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 L(P2,P3,…,Pn) be the differential operator generated in the space L2m(-∞,∞) of vector-valued functions by the differential expression
(-i)ny(n)(x)+(-i)n-2P2(x)y(n-2)(x)+∑v=3nPv(x)y(n-v)(x),
where n is an integer greater than 1 and Pk(x), for k=2,3,…,n, is the m×m matrix with the complex-valued summable entries pk,i,j(x) satisfying pk,i,j(x+1)=pk,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,…,Pn) is the union of the spectra of the operators Lt(P2,P3,…,Pn) for t∈[0,2π) generated in L2m(0,1) by expression (1) and the quasiperiodic conditions
Uν(y)≡y(ν)(1)-eity(ν)(0)=0,ν=0,1,…,(n-1).
Note that L2m(a,b) is the set of vector-valued functions f=(f1,f2,…,fm) with fk∈L2(a,b) for k=1,2,…,m. The norm ∥·∥ and inner product (·,·) in L2m(a,b) are defined by
∥f∥2=∫ab|f(x)|2dx,(f,g)=∫ab〈f(x),g(x)〉dx,
where |·| and 〈·,·〉 are the norm and inner product in ℂm.

The first works concerned with the differential operator Lt(P2,P3,…,Pn) were by Birkhoff [5], Tamarkin [6] in the beginning of 20th century. There exist enormously many papers concerning with the operators Lt(P2,P3,…,Pn) and L(P2,P3,…,Pn). 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)+∑v=3nPvY(n-v)=λY,
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 Lt(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 Lt(P2,P3,…,Pn) 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,…,Pn).

Let us introduce some preliminary results and describe the results of this paper. Clearly,
φk,1,t=(ei(2πk+t)x,0,…,0),φk,2,t=(0,ei(2πk+t)x,0,…,0),…,φk,m,t=(0,0,…,0,ei(2πk+t)x)
are the eigenfunctions of the operator Lt(0) corresponding to the eigenvalue (2πk+t)n, where k∈ℤ, and the operator Lt(P2,…,Pn) is denoted by Lt(0) when P2(x)=0,…,Pn(x)=0. Furthermore, for brevity of notation, the operators Lt(P2,…,Pn) and L(P2,…,Pn) are denoted by Lt(P) and L(P), respectively. It easily follows from the classical investigations [7, Chapter 3, Theorem 2] that the large eigenvalues of the operator Lt(P) consist of m sequences
{λk,1(t):|k|≥N},{λk,2(t):|k|≥N},…,{λk,m(t):|k|≥N},
satisfying the following, uniform with respect to t in [0,2π), asymptotic formulas:
λk,j(t)=(2πk+t)n+O(kn-1-1/2m)
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
|λk,j(t)-(2πk+t)n|<c1|k|n-1-1/2m,∀|k|≥N,∀t∈[0,2π).
In this paper, by the suggested method, we obtain the uniform asymptotic formulas of high accuracy for the eigenvalues λk,j(t) and for the corresponding normalized eigenfunctions Ψk,j,t(x) of Lt(P) when the entries p2,i,j(x),p3,i,j(x),…,pn,i,j(x) of P2(x),P3(x),…,Pn(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 λk,j(t) of Lt(P) is close to the eigenvalue (2kπ+t)n of Lt(0). To analyze the distance of the eigenvalue λk,j(t) of Lt(P) from the other eigenvalues (2pπ+t)n of Lt(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 Lt(0) lies far from the other eigenvalues (2pπ+t)n of Lt(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
T(k)=[-π2,3π2)∖((-1ln|k|,1ln|k|)∪(π-1ln|k|,π+1ln|k|)).

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,|λk,j(t)-(2πp+t)n|>c2(ln|k|)-1(||k|-|p||+1)(|k|+|p|)n-1,
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|>dpn-2|λk,j(t)-(2πp+t)n|=O(1d),∑p:p∉A(k,n,t)pn-2|λk,j(t)-(2πp+t)n|=O(ln|k|k),∑p:p∉A(k,n,t)k2n-4|λk,j(t)-(2πp+t)n|2=O((ln|k|)2k2),∑p:p∉A(k,n,t)p2n-4|λp,j(t)-(2πk+t)n|2=O((ln|k|)2k2)
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
|λk,j(t)-(2πp+t)n|>|p|n,∀t∈[-π2,3π2).
Therefore the left-hand side of (11) is less than
∑p:|p|>d1p2,∀t∈[-π2,3π2)
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),
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)pn-2|λk,j(t)-(2πp+t)n|,S(9,2)=∑p:p∈S(2)pn-2|λk,j(t)-(2πp+t)n|,S(9,3)=∑p:p∈S(3)pn-2|λk,j(t)-(2πp+t)n|.
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
|λk,j(t)-(2πp+t)n|>(||k|-|p||)(|k|+|p|)n-1,|pn-2||λk,j(t)-(2πp+t)n|<1(||k|-|p||)|k|
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
|S(9,2)|≤41|k|(∑s=3k1s)=O(ln|k|k).
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).
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).
These equalities imply the proof of (13) and (14).

