# On the real spectrum of differential operators with PT-symmetric periodic
matrix coefficients

O. A. Veliev
Dogus University, Istanbul, Turkey.
e-mail<EMAIL_ADDRESS>
###### Abstract

We study the spectrum of the differential operator $T$ generated by the
differential expression of order $n>2$ with the $m\times m$ PT-symmetric
periodic matrix coefficients. The case when $m$ and $n$ are the odd numbers
was investigated in [8]. In this paper, we consider the all remained cases:
(a) $n$ is an odd number and $m$ is an even number, (b) $n$ is an even number
and $m$ is an arbitrary positive integer. We find conditions on the
coefficients under which in the cases (a) and (b) the spectrum of $T$ contains
the sets $(-\infty,H]$ $\cup[H,\infty)$ and $[H,\infty)$ respectively for some
$H>0.$

Key Words: Non-self-adjoint differential operator, PT-symmetric periodic
matrix coefficients. Real spectrum.

AMS Mathematics Subject Classification: 34L05, 34L20.

##  1 Introduction and Preliminary Facts

In this paper, we consider the spectrum $\sigma(T)$ of the differential
operator $T$ generated in the space $L_{2}^{m}(-\infty,\infty)$ by the
differential expression

$(-i)^{n}y^{(n)}+(-i)^{n-2}P_{2}y^{(n-2)}+(-i)^{n-3}P_{3}y^{(n-3)}+...+P_{n}y,$
(1)

where $n>2,$ $P_{k}=(p_{k,i,j})$ for $k=2,3,...,n$ are the $m\times m$
matrices with the complex-valued PT-symmetric periodic entries

$p_{k,i,j}\left(x+1\right)=p_{k,i,j}\left(x\right),\text{
}p_{k,i,j}\left(-x\right)=\overline{p_{k,i,j}\left(x\right)},\text{
}p_{k,i,j}\in L_{2}[0,1]$ (2)

and $y=(y_{1},y_{2},...,y_{m})^{T}$ is a vector-valued function. Here
$L_{2}^{m}(a,b)$ for $-\infty\leq a<b\leq\infty$ is the space of the vector-
valued functions $f=\left(f_{1},f_{2},...,f_{m}\right)^{T}$ with the norm
$\left\|\cdot\right\|_{(a,b)}$ and inner product $(\cdot,\cdot)_{(a,b)}$
defined by

$\left\|f\right\|_{(a,b)}^{2}=\int_{a}^{b}\left|f\left(x\right)\right|^{2}dx,\text{
}(f,g)_{(a,b)}=\int_{a}^{b}\left\langle
f\left(x\right),g\left(x\right)\right\rangle dx,$

where $\left|\cdot\right|$ and $\left\langle\cdot,\cdot\right\rangle$ are the
norm and inner product in $\mathbb{C}^{m}.$

It is well known that (see for example [3, 5]) the spectrum $\sigma(T)$ of $T$
is the union of the spectra $\sigma(T_{t})$ of the operators $T_{t}$ for
$t\in[0,2\pi)$ generated in $L_{2}^{m}[0,1]$ by (1) and the boundary
conditions

$y^{(\mathbb{\nu})}\left(1\right)=e^{it}y^{(\mathbb{\nu})}\left(0\right),\text{
}\mathbb{\nu}=0,1,...,(n-1).$ (3)

The spectrum of $T_{t}$ consists of the eigenvalues. These eigenvalues are
known as Bloch eigenvalues of $T$ and are the roots of the characteristic
equation

$\Delta(\lambda,t):=\det(Y_{j}^{(\nu-1)}(1,\lambda)-e^{it}Y_{j}^{(\nu-1)}(0,\lambda))_{j,\nu=1}^{n}=$
(4)
$e^{inmt}+f_{1}(\lambda)e^{i(nm-1)t}+f_{2}(\lambda)e^{i(nm-2)t}+\cdot\cdot\cdot+f_{nm-1}(\lambda)e^{it}+1,$

where $f_{1}(\lambda),f_{2}(\lambda),...$ are the entire functions,
$Y_{1}(x,\lambda),Y_{2}(x,\lambda),\ldots,Y_{n}(x,\lambda)$ are the solutions
of the matrix equation

$(-i)^{n}Y^{(n)}+(-i)^{n-2}P_{2}Y^{(n-2)}+(-i)^{n-3}P_{3}Y^{(n-3)}+...+P_{n}Y=\lambda
Y$

satisfying $Y_{k}^{(j)}(0,\lambda)=O_{m}$ for $j\neq k-1$,
$Y_{k}^{(k-1)}(0,\lambda)=I_{m}$. Here $O_{m}$ and $I_{m}$ are the $m\times m$
zero and identity matrices (see [4, Chapter 3]).

