On the spectrum of the differential operators of even order with periodic matrix coefficients

TL;DR

This paper investigates the spectral properties of high-order self-adjoint differential operators with periodic matrix coefficients, identifying conditions that lead to a finite number of spectral gaps.

Contribution

It provides new criteria for the coefficients of such operators to ensure a finite number of spectral gaps, advancing understanding of their spectral structure.

Findings

Conditions for finite spectral gaps are established.

Analysis of band and Bloch functions for the operator.

Spectral spectrum structure characterized under specific coefficient conditions.

Abstract

In this paper, we consider the band functions, Bloch functions and spectrum of the self-adjoint differential operator L with periodic matrix coefficients. Conditions are found for the coefficients under which the number of gaps in the spectrum of the operator L is finite

Equations

$$(-i{)}^{2\nu }{y}^{(2\nu )}(x)+\sum _{k=2}^{2\nu -2}{P}_k(x)y^{(2\nu -k)}(x),$$

$$U_p(y):=y^{(p)}\left(1\right)-e^{it}{y}^{(p)}\left(0\right)=0,p=0,1,...,(2\nu -1).$$

$${\lambda }_1(t)\le {\lambda }_2(t)\le \cdot \cdot \cdot$$

$$I_n=\left\{{\lambda }_n(t):t\in (-\pi ,\pi ]\right\},$$

$$C={\int }_0^1{P}_2\left(x\right)dx$$

$$\Delta (\lambda ,t)=\det (Y_j^{(p-1)}(1,\lambda )-e^{it}{Y}_j^{(p-1)}(0,\lambda ){)}_{j,p=1}^{2\nu }=$$

$$e^{i2\nu mt}+f_1(\lambda )e^{i(2\nu m-1)t}+f_2(\lambda )e^{i(2\nu m-2)t}+...+f_{2\nu m-1}(\lambda )e^{it}+1$$

$$(-i{)}^{2\nu }{Y}^{(2\nu )}+P_2{Y}^{(2\nu -2)}+P_3{Y}^{(2\nu -3)}+...+P_{2\nu }Y=\lambda Y,$$

$$G(x,\xi ,\lambda ,t)=g(x,\xi ,\lambda )-\frac{1}{\Delta (\lambda ,t)}\sum _{j,p=1}^{2\nu }{Y}_j(x,\lambda )V_{jp}(x,\lambda )U_p(g),$$

$$r<\frac{1}{2}\underset{j=1,2,...p}{\min }\left({\Lambda }_{j+1}(a)-{\Lambda }_j(a)\right)$$

$$\left|\Delta (\lambda ,t)\right|>c/2$$

$${\left(L_t-\lambda I\right)}^{-1}f(x)={\int }_0^{1}G(x,\xi ,\lambda ,t)f(\xi )d\xi ,$$

$$\left|G(x,\xi ,\lambda ,t)-G(x,\xi ,\lambda ,a)\right|\le M\left|t-a\right|$$

$$P_t={\int }_{D_j(a)}{\left(L_t-\lambda I\right)}^{-1}d\lambda$$

$${\Vert f\Vert }_{\infty }={\sup }_{x\in [0,1]}\left|f(x)\right|.$$

$$\left|P_{t}f(x)-P_{a}f(x)\right|=\left|{\int }_{D_j(a)}{\int }_0^1\left(G(x,\xi ,\lambda ,t)-G(x,\xi ,\lambda ,a)\right)f(\xi )d\xi d\lambda \right|\le$$

$$2\pi rM\left|t-a\right|{\int }_0^1\left|f(\xi )\right|d\xi$$

$$u(x+\omega )=e^{i⟨t,\omega ⟩}u(x),\forall \omega \in \Omega ,$$

$$\Vert {\Psi }_{n,t}-{\Psi }_{n,a}\Vert \to 0$$

$$\arg ({\Psi }_{n,t},{\Psi }_{n,a})=0$$

$${\int }_D(T_t-\lambda I{)}^{-1}{e}^{i⟨a,x⟩}{e}_{k}d\lambda =(e^{i⟨a,x⟩}{e}_k,e^{i\alpha }{\Psi }_{n,t})e^{i\alpha }{\Psi }_{n,t}=(e^{i⟨a,x⟩}{e}_k,{\Psi }_{n,t}){\Psi }_{n,t}$$

$$\left|\left|({\Psi }_{n,t},e^{i⟨t,x⟩}{e}_k)\right|-\left|({\Psi }_{n,t},e^{i⟨a,x⟩}{e}_k)\right|\right|\le \Vert e^{i⟨t,x⟩}-e^{i⟨a,x⟩}\Vert$$