To obtain the asymptotic formulas we use (11)–(14) and consider the operator Lt(P) as perturbation of Lt(C) by Lt(P)-Lt(C), where C=∫01P2(x)dx, Lt(C) is the operator generated by (2) and by the expression
(-i)ny(n)(x)+(-i)n-2Cy(n-2)(x).
Therefore, first of all, we analyze the eigenvalues and eigenfunction of the operator Lt(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 Lt(C) is also self-adjoint. The eigenvalues of C, counted with multiplicity, and the corresponding orthonormal eigenvectors are denoted by μ1≤μ2≤⋯≤μm and v1,v2,…,vm. Thus
Cvj=μjvj,〈vi,vj〉=δi,j.
One can easily verify that the eigenvalues and eigenfunctions of Lt(C) are μk,j(t)=(2πk+t)n+μj(2πk+t)n-2, Φk,j,t(x)=vjei(2πk+t)x, that is,
(L(C)-μk,j(t))Φk,j,t(x)=0.

To prove the asymptotic formulas for the eigenvalues λk,j(t) and for the corresponding normalized eigenfunctions Ψk,j,t of Lt(P) we use the formula
(λk,j(t)-μp,s(t))(Ψk,j,t,Φp,s,t)=(-i)n-2((P2-C)Ψk,j,t(n-2),Φp,s,t)+∑ν=3n(PνΨk,j,tn-ν,Φp,s,t)
which can be obtained from
L(P)Ψk,j,t(x)=λk,j(t)Ψk,j,t(x)
by multiplying both sides by Φp,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 (Ψk,j,t,Φp,s,t) (see Lemma 4) and using these estimations in (26), we find the asymptotic formulas for the eigenvalues and eigenfunctions of Lt(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
(λk,j(t)-(2πp+t)n)(Ψk,j,t,φp,s,t)=(-i)n-2(P2Ψk,j,tn-2,φp,s,t)+∑ν=3n(PνΨk,j,tn-ν,φp,s,t)
which can be obtained from (27) by multiplying both sides by φp,s,t and using Lt(0)φp,s,t=(2πp+t)nφp,s,t.

Lemma 2.