Note that there are a large number of papers for the scalar case $m=1$ and
$n=2$, namely for the Schrödinger operator (see the monographs [1, Chapters 4
and 6] and [6, Chapters 3 and 5] and the papers they refer to). The results
and the method used in this paper are completely different from the results
and methods of those papers. Therefore we do not discuss the scalar case in
detail.

As far as I know, only the papers [7, 8] were devoted to the differential
operator with the periodic PT-symmetric matrix coefficients. In [7] the
Schrödinger operator with a PT-symmetric periodic matrix potential was
investigated, where $n=2$. In [8] we considered the following case.

###### Case 1

$m$ and $n$ are the odd numbers.

We proved that in Case 1, $\sigma(T)$ contains all real line $\mathbb{R}.$ In
this paper, we consider the others and all the remained cases, namely the
following cases:

###### Case 2

$n$ is an odd number and $m$ is an even number.

###### Case 3

$n$ is an even number and $m$ is an arbitrary positive integer.

Therefore, this paper can be considered as a continuation and completion of
the paper [8]. Moreover, the method used in [8] for Case 1 can note be used
for Cases 2 and 3, since the method of Case 1 passes through only if $nm$ is
an odd number. That is why, the methods used in [8] and in this paper are
completely different.

The paper is organized as follows. To study the spectrum of $T_{t},$ we
consider the family of the operators $T_{t}(\varepsilon,C)$ generated by the
differential expression

$(-i)^{n}y^{n}+(-i)^{n-2}Cy^{(n-2)}+\varepsilon\left((-i)^{n-2}(P_{2}-C)y^{(n-2)}+{\textstyle\sum\limits_{l=3}^{n}}(-i)^{n-l}P_{l}y^{(n-l)}\right)$
(5)

and boundary conditions (3), where $\varepsilon\in[0,1],$
$C=\int_{0}^{1}P_{2}\left(x\right)dx,$ $T_{t}(1,C)=T_{t}$ and
$T_{t}(0,C)=:T_{t}(C)$ is the operator generated by the expression

$(-i)^{n}y^{(n)}+(-i)^{n-2}Cy^{(n-2)}$

and boundary conditions (3). Thus $T_{t}(C)$ and
$T_{t}(\varepsilon,C)-T_{t}(C)$ are respectively the unperturbed operator and
perturbation. We prove that the large eigenvalues of $T_{t}(\varepsilon,C)$
are located in the small neighborhood of the eigenvalues of $T_{t}(C).$
Therefore, first of all, let us analyze the eigenvalues and eigenfunction of
the operator $T_{t}(C)$. Using (2) one can easily verify that the entries of
the matrix $C$ are the real numbers. Therefore, the eigenvalues of the matrix
$C$ consist of the real eigenvalues and the pairs of the conjugate complex
numbers. The distinct eigenvalues of $C$ are denoted by
$\mu_{1},\mu_{2},...,\mu_{p}.$ If the multiplicity of $\mu_{j}$ is $m_{j},$
then

$m_{1}+m_{2}+...+m_{p}=m.$ (6)

Without loss of generality, we denote the real eigenvalues by
$\mu_{1}<\mu_{2}<...<\mu_{s}$ and the nonreal eigenvalues by$\
\mu_{s+1},\mu_{s+2},...,\mu_{p}.$ One can easily verify that the eigenvalues
and eigenfunctions of $T_{t}(C)$ are respectively