$$\left|({\Psi }_{n,a},e^{i⟨\gamma +a,x⟩}{e}_k)\right|>\epsilon .$$

$$\left|({\Psi }_{n,t},e^{i⟨t,x⟩}{e}_k)\right|>\epsilon$$

$$\arg ({\Psi }_{n,t},e^{i⟨t,x⟩}{e}_k)=0.$$

$$\frac{({\Psi }_{n,t},e^{i⟨a,x⟩}{e}_k)}{({\Psi }_{n,a},e^{i⟨a,x⟩}{e}_k)}=:\alpha (t)\to 1$$

$$\Vert (e^{i⟨a,x⟩}{e}_k,{\Psi }_{n,t}){\Psi }_{n,t}-(e^{i⟨a,x⟩}{e}_k,{\Psi }_{n,a}){\Psi }_{n,a}\Vert \to 0$$

$$\Vert {\Psi }_{n,t}-{\Psi }_{n,a}\Vert \le \Vert (1-\alpha (t)){\Psi }_{n,t}\Vert +\Vert \alpha (t)){\Psi }_{n,t}-{\Psi }_{n,a}\Vert \to 0$$

$${\left|({\Psi }_{n,t},e^{i⟨t,x⟩})\right|}^2=1+O({\rho }^{-\delta })>\frac{1}{2}$$

$${\Vert (e^{iax}{e}_k,{\Psi }_{n,t}){\Psi }_{n,t}-(e^{iax}{e}_k,{\Psi }_{n,a}){\Psi }_{n,a}\Vert }_{\infty }\to 0$$

Full text

Key Words: Band functions, Bloch functions, Spectrum.

AMS Mathematics Subject Classification: 34L05, 34L20.

In this paper, we investigate the band functions, Bloch functions and spectrum of the differential operator $L,$ generated in the space $L_{2}^{m}(\mathbb{R})$ of vector-valued functions by formally self-adjoint differential expression

[TABLE]

where $\nu>1$ and $P_{k}\left(x\right),$ for each $k=2,3,...2\nu-2,$ is the $m\times m$ matrix with the summable entries $p_{k,i,j}$ satisfying $p_{k,i,j}\left(x+1\right)=p_{k,i,j}\left(x\right)$ for all $i$ $=1,2,...m$ and $j$ $=1,2,...m.$

To explain the results of this paper, let us introduce some notations. It is well-known that (see [1, 2, 4]) the spectrum $\sigma(L)$ of the operator $L$ is the union of the spectra $\sigma(L_{t})$ of the operators $L_{t},$ for $t\in(-\pi,\pi],$ generated in $L_{2}^{m}[0,1]$ by (1) and the quasiperiodic conditions

[TABLE]

For $t\in(-\pi,\pi]$ the spectra $\sigma(L_{t})$ of the operators $L_{t}$ consist of the eigenvalues

[TABLE]

called the Bloch eigenvalues of $L$. The eigenfunctions $\Psi_{n,t}$ corresponding to the Bloch eigenvalues $\lambda_{n}(t)$ are the Bloch functions of $L$.

In [10] the continuity of the band function $\lambda_{n}:t\rightarrow\lambda_{n}(t)$ and Bloch function of the operator $L$ was investigated. In Section 2, we improve these results as follows. We only assume that the entries of the coefficients of (1) are summable function, while in [10] it was assumed that they are bounded functions. In [10] the choice of the continuous Bloch functions was made in a non-constructive way. Here we constructively define the continuous Bloch functions. We prove that for each $f\in L_{2}^{m}[0,1]$ the function $P_{t}f(x)$ converges to $P_{a}f$ $(x)$ uniformly with respect to $x\in[0,1]$ as $t\rightarrow a$, while in [10] this convergence was done in the $L_{2}$ norm, where $P_{t}$ is the projection of $L_{t}$ corresponding to the eigenvalue $\lambda_{n}(t).$ Moreover, the methods used in Section 2 and [10] are completely different. Therefore, Section 2 can be considered as a continuation and completion of the paper [10].

In Section 3, we consider the spectrum of the operator $L.$ Since $\sigma(L)$ is the union of $\sigma(L_{t})$ for $t\in(-\pi,\pi],$  the spectrum of $L$ consists of the sets

[TABLE]

for $n=1,2,....$ The set $I_{n}$ is called the $n$th band of the spectrum. The band $I_{n}$ tends to infinity as $n\rightarrow\infty.$ The spaces between the bands $I_{k}$ and $I_{k+1}$, for $k=1,2,...,$ are called the gaps in the spectrum of $L.$ In Section 3, we prove that most of the positive real axis is overlapped by $m$ bands of the spectrum and consider the gaps (see Theorems 5). Then we find a condition on the eigenvalues of the matrix

[TABLE]

for which the number of the gaps in the spectrum is finite (see Theorem 6). Note that in [6], we proved Theorem 6 under the assumption that the matrix $C$ has three simple eigenvalues $\mu_{j_{1}},$ $\mu_{j_{2}}$ and $\mu_{j_{3}}$ satisfying (30). In this paper, we prove Theorem 6 without any conditions on the multiplicity of these eigenvalues. The case $\nu=1$ was investigated in [9]. Parts of the proofs of Theorems 5 and 6, similar to the proofs of the case $\nu=1,$ are omitted and references to [9] are given.