Let Ψk,j,t(x) be normalized eigenfunction of Lt(P). Then
supx∈0,1]|Ψk,j,t(ν)(x)|=O(kν)
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 Ψk,j,t corresponding to the eigenvalue λk,j(t) has the form
Ψk,j,t(x)=Y1(x,ρk,j)a1+Y2(x,ρk,j)a2+⋯+Yn(x,ρk,j)an,
where ak∈ℂm, ρk,j(t)=i(λk,j(t))1/n, Ys(x,ρk,j(t)) for s=1,2,…,n are linearly independent m×m matrix solutions of (4) for λ=λk,j(t) satisfying
dνYs(x,ρk,j(t))dxν=(ρk,j(t))νeρk,j(t)ωsx[ωsνI+O(1k)]
for ν=0,1,…,(n-1). Here I is unit matrix, ω1,ω2,…,ωn are the nth root of 1, and O(1/k) is an m×m matrix satisfying the following conditions:
O(1k)=A(x,t,k)k,|A(x,t,k)|<c3|k|,∀x∈[0,1],∀t∈[-π2,3π2),
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 (λk,j(t))1/n the root of λk,j(t) lying in O(k-1/2m) neighborhood of (2kπ+t) and put ρk,j(t)=i(λk,j(t))1/n. Then we have
ρk,j(t)=(2kπ+t)i+O(k-1/2m).
Suppose ω1,ω2,…,ωn are ordered in such a way that
ωr=1,ℛ(ρk,j(t)ωs)<0,∀s<r,ℛ(ρk,j(t)ωs)>0,∀s>r,
where ℛ(z) is the real part of z. Using (31), (34), (2), and (33), we get
Ys(ν-1)(1,ρk,j(t))=(ρk,j(t))ν-1eρk,j(t)ωs[ωsν-1I],Ys(ν-1)(0,ρk,j(t))=(ρk,j(t))ν-1[ωsν-1I],Uν(Ys(x,ρk,j(t)))=-(ρk,j(t))ν-1eit[ωsν-1I],∀s<r,Uν(Ys(x,ρk,j(t)))=(ρk,j(t))ν-1eρk,j(t)ωs[ωsν-1I],∀s>r,Uν(Yr(x,ρk,j(t)))=(ρk,j(t))ν-1O(k-1/2m),
where [ωsν-1I]=ωsν-1I+O(1/k) and O(1/k) satisfies the relation (32). Now using these relations and the notations of (30), we prove that
Ys(x,ρk,j(t))as=O((|ar|)k-1/2m),∀s≠r.
Since Ψk,j,t(x) satisfies (2) and (30), we have the system of equations
∑s≠rUν(Ys(x,ρk,j(t)))as=-Uν(Yr(x,ρk,j(t)))ar,ν=0,1,…,(n-2)
with respect to as,q for s≠r and q=1,2,…,m, where as,q are coordinates of the vector as. 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 (ρk,j(t))ν, we get the system of equations whose coefficient matrix A is
(-eit[I]⋯-eit[I]eρk,jωr+1[I]⋯eρk,jωn[I]-eit[ω1I]⋯-eit[ωr-1I]eρk,jωr+1[ωr+1I]⋯eρk,jωn[ωnI]⋯⋯⋯⋯⋯⋯-eit[ω1n-2I]⋯-eit[ωr-1n-2I]eρk,jωr+1[ωr+1n-2I]⋯eρk,jωn[ωnn-2I]),
and the right-hand side is O((|ar|)k-1/2m). To estimate detA let us denote by Ã(m) the matrix obtained from A by replacing [ωsjI] with ωsjI 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 -eit and by eρk,jωs, respectively. Clearly,
Ã(1)=(1⋯11⋯1ω1⋯ωr-1ωr+1⋯ωn⋯⋯⋯⋯⋯⋯ω1n-2⋯ωr-1n-2ωr+1n-2⋯ωnn-2)
and detÃ(1)≠0. Besides, interchanging the rows and then interchanging the columns of Ã(m), we obtain detÃ(m)=(Ã(1))m. Using this and solving (39) by Cramer's rule, we get
as,q=detAs,qdetA=O((|ar|)e-ρk,jωsk-1/2m),∀s>r,
since As,q is obtained from A by replacing the ((s-1)m+q)th column of A, which is the qth column of
(eρk,jωs[I]eρk,jωs[ωkI]⋮eρk,jωs[ωkn-2]),with(O(|ar|k-1/2m)O(|ar|k-1/2m)⋮O(|as|k-1/2m)).
In the same way, we obtain
as,q=O((|ar|)k-1/2m),∀s<r.
Now (38) follows from (44), (42), and (34). Therefore, the normalization condition ∥Ψk,j,t∥=1, and (38), (30), (31), (33), and (34) imply that
Ψk,j,t(x)=(Yr(x,ρk,j(t)))ar+O(k-1/2m)=ei(2kπ+t)xar+O(k-1/2m),
where |ar|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,…,ωn of 1 are ordered in such a way that
ωr=1,ωr+1=-1,ℛ(ρk,jωs)<0,∀s<r;ℛ(ρk,jωs)>0,∀s>r+1.
Hence we have
Uν(Ys(x,ρk,j(t)))=-(ρk,j(t))ν-1eit[ωsv-1I],∀s<r,Uν(Ys(x,ρk,j(t)))=(ρk,j(t))ν-1eρk,j(t)ωs[ωsv-1I],∀s>r+1.
Now using these equalities, we prove that
Ys(x,ρk,j(t))as=O((|ar|+|ar+1|)k-1/2m),∀s≠r,r+1.
Using (47) and arguing as in the case n=2r-1, we get the system of equations
∑s≠r,r+1Uν(Ys(x,ρk,j(t)))as=-∑s=r,r+1Uν(Ys(x,ρk,j(t)))as
for ν=0,1,2,…,(n-3). Arguing as in the proof of (42)–(45) and using (46), we get
as,q=O((|ar|+|ar+1|)e-ρk,jωsk-1/2m),∀s>r+1,as,q=O((|ar|+|ar+1|)k-1/2m),∀s<r,Ψk,j,t(x)=ei(2kπ+t)xar+e-i(2kπ+t)xar+1+O(k-1/2m),
where |ar|2+|ar+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
(PνΨk,j,tn-ν,φp,s,t)=O(kn-ν),(PνΨk,j,tn-ν,Φp,s,t)=O(kn-ν)
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
|(Ψk,j,t,φp,s,t)|≤c4|k|n-2|λk,j(t)-(2πp+t)n|
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 bs,q(x) be the entries of P2(x) and bs,q,p=∫01bs,q(x)e-2πipxdx. Then
(Ψk,j,t(n-2),P2φp,s,t)=∑q=1,2,…m;l∈ℤbs,q,p-l(Ψk,j,t(n-2),φl,q,t),(Ψk,j,t(n-2),(P2-C)Φp,s,t)=O(kn-3ln|k|)+O(kn-2bk)
for p∈A(k,n,t) and s=1,2,…,m, where
bk=max{|bi,j,p|:i,j=1,2,…,m;p=2k,-2k,2k+1,-2k-1},
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
(Ψk,j,t(n-2),(P2-C)Φk,s,t)=O(kn-3ln|k|)
holds. If n=2r-1, then (56) is uniform with respect to t in [-π/2,3π/2).

