak, Project no. 108T683).