1 On the band functions and Bloch functions

In this section, first, we study the continuity of the band functions and Bloch functions of $L$ with respect to the quasimomentum by using the following well-known statements (see for example [5] Chap. 3) formulated here as summary.

Summary 1

The eigenvalues of $L_{t}$ are the roots of the characteristic determinant

[TABLE]

[TABLE]

which is a polynomial of $e^{it}$ with entire coefficients $f_{1}(\lambda),f_{2}(\lambda),...$, where

$Y_{1}(x,\lambda),Y_{2}(x,\lambda),\ldots,Y_{2\nu}(x,\lambda)$* are the solutions of the matrix equation*

[TABLE]

satisfying $Y_{k}^{(j)}(0,\lambda)=O$ for $j\neq k-1$ and $Y_{k}^{(k-1)}(0,\lambda)=I$. Here, $O$ and $I$ are $m\times m$ zero and identity matrices, respectively. The Green’s function of $L_{t}-\lambda I$ is defined by formula

[TABLE]

where $g$ does not depend on $t$ and $V_{jp}$ is the transpose of that $m$th order matrix consisting of the cofactor of the element $U_{p}(Y_{j})$ in the determinant $\det(U_{p}(Y_{j}))_{j,p=1}^{2\nu}.$ Hence, the entries of the matrices $V_{jp}(x,\lambda)$ and $U_{p}(g)$ either do not depend on $t$ or have the forms $u^{(p)}(1,\lambda)-e^{it}u^{(p)}(0,\lambda)$ and $h(1,\xi,\lambda)-e^{it}h(0,\xi,\lambda)$ respectively, where the functions $u$ and $h$ do not depend on $t.$

Now, using this summary, we prove that for each $n$ the function $\lambda_{n}$ defined in (3) is continuous at each point $a\in(-\pi,\pi].$ For this we introduce the following notations.

Notation 1

Let $\Lambda_{1}(a)<\Lambda_{2}(a)<\cdot\cdot\cdot$ be the distinct eigenvalues of $L_{a}$ with the multiplicities $k_{1},k_{2},...,$ respectively. For each $n$ there exists $p$ such that $n\leq k_{1}+k_{2}+\cdot\cdot\cdot+k_{p}.$ These notations with the notation (3) imply that $\lambda_{s_{j-1}+1}(a)=\lambda_{s_{j-1}+2}(a)=\cdot\cdot\cdot=\lambda_{s_{j}}(a)=\Lambda_{j}(a),$ where $s_{0}=0,$ $s_{j}=k_{1}+k_{2}+\cdot\cdot\cdot+k_{j}$ for $j=1,2,...,p$ and $n\leq s_{p}.$ Since $L$ is below-bounded operator, there exists $b\in\mathbb{R}$ such that $\lambda_{1}(t)>b$ for all $t\in(-\pi,\pi].$

Now, we are ready to prove the following theorem.

Theorem 1

$(a)$* For every $r>0$ satisfying the inequality*

[TABLE]

there exists $\delta>0$ such that the operator $L_{t}$, for $t\in(a-\delta,a+\delta),$ has  $k_{j}$ eigenvalues in the interval $\left(\Lambda_{j}(a)-r,\Lambda_{j}(a)+r\right)$, where $j=1,2,...,p$ and $a\in(-\pi,\pi].$

$(b)$* The eigenvalues of $L_{t}$ for $t\in(a-\delta,a+\delta)$ lying in $\left(\Lambda_{j}(a)-r,\Lambda_{j}(a)+r\right)$ are $\lambda_{s_{j-1}+1}(t),\lambda_{s_{j-1}+2}(t),...,\lambda_{s_{j}}(t),$ where $s_{j}$ is defined in Notation 1.*

Proof. $(a)$ By (8), the circle $D_{j}(a)=\left\{z\in\mathbb{C}:\left|z-\Lambda_{j}(a)\right|=r\right\}$ belongs to the resolvent set of the operator $L_{a}.$ It means that $\Delta(\lambda,a)\neq 0$ for each $\lambda\in D_{j}(a).$ Since $\Delta(\lambda,a)$ is a continuous function on the compact $D_{j}(a),$ there exists $c>0$ such that $\left|\Delta(\lambda,a)\right|>c$ for all $\lambda\in D_{j}(a).$ Moreover, by (6), $\Delta(\lambda,t)$ is a polynomial of $e^{it}$ with entire coefficients. Therefore, there exists $\delta>0$ such that