$\mu_{k,j}(t)=\left(2\pi k+t\right)^{n}+\mu_{j}\left(2\pi k+t\right)^{n-2}$
(7)

and

$\Phi_{k,j,l,t}(x)=u_{j,l}e^{i\left(2\pi k+t\right)x}$ (8)

for $k\in\mathbb{Z},$ $j=1,2,...,p$ and $l=1,2,...,l_{j\text{ }},$ where
$u_{j,1},$ $u_{j,2},...u_{j,l_{j}}$ are the linearly independent eigenvectors
corresponding to the eigenvalue $\mu_{j}$ of $C.$ Therefore
$\sigma\left(T(C)\right)$ consists of the lines and half lines, respectively
if $n$ is an odd and even number. To see the difference of the investigations
in Cases 2 and 3 let us discuss the exceptional points of $\sigma(T(C)),$
since the perturbation $T(\varepsilon,C)-T(C)$ for the small values of
$\varepsilon$ may generate the gaps in $\sigma\left(T(C)\right)$ only at the
neighborhoods of the exceptional Bloch eigenvalues. Note that the exceptional
points of $\sigma\left(T(C)\right)$ are the points
$\mu_{k,j}(t_{0})\in\sigma\left(T(C)\right),$ where the multiplicity of the
eigenvalues $\mu_{k,j}(t)$ varies in any neighborhood of $t_{0}$. In
Proposition 1$(a)$, we prove that if $n$ is an odd number, then the
multiplicity of the eigenvalues $\mu_{k,j}(t)$ is equal to $m_{j}$ for all
$t\in\mathbb{R}$. Therefore, in Case 2 $\sigma\left(T(C)\right)$ has no
exceptional points. This situation simplifies the study of Case 2. In this
case we prove that if the matrix $C$ has at least one real eigenvalue of odd
multiplicity, then $\sigma(T)$ contains the set $(-\infty,-H]\cup[H,\infty)$
for some $H\geq 0$. However, if $n$ is an even number, then there exist the
points $t\in[0,2\pi)$ (see Proposition 1$(b)$) such that
$\mu_{k,j}(t)=\mu_{l,i}(t)$ for some $(l,i)\neq(k,j),$ that is, the
multiplicity of the eigenvalues $\mu_{k,j}(t)$ varies. Therefore, in this
case, $\sigma\left(T(C)\right)$ may have infinitely many exceptional points.
This situation complicate the investigation of Case 3. In this case we prove
that if the matrix $C$ has at least three real eigenvalues of odd multiplicity
satisfying some conditions (see (27)), then $\sigma(T)$ contains the set
$[H,\infty)$ for some $H\geq 0$. Fortunately, the investigations [7] of the
case $n=2$ helps us to consider this complicated case.

Note that we use the following theorem, which can be proved by repeating the
proof of the Theorem 1 of [7].

###### Theorem 1

If $\lambda$ is an eigenvalue of multiplicity $v$ of the operator $T_{t},$
then $\overline{\lambda}$ is also an eigenvalue of the same multiplicity of
$T_{t}$

To formulate the following theorem, which is essentially used in this paper,
we introduce the following notation. Denote by $u_{j,l,1},$
$u_{j,l,2},...u_{j,l,r_{j,l}-1}$ the associated vectors corresponding to the
eigenvector $u_{j,l},$ such that
$\left(C-\mu_{j}I\right)u_{j,l,q}=u_{j,l,q-1}$ for $q=1,2,...,r_{j,l}-1,$
where $u_{j,l,0}=u_{j,l}$ and recall that $u_{j,1},$ $u_{j,2},...u_{j,l_{j}}$
are the linearly independent eigenvectors corresponding to the eigenvalue
$\mu_{j}$ of $C$. Then it is not hard to verify that

$\left(T_{t}(C)-\mu_{k,j}(t)I\right)u_{j,l,q}e^{i\left(2\pi
k+t\right)x}=u_{j,l,q-1}e^{i\left(2\pi k+t\right)x}$

and

$r_{j,1}+r_{j,2}+...+r_{j,l_{j}}=m_{j},$ (9)

where $m_{j}$ is the multiplicity of the eigenvalue $\mu_{j}.$ It means that
the associated functions of $T_{t}(C)$ corresponding to the eigenfunction
$\Phi_{k,j,l,t}(x)=u_{j,l}e^{i\left(2\pi k+t\right)x}$ are

$\Phi_{k,j,l,q,t}(x)=u_{j,l,q}e^{i\left(2\pi k+t\right)x}$ (10)

for $q=1,2,...,r_{j,l}-1.$ Thus the dimension of the space generated by the
functions (8) and (10) is the multiplicity $m_{j}$ of the eigenvalue
$\mu_{j},$ due to (9).