Proof.

Note that if the entries bs,q of P2 belong to L2[0,1], then (53) is obvious, since {φl,q,t:l∈ℤ,q=1,2,…,m} is an orthonormal basis in L2m[0,1]. Now we prove (53) in case bs,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
|(Ψk,j,t(n-2),φl,q,t)|=|(2πl+t)n-2(Ψk,j,t,φl,q,t)|≤c5|k|n-2|l|n-2|λk,j(t)-(2πl+t)n|,
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|(Ψk,j,t(n-2),φl,q,t)|<c6|k|n-2d,
where d≥2|k|, t∈[-π/2,3π/2). Hence the decomposition of Ψk,j,t(n-2) by the basis {φl,q,t:l∈ℤ,q=1,2,…,m} has the form

Ψk,j,t(n-2)(x)=∑|l|≤d;q=1,2,…,m(Ψk,j,t(n-2),φl,q,t)φl,q,t(x)+gd(x),wheresupx∈0,1]|gd(x)|<c6|k|n-2d.
Using (59) in (Ψk,j,t(n-2),P2φp,s,t) and letting d tend to ∞, we obtain (53).

Since Φp,s,t(x)≡vsei(2πp+t)x, to prove (54), it is enough to show that
(Ψk,j,t(n-2),(P2-C)φp,s,t)=O(kn-3ln|k|)+O(kn-2bk)
for s=1,2,…,m and p∈A(k,n,t). Using the obvious relation
(Ψk,j,t(n-2),Cφp,s,t)=∑q=1,2,…,mbs,q,0(Ψk,j,t(n-2),φp,q,t)
and (53), we see that
(Ψk,j,t(n-2),(P2-C)φp,s,t)=∑l:l∈A(k,n,t)∖p;q=1,2,…,mbs,q,p-l(Ψk,j,t(n-2),φl,q,t)+∑l:l∉A(k,n,t);q=1,2,…,mbs,q,p-l(Ψk,j,t(n-2),φl,q,t).
Since
|bj,i,s|≤maxp,q=1,2,…,m∫01|bp,q(x)|dx=O(1)
for all j, i, s, using (57) and (12), we see that the second summation of the right-hand side of (62) is O((kn-3ln|k|). Besides, it follows from (29) and (55) that the first summation of the right-hand side of (62) is O(kn-2bk), 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.

If C is Hermitian matrix, then for each eigenfunction Ψk,j,t of Lt(P) there exists an eigenfunction Φp,s,t of Lt(C) satisfying
|(Ψk,j,t,Φp,s,t)|>13m.

If Lt(P) is self-adjoint operator, then for each eigenfunction Φk,j,t of Lt(C) there exists an eigenfunction Ψp,s,t of Lt(P) satisfying
|(Φk,j,t,Ψp,s,t)|>13m.

Proof.

It follows from (52) and (13) that
∑s=1,2,…,m(∑p:p∉A(k,n,t)|(Ψk,j,t,φp,s,t)|2)=O((ln|k|)2k2).
Hence using the equality Φp,s,t(x)=vsei(2πp+t)x, where vs is the normalized eigenvectors of C, and the Parseval equality, we get
∑s=1,2,…,m(∑p:p∉A(k,n,t)|(Ψk,j,t,Φp,s,t)|2)=O((ln|k|)2k2),∑s=1,2,…,m;p∈A(k,n,t)|(Ψk,j,t,Φp,s,t)|2=1+O((ln|k|)2k2).
Since the number of the eigenfunctions Φp,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)|(φk,j,t,Ψp,s,t)|2)=O((ln|k|)2k2).
Therefore, arguing as in the proof of (64) and taking into account that the eigenfunctions of the self-adjoint operator Lt(P) form an orthonormal basis in L2m(0,1), we get the proof of (65).

Theorem 5.