[TABLE]

for all $t\in(a-\delta,a+\delta)$ and $\lambda\in D_{j}(a).$ It implies that $D_{j}(a)$ belongs to the resolvent set of $L_{t}$ for all $t\in(a-\delta,a+\delta).$ On the other hand, it is well-known that

[TABLE]

where $G(x,\xi,\lambda,t)$ is the Green’s function of $L_{t}$ defined in (7). Moreover, it easily follows from Summary 1 and (9) that there exists $M$ such that

[TABLE]

for all $x\in[0,1],$ $\xi\in[0,1]$, $\lambda\in D_{j}(a)$ and $t\in(a-\delta,a+\delta).$ Therefore, using (10) and Summary 1, one can easily verify that $\left(L_{t}-\lambda I\right)^{-1}$ for $\lambda\in D_{j}(a)$ and the projection

[TABLE]

continuously depend on $t\in(a-\delta,a+\delta).$ This implies that the operator $L_{t}$ for each $t\in(a-\delta,a+\delta)$ has  $k_{j}$ eigenvalues inside $D_{j}(a)$ and, therefore, in the interval $\left(\Lambda_{j}(a)-r,\Lambda_{j}(a)-r\right),$ since $L_{a}$ has  $k_{j}$ eigenvalues (counting multiplicity) inside $D_{j}(a)$.

$(b)$ Since $L_{a}$ has no eigenvalues in the intervals $[b,\Lambda_{1}(a)-r]$ and $\left[\Lambda_{j}(a)+r,\Lambda_{j+1}(a)-r\right]$ for $j=1,2,...,p$, arguing as above we obtain that $L_{t}$ for $t\in(a-\delta,a+\delta)$ also has no eigenvalues in these closed intervals. Therefore, the eigenvalues of $L_{t}$ for $t\in(a-\delta,a+\delta)$ lying in $\left(\Lambda_{j}(a)-r,\Lambda_{j}(a)+r\right)$ are $\lambda_{s_{j-1}+1}(t),\lambda_{s_{j-1}+2}(t),...,\lambda_{s_{j}}(t)$ for $j=1,2,...,p.$

Now, using these statements, we prove the main results of this section.

Theorem 2

$(a)$* For each $n$ the function $\lambda_{n}$ defined in (3) is continuous at $a\in(-\pi,\pi].$*

$(b)$* For each $f\in L_{2}^{m}[0,1]$ we have $\left\|P_{t}f-P_{a}f\right\|_{\infty}\rightarrow 0$ as $t\rightarrow a$, where*

[TABLE]

Proof. $(a)$ Consider any sequence $\left\{\left(\lambda_{n}(t_{k}),t_{k}\right):k\in\mathbb{N}\right\}$ such that $t_{k}\in(a-\delta,a+\delta)$ for all $k\in\mathbb{N}$ and $t_{k}\rightarrow a$ as $k\rightarrow\infty,$ where $\delta$ is defined in Theorem 1. Let $\left(\lambda,a\right)$ be any limit point of the sequence $\left\{\left(\lambda_{n}(t_{k}),t_{k}\right):k\in\mathbb{N}\right\}.$ Since $\Delta$ is a continuous function with respect to the pair $(\lambda,t)$ and $\Delta\left(\lambda_{n}(t_{k}),t_{k}\right)=0$ for all $k,$ we have $\Delta(\lambda,a)=0.$ This means that $\lambda$ is an eigenvalue of $L_{a}$ lying in $\left(\Lambda_{j}(a)-r,\Lambda_{j}(a)+r\right)$. Hence, by Theorem 1$(b),$ we have $\lambda=\lambda_{s_{j-1}+1}(a)=$ $\lambda_{s_{j-1}+2}(a)=...=\lambda_{s_{j}}(a),$ where $n\in[s_{j-1}+1,s_{j}].$ Thus, $\lambda_{n}(t_{k})\rightarrow\lambda_{n}(a)$ as $k\rightarrow\infty$ for any sequence $\left\{t_{k}:k\in\mathbb{N}\right\}$ converging to $a$ and $\lambda_{n}$ is continuous at $a$.