###### Theorem 2

There exist positive numbers $N$ and $c$ such that the large eigenvalues of
$T_{t}(\varepsilon,C)$ lie in the disks

$U_{\varepsilon_{k}}(\mu_{k,j}(t)):=\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}(t)\right|<\varepsilon_{k}\right\\}$

for $\left|k\right|\geq N$ and $j=1,2,...,p,$ where
$\varepsilon_{k}=c\left(\mid k\mid^{n-3}\right)^{1/r}$ if $n$ is an odd
number,

$\varepsilon_{k}=c\left(\left(\mid
k^{-1}\mid+q_{k}\right)\left|k\right|^{n-2}\right)^{1/r},$

if $n$ is an even number,
$r=\max\limits_{j=1,2,...,p}\left\\{r_{j,1},r_{j,2},...,r_{j,l_{j}}\right\\},$
$t\in[-1,2\pi-1)$, $\varepsilon\in[0,1]$,

$q_{k}=\max\left\\{\left|p_{2,i,j,l}\right|:i,j=1,2,...,m;\text{ }l=\pm
2k,\pm(2k+1)\right\\}$

and $p_{2,i,j,l}=\int\nolimits_{[0,1]}p_{2,i,j}(x)e^{-2\pi ilx}dx.$

Theorems analogous to Theorem 2 in the cases: (i) $n=2$ and (ii) $T_{t}$ is a
self-adjoint operator were proved in [7] and [9], respectively. Since these
cases do not cover the operator $T_{t}$, we cannot directly refer to these
papers. However, the proof of this theorem is similar to the proofs of the
corresponding Theorems 3 and 5 in [7] and [9], respectively. Therefore, we
present the proof of Theorem 2 in the Appendix.

##  2 Main Results

First, let us consider the eigenvalues and root functions of $T_{t}(C)$ for
$t\in[0,2\pi).$

###### Proposition 1

Let $\mu_{j}$ be an eigenvalue of $C$ of multiplicity $m_{j}.$

$(a)$ If $n$ is an odd number, then the multiplicity of the eigenvalue
$\mu_{k,j}(t)$ of $T_{t}(C)$ is $m_{j}$ for all $k\in\mathbb{Z}$ and
$t\in[0,2\pi),$ where $\mu_{k,j}(t)$ is defined in (7).

$(b)$ If $n$ is an even number, then the multiplicity of $\mu_{k,j}(t)$ is
$m_{j}$ for $t\in[0,2\pi)\backslash A(k,j),$ where

$A(k,j)={\textstyle\bigcup\limits_{l\in\mathbb{Z},i=1,2,...,p}}\left\\{t_{l,i,q}:q=1,2,...,n\right\\}$
(11)

and $t_{l,i,1},$ $t_{l,i,2},...,t_{l,i,n}$ are the roots of the equation

$\left(2\pi k+t\right)^{n}+\mu_{j}\left(2\pi k+t\right)^{n-2}=\left(2\pi
l+t\right)^{n}+\mu_{i}\left(2\pi l+t\right)^{n-2}.$

Proof. It follows from (8)-(10) that, if

$\mu_{k,j}(t)\neq\mu_{l,i}(t)$ (12)

for $(l,i)\neq(k,j),$ then $\mu_{k,j}(t)$ is an eigenvalue of $T_{t}(C)$ of
the multiplicity $m_{j}.$ On the other hand, by (7), if $n$ is an odd and even
number respectively, then (12) holds for all $t\in[0,2\pi)$ and
$t\in[0,2\pi)\backslash A(k,j).$ Therefore, the proposition is true.