Let Lt(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 Lt(P) consist of m sequences (6) satisfying
λk,j(t)=(2πk+t)n+μj(2πk+t)n-2+O(kn-3ln|k|),
and the normalized eigenfunction Ψk,j,t corresponding to λk,j(t) satisfies
∥Ψk,j,t-EΨk,j,t∥=O((ln|k|)k)
for j=1,2,…,m, where μ1≤μ2≤⋯≤μm are the eigenvalues of C and E is the orthogonal projection onto the eigenspace of Lt(C) corresponding to μk,j(t). If μj is a simple eigenvalue of C, then the eigenvalue λk,j(t) satisfying (70) is a simple eigenvalue, and the corresponding eigenfunction satisfies
Ψk,j,t(x)=vjei(2πk+t)x+O((ln|k|)k),
where vj is the eigenvector of C corresponding to the eigenvalue μj. 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(kn-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 (Ψk,j,t,Φp,s,t), where p∈A(k,n,t), and hence p=k, and using (64), we get
{λk,1(t),λk,2(t),…,λk,m(t)}⊂⋃j=1m(U(μk,j(t),δk)),
where U(μ,δ)={z∈ℝ:|μ-z|<δ}, |k|≥max{N,N0}, δk=O(|k|n-3ln|k|). Instead of (64) using (65), in the same way, we obtain
U(μk,s(t),δk)∩{λk,1(t),λk,2(t),…,λk,m(t)}≠∅
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 μj is q then there exist precisely q eigenvalues of Lt(P) lying in U(μk,j(t),δk) for |k|≥max{N,N0}. The eigenvalues of Lt(P) and Lt(C) can be numbered in the following way: λk,1(t)≤λk,2(t)≤⋯≤λk,m(t) and μk,1(t)≤μk,2(t)≤⋯≤μk,m(t). If C has ν different eigenvalues μj1,μj2,…,μjν with multiplicities j1,j2-j1,…,jν-jν-1, then we have
j1<j2<⋯<jν=m,μj1<μj2<⋯<μjν,μ1=μ2=⋯=μj1,μj1+1=μj1+2=⋯=μj2,…,μjν-1+1=μjν-2+2=⋯=μjν.
Suppose there exist precisely s1,s2,…,sν eigenvalues of Lt(P) lying in the intervals
U(μk,j1(t),δk),U(μk,j2(t),δk),…,U(μk,jν(t),δk),
respectively. Since
δk≪(minp=1,2,…,ν-1|(μjp+1-μjp)(2πk+t)n-2|)for|k|≫1,
these intervals are pairwise disjoints. Therefore using (6) and (7), we get
s1+s2+⋯+sν=m.
Now let us prove that s1=j1,s2=j2-j1,…,sν=jν-jν-1. Due to the notations the eigenvalues λk,1(t),λk,2(t),…,λk,s1(t) of the operator Lt(P) lie in U(μk,1(t),δk) and by the definition of δk we have
|λk,j(t)-μk,s(t)|>12(minp:p>j1|(μ1-μp)(2πk+t)n-2|)
for j≤s1 and s>j1. Hence using (26) for p=k and (56), (51), we get
∑s:s>j1|(Ψk,j,t,Φk,s,t)|2=O((ln|k|)2k2),∀j≤s1.
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 Φk,j,t(x), of Lt(P) corresponding to μk,1(t)=μk,2(t)=⋯=μk,j1(t) such that
Ψk,j,t(x)=Φk,j,t(x)+O(k-1ln|k|)
for j≤s1. Since Ψk,1,t,Ψk,2,t,…,Ψk,s1,t are orthonormal system we have
(Φk,j,t,Φk,s,t)=δs,j+O(k-1ln|k|),∀s,j=1,2,…,s1.
This formula implies that the dimension j1 of the eigenspace of Lt(C) corresponding to the eigenvalue μk,1(t) is not less than s1. Thus s1≤j1. In the same way we prove that s2≤j2-j1,…,sν≤jν-jν-1. Now (78) and the equality jν=m (see (75)) imply that s1=j1,s2=j2-j1,…,sν=jν-jν-1. Therefore, taking into account that, the eigenvalues of Lt(P) consist of m sequences satisfying (7), we get (70). The proof of (71) follows from (81).

Now suppose that μj is a simple eigenvalue of C. Then μk,j(t) is a simple eigenvalues of Lt(C) and, as it was proved above, there exists unique eigenvalues λk,j(t) of Lt(P) lying in U(μk,s(t),δk), where |k|≥max{N,N0}, and the eigenvalues λk,j(t) for |k|≥max{N,N0} are the simple eigenvalues. Hence (72) is the consequence of (71), since there exists unique eigenfunction Φk,j,t(x)=vjei(2πk+t)x corresponding to the eigenvalue μk,j(t). The uniformity of the formulas (70)–(72) follows from the uniformity of (56), (51), (64), and (65).