$(b)$ Using (10)-(12) we obtain the following estimation

[TABLE]

[TABLE]

for all $x\in[0,1].$ This estimation implies the proof of $(b).$

Note that in [9], the continuity of the band function for the case $\nu=1$ was proved by using the perturbation theory from [3]. In [10], we investigated the differential operator $T$, generated in the space $L_{2}^{m}(\mathbb{R}^{d})$ by formally self-adjoint differential expression of order $2\nu$ with matrix coefficients, whose entries are periodic with respect to the lattice $\Omega$, where $d\geq 1$. Note that the band functions \lambda_{1}(t)\leq\lambda_{2}(t)\leq\cdot\cdot\cdot\ and Bloch functions $\Psi_{1,t},\Psi_{2,t},...$ of $T$ are the eigenvalues and normalized eigenfunctions of the operator $T_{t}$ generated in $L_{2}^{m}(F)$ by by the same differential expression and the quasiperiodic conditions

[TABLE]

where $t\in F^{\star}$, $\left\langle\cdot,\cdot\right\rangle$ is the inner product in $\mathbb{R}^{d}$, $F$ and $F^{\star}$ are the fundamental domains of the lattice $\Omega$ and dual lattice $\Gamma$, respectively. It was proved, in [10], that the Bloch eigenvalues and corresponding projections of the differential operator $T_{t}$ depend continuously on $t\in F^{\ast}$. Moreover, if $\lambda_{n}(a)$ is a simple eigenvalue, then the eigenvalues $\lambda_{n}(t)$ are simple in some neighborhood of $a$ and the corresponding eigenfunctions $\Psi_{n,t}$ can be chosen so that

[TABLE]

as $t\rightarrow a$. In [10], the Bloch function $\Psi_{n,t}$ was chosen so that

[TABLE]

which is not a constructive choice.

Now, instead of (13), we constructively define the normalized eigenfunctions $\Psi_{n,t}$ that depend continuously on $t.$ First of all, let us note the following obvious statement. If $\lambda_{n}(t)$ is a simple eigenvalue, then the set of all normalized eigenfunctions corresponding to $\lambda_{n}(t)$ is $\left\{e^{i\alpha}\Psi_{n,t}:\alpha\in[0,2\pi)\right\},$ where $\Psi_{n,t}$ is a fixed normalized eigenfunction. If the eigenvalue $\lambda_{n}(a)$ is simple, then there exists a neighborhood $U(a)$ of the point $a\in F^{\star}$ such that for $t\in U(a)$ the eigenvalue $\lambda_{n}(t)$ is also simple and the equality

[TABLE]

is true for any choice of the normalized eigenfunction $\Psi_{n,t},$ where $D$ is a closed curve enclosing only the eigenvalue $\lambda_{n}(t),$ and $e_{1},e_{2},...,e_{m}$ is the standard basis of $\mathbb{C}^{m}.$ Since the projection operator onto the subspace corresponding to the eigenvalue $\lambda_{n}(t)$  depends continuously on $t\in U(a),$ and the norm is a continuous function, it follows from (14) that $\left|(\Psi_{n,t},e^{i\left\langle a,x\right\rangle}e_{k})\right|$ is also a continuous function with respect to $t$ in $U(a)$ for any normalized eigenfunction $\Psi_{n,t}.$ This and the inequality

[TABLE]

give the following obvious statement.

Proposition 1

If $\lambda_{n}(a)$ is a simple eigenvalue, then the function $\left|(\Psi_{n,t},e^{i\left\langle t,x\right\rangle}e_{k})\right|$ does not depend on choice of the normalized eigenfunction $\Psi_{n,t}$ and is continuous in some neighborhood $U(a)$ of $a$, where $U(a)\subset F^{\ast}$ and any set $E$ satisfying the conditions:

$(a)$* $\left\{\gamma+t:t\in E,\text{ }\gamma\in\Gamma\right\}=\mathbb{R}^{d}$ and*

$(b)$* if $t\in E,$ then $\gamma+t\notin E$ for any $\gamma\in\Gamma\backslash\left\{0\right\},$*

can be used as the fundamental domain $F^{\star}$ of $\Gamma.$

Since $\left\{e^{i\left\langle\gamma+t,x\right\rangle}e_{k}:\gamma\in\Gamma,\text{ }k=1,2,...,m\right\}$ is an orthonormal basis of $L_{2}^{m}(F)$ there exist $\gamma\in\Gamma,$ $k\in\left\{1,2,...,m\right\}$ and $\varepsilon>0$ such that

[TABLE]

For example if $\left|(\Psi_{n,a},e^{i\left\langle\gamma+a,x\right\rangle}e_{k})\right|=\max_{\beta\in\Gamma}\left|(\Psi_{n,a},e^{i\left\langle\beta+a,x\right\rangle}e_{k})\right|,$ then (16) holds. Note that if $F^{\ast}$ is a fundamental domain of $\Gamma,$ then $\gamma+F^{\ast}$ for any $\gamma\in\Gamma$ and even $b+F^{\ast}$ for any $b\in\mathbb{R}^{d}$ is a fundamental domain of the lattice $\Gamma.$ Therefore, without loss of generality and for the simplicity of the notation, we will use $a$ instead of $\gamma+a$ and assume that $a$ is an interior point of $F^{\ast}.$ Therefore, by Proposition 1 and (16) there exists a neighborhood $U(a)$ of $a$ such that