Thus the spectrum $\sigma\left(T(C)\right)$ has no exceptional points if $n$
is an odd number, while the spectrum $\sigma\left(T(C)\right)$ has infinitely
many exceptional points if $n$ is an even number. Moreover, the set of
$t\in[0,2\pi)$ for which $\mu_{k,j}(t)$ are the exceptional points has the
accumulation points $0,\pi$ and $2\pi.$ Since $T_{t}=T_{t+2\pi},$ sometimes,
instead of $t\in[0,2\pi)$ we use $t\in[-h,2\pi-h)$ for some $h\in(0,\pi)$ in
order to get two accumulation points. Note that in Theorem 2 and Proposition 1
one can replace $[-1,2\pi-1)$ and $[0,2\pi)$ by $[-h,2\pi-h).$

Now consider the large eigenvalues of $T_{t}$, by using Proposition 1, Theorem
2 and the notation $a_{k}\asymp b_{k}$ which means that there exist constants
$c_{1},$ $c_{2}$ and $c_{3},$ independent of $t$ and $\varepsilon,$ such that
$c_{1}|a_{k}|<\left|b_{k}\right|<c_{2}|a_{k}|$ for all $\left|k\right|>c_{3}.$
Note that in the forthcoming inequalities we denote by $c_{1},$ $c_{2},...$
positive constants independent of $t$ and $\varepsilon.$

###### Theorem 3

Let $\mu_{j}$ be an eigenvalue of $C$ of multiplicity $m_{j}.$

$(a)$ If $n$ is an odd number, then the operator $T_{t}$ has only $m_{j}$
eigenvalues lying in $U_{\varepsilon_{k}}(\mu_{k,j}(t))$ for
$\left|k\right|\geq N$ and $t\in[-h,2\pi-h),$ where $N$ and $\varepsilon_{k}$
are defined in Theorem 2 and $h\in(0,\pi).$

$(b)$ If $n$ is an even number, then there exists
$\delta_{k}\asymp\left(\varepsilon_{k}+\varepsilon_{-k}+\varepsilon_{-k-1}\right)k^{1-n}$
such that the operator $T_{t}$ has only $m_{j}$ eigenvalues lying in
$U_{\varepsilon_{k}}(\mu_{k,j}(t))$ for $\left|k\right|\geq N$ and

$t\in[-h,2\pi-h)\backslash U_{\delta_{k}}\left(A(k,j)\right),$ (13)

where $U_{\delta_{k}}\left(E\right)$ denotes the open $\delta_{k}$
neighborhood of the set $E$ and $A(k,j)$ is defined in (11).

Proof. $(a)$ Using (7) one can easily verify that if $\left|k\right|\geq N$,
then there exists a constant $c_{1}>0$ such that

$\left|\mu_{k,j}(t)-\mu_{l,i}(t)\right|\geq c_{1}\left|k\right|^{n-2}$ (14)

for all $(l,i)\neq(k,j).$ On the other hand $\varepsilon_{k}=o(k^{n-2}).$
Therefore, from Theorem 2 we obtain that the circle
$D\left(\mu_{k,j},\varepsilon_{k}\right)=\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}\right|=\varepsilon_{k}\right\\}$
belong to the resolvent set of the operators $T_{t}(\varepsilon,C)$ for all
$\varepsilon\in[0,1],$ where $T_{t}(\varepsilon,C)$ is generated by (5). This
implies that the operators $T_{t}:=T_{t}(1,C)$ and $T_{t}(C):=T_{t}(0,C)$ have
the same number of eigenvalues (counting the multiplicity) inside
$D\left(\mu_{k,j},\varepsilon_{k}\right),$ since $T_{t}(\varepsilon,C)$ is the
halomorphic family (with respect to $\varepsilon,$ in the sense of [2] (see
[2, Chapter 7])) of operators. Thus the proof of this theorem follows from
Proposition 1$(a),$ because the operator $T_{t}(C)$ has only one eigenvalue
$\mu_{k,j}$ inside $D\left(\mu_{k,j},\varepsilon_{k}\right)$ and the
multiplicity of $\mu_{k,j}$ is $m_{j}.$