Theorem 6.

Let Lt(P) be a self-adjoint operator, let C be a Hermitian matrix, let n=2r, μj be a simple eigenvalue of C, let αj be a positive constant satisfying αj<minq:q≠j|μj-μq|, and let B(αj,k,μj) be a set defined by B(αj,k,μj)=B(0,αj,k,μj)∪B(π,αj,k,μj), where
B(0,αj,k,μj)=⋃s=1,2…,m(μs-μj-αj4nπk,μs-μj+αj4nπk),B(π,αj,k,μj)=⋃s=1,2…,m(π+μs-μj-αj2nπ(2k+n-1),π+μs-μj+αj2nπ(2k+n-1)).
There exist a positive number N1 such that if |k|≥N1 and t∉B(αj,k,μj), then there exists a unique eigenvalue, denoted by λk,j(t), of Lt(P) lying in U(μk,j,εk), where εk=c7(|k|n-3ln|k|)+|k|n-2bk, bk is defined by (55), and c7 is a positive constant, independent of t. The eigenvalue λk,j(t) is a simple eigenvalue of Lt(P) and the corresponding normalized eigenfunction Ψk,j,t(x) satisfies
Ψk,j,t(x)=vjei(2πk+t)x+O(k-1ln|k|)+O(bk).

Proof.

To consider the simplicity of μk,j(t) and λk,j(t) we introduce the set
S(k,j,p,s)={t∈[-π2,3π2):|μk,j(t)-μp,s(t)|<αj|k|n-2}
for (p,s)≠(k,j). It follows from (10) that S(k,j,p,s)=∅ for p≠k,-k,-k-1. Moreover, if μj is a simple eigenvalue, then S(k,j,k,s)=∅ for s≠j, since
|μk,j(t)-μk,s(t)|=|(μj-μs)(2πk+t)n-2|>αj|k|n-2.
It remains to consider the sets S(k,j,-k,s), S(k,j,-k-1,s). Using the equality μk,j(t)-μ-k,s(t)=(2πk)n-2(4nkπt+μj-μs)+O(kn-3), we see that
S(k,j,-k,s)⊂(μs-μj-αj4nπk,μs-μj+αj4nπk).
Similarly, by using the obvious equality
μk,j(t)-μ-k-1,s(t)=(2πk+t)n+μj(2πk+t)n-2-(2πk+2π-t)n-μj(2πk+2π-t)n-2=(2πk)n-2(n2πkt-n2πk(2π-t)+12n(n-1)(t2-(2π-t)2)+μj-μs)+O(kn-3)=(2πk)n-2[(t-π)(2k+(n-1))2πn+μj-μs]+O(kn-3),
we get
S(k,j,-k-1,s)⊂(π+μs-μj-αj2nπ(2k+n-1),π+μs-μj+αj2nπ(2k+n-1)).
Using these relations and the definition of B(αj,k,μj), we obtain
⋃p∈ℤ,s=1,2,…,m,(p,s)≠(k,j)S(k,j,p,s)=⋃p=-k,-k-1,s=1,2,…,mS(k,j,p,s)⊂B(αj,k,μj).
Therefore it follows from (85) that if t∉B(αj,k,μj), then
|μk,j(t)-μp,s(t)|≥αj|k|n-2
for all (p,s)≠(k,j). Hence μk,j(t) is a simple eigenvalue of Lt(C) for t∉B(αj,k,μj). 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 λk,j(t), of Lt(P) lying in U(μk,j(t),εk). Now using the definition of εk and then (91), we see that
|λk,j(t)-μk,j(t)|<εk=o(kn-2),|λk,j(t)-μp,s(t)|>12αj|k|n-2
for |k|≥N1, s=1,2,…,m, (p,s)≠(k,j) and for any eigenvalue λk,j(t) lying in U(μk,j(t),εk). Let Ψk,j,t(x) be any normalized eigenfunction corresponding to λk,j(t). Dividing both sides of (26) by λk,j(t)-μp,s(t) and using (54), (51), and (92), we get
(Ψk,j,t,Φp,s,t)=O(k-1ln|k|)+O(bk)
for (p,s)≠(k,j) and p∈A(k,n,t). This, (67) and (68) imply that Ψk,j,t(x) satisfies (84). Thus we have proved that (84) holds for any normalized eigenfunction of Lt(P) corresponding to any eigenvalue lying in U(μk,j(t),εk). If there exist two different eigenvalues of Lt(P) lying in U(μk,j(t),εk) or if there exists a multiple eigenvalue of Lt(P) lying in U(μk,j(t),εk), then we obtain that there exist two orthonormal eigenfunctions satisfying (84) which is impossible. Therefore there exists unique eigenvalue λk,j(t) of Lt(P) lying in U(μk,j(t),εk) and λk,j(t) is a simple eigenvalue of Lt(P).