[TABLE]

for all $t\in U(a)$ and the normalized eigenfunction $\Psi_{n,t}$ can be chosen so that

[TABLE]

Then $(\Psi_{n,t},e^{i(t,x)}e_{k})=\left|(\Psi_{n,t},e^{i(t,x)}e_{k})\right|,$ and hence by Proposition 1, $(\Psi_{n,t},e^{i\left\langle t,x\right\rangle}e_{k})$ depends continuously on $t$ in some neighborhood of $a.$ From this, taking into account (15) and (17), it follows that

[TABLE]

as $t\rightarrow a.$ Therefore, using the continuity of the right side of (14) and then (17) we obtain

[TABLE]

and $\left\|\alpha(t))\Psi_{n,t}-\Psi_{n,a}\right\|\rightarrow 0$ as $t\rightarrow a.$ Thus, we have

[TABLE]

as $t\rightarrow a.$ In other words, the following statement is proved.

Proposition 2

If $\lambda_{n}(a)$ is a simple eigenvalue and (17) holds, then the normalized eigenfunction $\Psi_{n,t}$ satisfying (18) depends continuously on $t$ in $U(a).$

Remark 1

Note that the constructive choice (18) is also used in [7, 8]. Namely, in [8] for the case when the right side of (1) is equal to $-\Delta+q$ and $m=1$ (for the Schrödinger operator) in the neighborhood of the sphere $\left\{t\in\mathbb{R}^{d}:\left|t\right|=\rho\right\},$ where $\rho$ is a large number, I constructed a set $B$ such that if $t\in B,$ then there exists a unique eigenvalue $\lambda_{n(t)}(t)$ that is simple and close to $\left|t\right|^{2}$ and the corresponding normalized eigenfunction $\Psi_{n,t}$ satisfies the asymptotic formula

[TABLE]

for some $\delta>0.$ Moreover, the normalized eigenfunction was chosen so that (18) holds (see [8], p. 55). In [8] the choice (18) was made in order to write (20) in the elegant form $\Psi_{n,t}=e^{i\left\langle t,x\right\rangle}+O(\rho^{-\delta})$. However, in Proposition 2 we show that the choice (18) ensures the continuity of $\Psi_{n,t}.$ Note that Proposition 2 is also new for the Schrödinger operator. However, Proposition 1 for the Schrödinger operator is obvious, since it follows directly from the continuity of the function $e^{i\left\langle t,x\right\rangle},$ the projection operator, and the norm. Proposition 1 and (20) were used in [8] to prove that $n(t)=n(a)$ for all $t\in U(a)$ (see (5.11) of [8]), where $U(a)\subset B$ and the condition $(b)$ of Proposition 1 holds (see Lemma 5.1$(b)$ of [8]), i.e., $U(a)\subset\left(B\cap F^{\star}\right)$ for some fundamental domain $F^{\ast}.$

Now let us return to the study of $L.$

Theorem 3

If $\lambda_{n}(a)$ is a simple eigenvalue, then there exists $\beta>0$ such that the eigenvalues $\lambda_{n}(t)$ for $\left|t-a\right|<\beta$ are also simple eigenvalues and the normalized eigenfunctions $\Psi_{n,t}$ of $L_{t}$ satisfying (17) and (18) for $\left\langle t,x\right\rangle=tx$ converges to $\Psi_{n,a}(x)$ uniformly with respect to $x\in[0,1]$ as $t\rightarrow a$.

Proof. If $\lambda_{n}(a)$ is a simple eigenvalue, then $\frac{d\Delta(\lambda,a)}{d\lambda}\neq 0$ for $\lambda=\lambda_{n}(a).$ Then by Summary 1, Theorem 2$(b)$ and (14) there exists $\beta>0$ such that $\lambda_{n}(t)$ is also a simple eigenvalue for $\left|t-a\right|<\beta$ and

[TABLE]

as $t\rightarrow a.$ Moreover, from (21) and (17) it follows that there exist $\varepsilon>0$ and $M$ such that $\left|\Psi_{n,t}(x)\right|\leq M$ for all $\left|t-a\right|<\varepsilon$ and $x\in[0,1].$ Therefore, replacing $L_{2}$ norm $\left\|\cdot\right\|$ everywhere by the norm $\left\|\cdot\right\|_{\infty}$ and repeating the proof of (19) we obtain the proof of the theorem.