$(b)$ First we prove that if (13) holds, then

$D\left(\mu_{k,j}(t),\varepsilon_{k}\right)\cap
D\left(\mu_{l,i}(t),\varepsilon_{l}\right)=\varnothing$ (15)

for $(l,i)\neq(k,j),$ where $\left|k\right|\geq N.$ It follows from (7) that
if $t\in[-h,2\pi-h),$ then $\mu_{k,j}(t)-\mu_{k,i}(t)\asymp k^{n-2}$ for
$j\neq i$ and $\left|\mu_{k,j}(t)-\mu_{l,i}(t)\right|>d_{k}$ for $l\neq
k,-k,-(k+1),$ where $d_{k}\asymp k^{n-1}.$ Thus, (15) holds for
$t\in[-h,2\pi-h)$ and $l\neq k,-k,-(k+1).$

The validity of (15) for the case $l=k$ follows from (14) and the definition
$\varepsilon_{k}.$ To prove (15) for the cases $l=-k$ and $l=-(k+1),$ let us
consider the functions $f(t)=\mu_{k,j}(t)-\mu_{-k,i}(t),$
$g(t)=\mu_{k,j}(t)-\mu_{-k-1,i}(t).$ Using (7) and the binomial expansion of
$(a+b)^{n}$ we obtain

$f(t)=(2\pi k)^{n-2}(4nk\pi t+\mu_{j}-\mu_{i})+O(k^{n-3}),\text{
}f\left(\frac{\mu_{i}-\mu_{j}}{4nk\pi}\right)=O(k^{n-3}).$

On the other hand, one can easily verify that $f^{{}^{\prime}}(t)\asymp
k^{n-1}.$ Therefore, there exists
$\delta_{k}\asymp\left(\varepsilon_{k}+\varepsilon_{-k}+\varepsilon_{-k-1}\right)k^{1-n}$
such that if $t$ does not belong to the interval

$U(i,j,k,\delta_{k})=\left(\frac{\mu_{i}-\mu_{j}}{4nk\pi}-\delta_{k},\frac{\mu_{i}-\mu_{j}}{4nk\pi}+\delta_{k}\right),$
(16)

then $\left|f(t)\right|>\varepsilon_{k}+\varepsilon_{-k}$. In the same way we
prove that if $t$ does not belong to the interval

$U(i,j,-k-1,\delta_{k})=\left(\pi+\frac{\mu_{i}-\mu_{j}}{2\pi
n(2k+n-1)}-\delta_{k},\pi+\frac{\mu_{i}-\mu_{j}}{2\pi
n(2k+n-1)}+\delta_{k}\right),$ (17)

then $\left|g(t)\right|>\varepsilon_{k}+\varepsilon_{-k-1}.$ Thus, using (11)
we obtain that

$\left(U_{\delta_{k}}\left(A(k,j)\right)\cap[-1,2\pi-1)\right)\subset\left({\textstyle\bigcup\limits_{i=1}^{s}}\left(U(i,j,k,\delta_{k})\cup
U(i,j,-k-1,\delta_{k})\right)\right).$ (18)

Therefore, (15) is true if (13) holds.

Now, (15) with Theorem 2 implies that the circle
$D\left(\mu_{k,j}(t),\varepsilon_{k}\right)$ belong to the resolvent set of
the operators $T_{t}(\varepsilon,C)$ for all $\varepsilon\in[0,1],$ if (13)
holds. Therefore instead of Proposition 1$(a)$ using Proposition 1$(b)$ and
repeating the last statements of the proof of $(a)$, we get the proof of
$(b).$

Now, using Theorem 3 and the following arguments we consider the gaps in
$\sigma(T)$. The substitution $y(x)=e^{itx}\widetilde{y}(x)$ implies that the
operator $T_{t}$ is generated by the differential operation

$(-i)^{n}\left(\frac{\partial}{\partial
x}+it\right)^{n}+(-i)^{n-2}P_{2}\left(\frac{\partial}{\partial
x}+it\right)^{n-2}+...+P_{n}y$

and the periodic boundary conditions. Then the domain of the definition of
$T_{t}$ does not depend on $t$ and hence $\\{T_{t}:$ $t\in[-h,2\pi-h)\\}$ is a
halomorphic family (in the sense of [2] (see [2, Chapter 7])) of operators
with compact resolvent for each $h\in[0,\pi]$.