Theorem 7.

Let L(P) be self-adjoint operator generated in L2m(-∞,∞) by the differential expression (1), and let C be Hermitian matrix.

If n and m are odd numbers then the spectrum σ(L(P)) of L(P) coincides with (-∞,∞).

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.

Suppose that n is even number, and the matrix C has at least three simple eigenvalues μj1<μj2<μj3 such that diam({μj1+μi1,μj2+μi2,μj3+μi3})≠0 for each triple (i1,i2,i3), where ip=1,2,…,m for p=1,2,3 and diam(A) is the diameter supx,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(Uν(Ys(x,λ))) of Lt(P) has the form
Δ(λ,t)=einmt+a1(λ)ei(nm-1)t+a2(λ)ei(nm-2)t+⋯+anm(λ),
that is, Δ(λ,t) is a polynomial Sλ(u) of u=eit 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=eit1,u2=eit2,…,unm=eitnm of Sλ(u)=0 differ from 1, that is, tk≠tk¯ and λ is the eigenvalue of Ltk(P) for k=1,2,…,nm. It is not hard to see that Ltk*=Ltk¯, λ=λ¯∈σ(Ltk¯). Moreover, if λ is the eigenvalue of Ltk(P) of multiplicity mk then λ¯ is the eigenvalue of Ltk¯(P) of the same multiplicity mk. Now taking into account that uk=eitk is the root of Sλ(u)=0 of multiplicity mk if and only if λ is the eigenvalue of Ltk(P) of multiplicity mk, we obtain that eitk¯ is also root of Sλ(u)=0 of the same multiplicity mk. Since eitk¯≠eitk, we see that the number nm of the roots of Sλ(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 μj is a simple eigenvalue of the matrix C then there exists unique simple eigenvalue λk,j(t) of Lt(P) lying in U(μk,j(t),δk), where δk=c8|k|n-3ln|k| and t∈[-π/2,3π/2). Therefore λk,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)=λk,j(t0), Λ(t) is an eigenvalue of Lt(P) for t∈U(t0) and |Λ(t)-μk,j(t)|<δk, for all t∈U(t0), since |Λ(t0)-μk,j(t0)|=|λk,j(t0)-μk,j(t0)|<δk and the functions Λ(t), μk,j(t) are continuous. Now taking into account that there exists unique eigenvalue of Lt(P) lying in U(μk,j(t),δk), we obtain that Λ(t)=λk,j(t) for t∈U(t0), and hence λk,j(t) is continuous at t0∈(-π/2,3π/2). Therefore the sets Γk,j={λk,j(t):t∈(-π/2,3π/2)} for |k|≥N2 are intervals and Γk,j⊂σ(L(P)). Similarly there exists a neighborhood
U(-π2)=(-π2-β,-π2+β)
of -π/2 and a continuous function A(t) such that A(-π/2)=λk+1,j(-π/2), where k≥N2, A(t) is an eigenvalue of Lt+2π(P)(Lt(P)=Lt+2π(P)) for t∈(-π/2-β,-π/2] and A(t) is an eigenvalue of Lt(P) for t∈[-π/2,-π/2+β) and
|A(t)-μk+1,j(t)|<δk∀t∈[-π2,-π2+β),|A(t)-μk,j(t+2π)|<δk∀t∈(-π2-β,-π2),
since |A(-π/2)-μk+1,j(-π/2)|=|λk+1,j(-π/2)-μk+1,j(-π/2)|<δk, μk,j(t+2π)=μk+1,j(t) and the functions Λ(t), μk,j(t) are continuous. Again taking into account that there exists unique eigenvalue of Lt(P) lying in U(μk+1,j(t),δk) for t∈[-π/2,-π/2+β) and lying in U(μk,j(t),δk) for t∈(3π/2-β,3π/2), we obtain that
A(t)=λk+1,j(t),∀t∈[-π2,-π2+β),A(t)=λk,j(t+2π),∀t∈(-π2-β,-π2).
Thus one part of the interval {A(t):t∈(-π/2-β,-π/2+β)} lies in Γk,j and the other part lies in Γk+1,j, that is, the interval Γk,j and Γk+1,j are connected for k≥N2. Similarly the interval Γk,j and Γk-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(αjp,k,μjp), where p=1,2,3, then there exists a unique eigenvalue, denoted by λk,jp(t), of Lt(P) lying in U(μk,jp(t),εk) and it is a simple eigenvalue. Let us prove that λk,jp(t) is continuous at