2 On the spectrum of $L$

In this section, we study the spectrum of $L$. For this we consider the operator $L_{t}(\varepsilon,C)$ generated by the differential expression

[TABLE]

and boundary conditions (2), where $\varepsilon\in[0,1],$ and $C$ is defined in (5). We consider the operator $L_{t}(\varepsilon,C)$ as perturbation of $L_{t}(C)$ by $L_{t}(\varepsilon,C)-L_{t}(C),$ where $L_{t}(C)$ is the operator generated by the expression

[TABLE]

and boundary condition (2). Therefore, first of all, let us analyze the eigenvalues and eigenfunction of the operator $L_{t}(C)$. We assume that $C$ is the Hermitian matrix. Then $L_{t}(C)$ is the self-adjoint operator, since the expression (22) and boundary conditions (2) are self-adjoint. The distinct eigenvalues of $C$ are denoted by $\mu_{1}<\mu_{2}<...<\mu_{p}$. If the multiplicity of $\mu_{j}$ is $m_{j},$ then $m_{1}+m_{2}+...+m_{p}=m$. Let $u_{j,1},$ $u_{j,2},...,u_{j,m_{j}}$ be the normalized eigenvectors of the matrix $C$ corresponding to the eigenvalue $\mu_{j}.$ The functions $\Phi_{k,j,s,t}(x)=u_{j,s}e^{i\left(2\pi k+t\right)x}$ for $s=1,2,...,m_{j}$ are the eigenfunctions of $L_{t}(C)$ corresponding to the eigenvalue

[TABLE]

since

[TABLE]

Now we consider the large eigenvalues of $L_{t}(\varepsilon,C)$. In the forthcoming inequalities we denote by $c_{1},$ $c_{2},...$ the positive constants that do not depend on* $t\in(-\pi,\pi]$ and $\varepsilon\in[0,1]$*.

Theorem 4

There exists a positive number $N$ such that the eigenvalues of $L_{t}(\varepsilon,C)$ lying in $\left(\mu_{N,1}(t)-\varepsilon_{N},\infty\right)$ lie in $\varepsilon_{k}$ neighborhood $U_{\varepsilon_{k}}(\mu_{k,j}(t)):=(\mu_{k,j}(t)-\varepsilon_{k},\mu_{k,j}(t)+\varepsilon_{k})$ of $\mu_{k,j}(t)$ for $\left|k\right|\geq N$ and $j=1,2,...,p,$ where $\varepsilon_{k}=c_{1}\left(\mid k^{-1}\ln|k|\mid+q_{k}\right)\left(2\pi k\right)^{2\nu-2}$ and

[TABLE]

Moreover, for each $\left|k\right|\geq N$ and $j=1,2,...,p$, there exists an eigenvalue of $L_{t}(\varepsilon,C)$ lying in $U_{\varepsilon_{k}}(\mu_{k,j}(t))$.

Proof. Let $\lambda$ be the eigenvalue of $L_{t}(\varepsilon,C)$* lying in $\left(\mu_{N,1}(t)-\varepsilon_{N},\infty\right)$* and $\mu_{k,j}(t)$ be an eigenvalue of $L_{t}(C)$ closest to $\lambda.$ We prove that $\lambda\in U_{\varepsilon_{k}}(\mu_{k,j}(t)).$ For this we use the formula

[TABLE]

which can be obtained from $L_{t}(\varepsilon,C)\Psi=\lambda\Psi$ by multiplying both sides by $\Phi_{k,j,s,t}(x)$ and using (24), where $\Psi$ is a normalized eigenfunction of $L_{t}(\varepsilon,C)$ corresponding to the eigenvalue $\lambda.$ It was proved in [6] that there exists $c_{2}$ such that

[TABLE]

for $\left|k\right|\geq N$ (see (51) and (54) of [6]). Moreover, by Lemma 4 of [6], for each eigenfunction $\Psi_{k,j,s,t}$ of $L_{t}(C)$ such that $\left|k\right|\geq N$ there exists an eigenfunction $\Phi$ of $L_{t}(\varepsilon,C)$ satisfying

[TABLE]

and conversely for each eigenfunction $\Phi$ corresponding to the eigenvalue of $L_{t}(\varepsilon,C)$* lying in $\left(\mu_{N,1}(t)-\varepsilon_{N},\infty\right)$* there exists $\Psi_{k,j,s,t}$ satisfying (27). Therefore, using (26) and (27) in (25) we get the proof of the theorem.

Now, using Theorem 4 and repeating the proof of Theorem 2.3, Corollary 2.4, and Theorem 2.5 of [9] we obtain the following theorem about the bands and gaps.