It follows from Theorem 2 and the proof of Theorem 3 that, if
$\left|k\right|\geq N$, $t_{0}\in[-h,2\pi-h)$ and
$t_{0}\in[-h,2\pi-h)\backslash U_{\delta_{k}}\left(A(k,j)\right)$ respectively
for odd and even $n,$ then the circle$\ D_{\varepsilon_{k}}(\mu_{k,j}(t_{0}))$
belong to the resolvent set of the operator $T_{t_{0}}$. It means that
$\Delta(\lambda,t_{0})\neq 0$ for each $\lambda\in
D_{\varepsilon_{k}}(\mu_{k,j}(t_{0})),$ where $\Delta(\lambda,t)$ is defined
in (4). Since $\Delta(\lambda,t_{0})$ is a continuous function on the compact
$D_{\varepsilon_{k}}(\mu_{k,j}(t_{0})),$ there exists $a>0$ such that
$\left|\Delta(\lambda,t_{0})\right|>a$ for all $\lambda\in
D_{\varepsilon_{k}}(\mu_{k,j}(t_{0})).$ Moreover, by (4), $\Delta(\lambda,t)$
is a polynomial of $e^{it}$ with entire coefficients. Therefore, there exists
$\delta$ such that $\left|\Delta(\lambda,t)\right|>a/2$ for all
$t\in(t_{0}-\delta,t_{0}+\delta)$ and $\lambda\in
D_{\varepsilon_{k}}(\mu_{k,j}(t_{0})).$ It means that
$D_{\varepsilon_{k}}(\mu_{k,j}(t_{0}))$ belong to the resolvent set of $T_{t}$
for all $t\in(t_{0}-\delta,t_{0}+\delta).$ Hence, the spectrum of $T_{t}$ is
separated by $D_{\varepsilon_{k}}(\mu_{k,j}(t_{0}))$ into two parts in the
sense of [2] (see [2, Chapter 3, Section 6.4]). Therefore, the theory of
halomorphic family of the finite dimensional operators [2, Chapter 2] can be
applied to the part of $T_{t}$ for $t\in(t_{0}-\delta,t_{0}+\delta)$
corresponding to the inside of $D_{\varepsilon_{k}}(\mu_{k,j}(t_{0})).$ Now,
using these arguments we prove the following lemma which plays the crucial
role in the prove of the main results of this paper.

###### Lemma 1

Suppose that the matrix $C$ has a real eigenvalue $\mu_{j}$ of odd
multiplicity $m_{j}$. If $\lambda=\mu_{k,j}(t_{0})$ for some
$t_{0}\in[-1,2\pi-1)$ and the disk $U_{\varepsilon_{k}}(\mu_{k,j}(t))$ for all
$t\in\left[t_{0}-h_{k},t_{0}+h_{k}\right]$ contains only $m_{j}$ eigenvalues
(counting the multiplicity) of $T_{t},$ then the spectrum of $T$ contains the
point $\lambda,$ where $k\geq N$, $h_{k}=2\varepsilon_{k}(2\pi(k-1))^{1-n},$
$\varepsilon_{k}$ and $N$ are defined in Theorem 2.

Proof. It follows from (7) that

$(2\pi(k-1))^{n-1}\leq\left|\frac{d\mu_{k,j}(t)}{dt}\right|\leq(2\pi(k+2))^{n-1}$
(19)

for all $t\in[-\pi,2\pi).$ Therefore

$\mu_{k,j}(t_{0}-h_{k})\leq\lambda-2\varepsilon_{k},\text{
}\mu_{k,j}(t_{0}+h_{k})\geq\lambda+2\varepsilon_{k}.$ (20)