t0∈[-π2,3π2)∖B(αjp,k,μjp).
Since λk,jp(t0) is a simple eigenvalue it is a simple zero of the characteristic determinant Δ(λ,t) of the operator Lt(P). Therefore repeating the argument of the proof of the continuity of λk,j(t) in the proof of (b), we obtain that λk,jp(t) is continuous at t0 for |k|≥N1. Now we prove that there exists H such that
(H,∞)⊂{λk,jp(t):t∈[-π2,3π2)∖B(αjp,k,μjp),k=N1,N1+1,…}.
It is clear that
(h,∞)⊂{μk,jp(t):t∈[-π2,3π2),k=N1,N1+1,…},∀p=1,2,3,
where h=μN1,j3(-π/2). Since μk,jp(t) is increasing function for k≥N1, it follows from the obvious equality
μk,jp(μs-μjp∓αjp4nπk)=(2πk)n+n(2πk)n-1μs-μjp∓αjp4nπk+μjp(2πk)n-2+O(kn-4)=(2πk)n+(2πk)n-2μs+μjp∓αjp2+O(kn-4)
and from the definition of B(0,αjp,k,μjp) that
{μk,jp(t):t∈B(0,αjp,k,μjp)}⊂⋃s=1,2,…,mC(0,k,jp,s,αjp),
where C(0,k,jp,s,αjp)={x∈ℝ:|x-(2πk)n+(2πk)n-2((μs+μjp)/2)|<αjp(2πk)n-2}. This inclusion with (100) implies that the set
(h,∞)∖⋃k:k≥N1;s=1,2,…,mC(0,k,jp,s,αjp)
is a subset of the set {μk,jp(t):t∈[-π/2,3π/2)∖B(0,αjp,k,μjp),k≥N1}. Similarly, using
μk,jp(π+μs-μjp∓αjp4nπk)=(2πk+π)n+(2πk)n-2μs+μjp∓αjp2+O(kn-3),
which can be proved by direct calculations, we obtain that the set
(h,∞)∖⋃k:k≥N1;s=1,2,…,mC(π,k,jp,s,αjp),
where C(π,k,jp,s,αjp)={x∈ℝ:|x-(2πk+π)n+(2πk)n-2((μs+μjp)/2)|<αjp(2πk)n-2}, is a subset of
{μk,jp(t):t∈[-π2,3π2)∖B(π,αjp,k,μjp),k≥N1}.
Now using (92) and the continuity of λk,jp(t) on [-π/2,3π/2)∖B(αjp,k,μjp), we see that the set
(H,∞)∖(⋃k:k≥N1;s=1,2,…,mC(k,jp,s,2αjp)),
where H=h+1, C(k,jp,s,2αjp)=C(0,k,jp,s,2αjp)∪C(π,k,jp,s,2αjp), is a subset of the set {λk,jp(t):t∈[-π/2,3π/2)∖B(αjp,k,μjp),k≥N1}. Thus we have
⋃p=1,2,3((H,∞)∖(⋃k≥N1;s=1,2,…,mC(k,jp,s,2αjp)))⊂σ(L(P)).
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,…,mC(k,jp,s,2αjp))
is empty. If this set contains an element x, then
x∈⋃k≥N1;s=1,2,…,mC(k,jp,s,αjp)
for all p=1,2,3. Using this and the definition of C(k,jp,s,αjp), we obtain that there exist k≥N1; ν=0,1 and s=ip such that
|x-(π(2k+ν))n-μjp+μip2(2πk)n-2|<2αjp(2πk)n-2
for all p=1,2,3 and hence
|μjq+μiq2-μjp+μip2|<4αjp
for all p,q=1,2,3. Clearly, the constant αjp can be chosen so that
8αjp<mini1,i2,i3(diam({μj1+μi1,μj2+μi2,μj3+μi3})),
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).

EasthamM. S. P.GelfandI. M.Expansion in characteristic functions of an equation with periodic coefficientsMaksudovF. G.VelievO. A.Spectral analysis of differential operators with periodic matrix coefficientsMcGarveyD. C.Operators commuting with translation by one. II. Differential operators with periodic coefficients in Lp(−∞,∞)BirkhoffG. D.Boundary value and expansion problems of ordinary linear differential equationsTamarkinJ. D.NaimarkM. A.MarchenkoV. A.DunfordN.SchwartzJ. T.YakubovichV. A.StarzhinskiV. M.VelievO. A.On the spectrum of the Schrödinger operator with periodic potentialVelievO. A.On Hill's operator with a matrix potentialChelkakD.KorotyaevE.Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line