Theorem 5

$(a)$* There exists a positive integer $N_{1}$ such that if $s\geq N_{1}$ then the interval $\left[a(s),b(s)\right]$ is contained in each of the bands $I_{sm+1},I_{sm+2},...,I_{sm+m},$ where*

[TABLE]

$I_{n}$* is defined in (4), $\varepsilon(s)=\varepsilon_{k}$ if $s\in\left\{2k,2k+1\right\}$ and $\varepsilon_{k}$ is defined in Theorem 4.*

$(b)$* Let $(\alpha,\beta)$ be the spectral gap of $L$ such that $\alpha>b(N_{1}).$ Then $(\alpha,\beta)$ is contained in the interval $U(s):=(b(s),a(s+1))$ for some $s\geq N_{1}$. Moreover, the spectral gap $(\alpha,\beta)\subset U(s)$ lies between the bands $I_{sm+m}(Q)$ and $I_{sm+m+1}(Q)$ and its length does not exceed $2\max\left\{\varepsilon(s),\varepsilon(s+1)\right\}.$*

For a detailed study of $\sigma(L),$ by using the asymptotic formulas, we need to consider the multiplicities of the eigenvalues of $L_{t}(C)$ and the exceptional points of the spectrum of $L(C).$ The multiplicity of $\mu_{k,j}(t)$ is $m_{j}$ if $\mu_{k,j}(t)\neq\mu_{n,i}(t)$ for all $(n,i)\neq(k,j).$ The multiplicity of $\mu_{k,j}(t)$ is changed, that is, $\mu_{k,j}(t)$ is an exceptional point of $\sigma(L(C))$ if $\mu_{k,j}(t)=\mu_{n,i}(t)$ for some $(n,i)\neq(k,j).$ To consider the exceptional points of $\sigma(L(C))$ and $\sigma(L)$ we use the notation $a_{k}\asymp b_{k}$ which means that there exist constants $c_{4},$ $c_{5},$ $c_{6}$ such that $c_{4}|a_{k}|<\left|b_{k}\right|<c_{5}|a_{k}|$ for all $\left|k\right|>c_{6}.$ It follows from (23) that if $t\in[-\frac{\pi}{2},\frac{3\pi}{2}),$ then $\mu_{k,j}(t)-\mu_{k,i}(t)\asymp k^{2\nu-2}$ for $j\neq i$ and $\left|\mu_{k,j}(t)-\mu_{n,i}(t)\right|\geq d_{k}$ for $n\neq k,-k,-(k+1),$ where $d_{k}\asymp k^{2\nu-1}.$ Thus, the large eigenvalue $\mu_{k,j}(t)$ for $t\in[-\frac{\pi}{2},\frac{3\pi}{2})$ may become an exceptional Bloch eigenvalue of $L(C)$ if at least one of the following equalities holds

[TABLE]

Therefore we need to consider the points $t\in[-\frac{\pi}{2},\frac{3\pi}{2})$ for which the equalities in (28) do not hold. Moreover, to prove that the eigenvalues of $L_{t}$ lying in $\varepsilon_{k}=o(k^{2\nu-2})$ neighborhood of $\mu_{k,j}(t)$ (see Theorem 4) do not coincide with the eigenvalues lying in $\varepsilon_{-k}$ and $\varepsilon_{-k-1}$ neighborhood of $\mu_{-k,i}(t)$ and $\mu_{-k-1,i}(t)$ we consider the points $t\in[-\frac{\pi}{2},\frac{3\pi}{2})$ for which

[TABLE]

where $f(t)=\mu_{k,j}(t)-\mu_{-k,i}(t),$ $g(t)=\mu_{k,j}(t)-\mu_{-k-1,i}(t).$ Using (23) and the binomial expansion of $(a+b)^{n}$ for $n=2\nu$ and $n=2\nu-2$ we obtain

[TABLE]

On the other hand, one can easily verify that $f^{{}^{\prime}}(t)\asymp k^{2\nu-1}.$ Therefore, there exists $\delta_{k}=o(k^{-1})$ such that the first inequality of (29) holds if $t$ does not belong to the interval

[TABLE]

In the same way we prove that if $t$ does not belong to the interval

[TABLE]

then the second inequality of (29) holds. Therefore, using (29) and Theorem 4 and repeating the proof of Corollary 2.8  and Theorem 2.10 of [9] we obtain.

Theorem 6

$(a)$* There exist $N_{2}>N_{1}$ and $\gamma_{k}=o(k^{2\nu-2})$ such that the spectral gap $(\alpha,\beta)$ defined in Theorem 5 and lying in $U(k)$ for $k>N_{2}$ is contained in the intersection of the sets $S(1,k),S(2,k),...,S(p,k),$ where*

[TABLE]

$(b)$* If there exists a triple $(j_{1},j_{2},j_{3})$ such that*

[TABLE]

where minimum is taken under condition $i_{s}\in\left\{1,2,...,p\right\}$ for $s=1,2,3$ and

[TABLE]

then there exists a number $H$ such that $(H,\infty)\subset\sigma(L)$ and the number of the gaps in $\sigma(L)$ is finite.