Denote by $\lambda_{k,1}(t),\lambda_{k,2}(t),...,\lambda_{k,m_{j}}(t)$ the
eigenvalues of $T_{t}$ lying in $U_{\varepsilon_{k}}(\mu_{k,j}(t))$ and
consider the unordered $m_{j}$-tuple
$\Omega(t):=\left\\{\lambda_{k,1}(t),\lambda_{k,2}(t),...,\lambda_{k,m_{j}}(t)\right\\}.$
As is explained above the theory of continuos family of the finite dimensional
operators [2, Chapter 2] can be applied to the part $\Omega(t)$ of the
spectrum of $T_{t}.$ Therefore, the unordered $m_{j}$-tuple $\Omega(t)$ depend
continuously (see page 108 of [2]) on the parameter
$t\in\left[t_{0}-h_{k},t_{0}+h_{k}\right].$ Then by Theorem 5.2 of [2] (see
page 109) there exist $p$ single-valued continuous functions
$\lambda_{1}(t),\lambda_{2}(t),...,\lambda_{m_{j}}(t)$ the value of which
constitute the $m_{j}$-tuple $\Omega(t)$ for
$t\in\left[t_{0}-h_{k},t_{0}+h_{k}\right].$ Moreover, it follows from (20) and
Theorem 2 that

$\operatorname{Re}\lambda_{l}(t_{0}-h_{k})<\lambda-\varepsilon_{k},\text{
}\operatorname{Re}\lambda_{l}(t_{0}+h_{k})>\lambda+\varepsilon_{k}$ (21)

for $l=1,2,...,m_{j}.$ Now, we prove that
$\lambda\in\left(\cup_{l=1}^{m_{j}}\gamma_{l}\right),$ where $\gamma_{l}$ is
the curve
$\left\\{\lambda_{l}(t):t\in\left[t_{0}-h_{k},t_{0}+h_{k}\right]\right\\}.$
Assume the converse. Then by (21) the continuous curves
$\gamma_{l}=\left\\{\lambda_{s}(t):t\in\left[t_{0}-h_{k},t_{0}+h_{k}\right]\right\\}$
extend from $\lambda_{l}(t_{0}-h_{k})\ $to $\lambda_{l}(t_{0}+h_{k})$ pas
above or below of the point $\lambda$ for each $l=1,2,...,m_{j}.$ On the other
hand, by Theorem 1 if $\gamma_{l}$ passes above of $\lambda$ then there exist
$s\in\left\\{1,2,...,m_{j}\right\\}$ such that $s\neq l$ and $\gamma_{s}$
passes below of$\ \lambda.$ It implies that the number $m_{j}$ of the curves
$\gamma_{1},\gamma_{2},...,\gamma_{m_{j}}$ is an even number. It contradicts
to the assumption that $m_{j}$ is an odd number. Thus, there exists $l$ such
that $\lambda\in\gamma_{l}.$ Since $\gamma_{l}\subset$ $\sigma(T),$ the
theorem is proved.

Now we are ready to prove the main results of this paper. First let us
consider the Case 2

###### Theorem 4

Suppose that the matrix $C$ has a real eigenvalue $\mu_{j}$ of odd
multiplicity $m_{j}.$ If $n$ is odd number, then $\sigma(T)$ contains the set
$(-\infty,-H]\cup[H,\infty)$ for some $H\geq 0.$

Proof. Let $\lambda$ be large real number. Without loss of generality assume
that $\lambda>\mu_{N,j}(-1),$ where $\mu_{k,j}(t)$ and $N$ are defined in (7)
and Theorem 2, respectively. Then there exists $t_{0}\in[-1,2\pi-1)$ such that
$\lambda=\mu_{k,j}(t_{0}))$ for some $k\geq N$. It is clear that there exists
$h\in(0,\pi)$ such that
$\left[t_{0}-h_{k},t_{0}+h_{k}\right]\subset[-h,2\pi-h).$ Therefore, it
follows from Theorem 3$(a)$ that the conditions of Lemma 1 holds. Then
$\lambda\in\sigma(T)$ and the theorem is proved.

Now, we study the complicated Case 3. Consider the intervals
$U(i,j,k,\delta_{k}+h_{k})$ and $U(i,j,-k-1,\delta_{k}+h_{k})$ for
$i\in\left\\{1,2,...,s\right\\},$ where these intervals are obtained from the
intervals (16) and (17) by replacing $\delta_{k}$ with $\delta_{k}+h_{k}.$ By
definitions of $\delta_{k}$, $h_{k}$ and $\varepsilon_{k}$ we have

$\delta_{k}+h_{k}=o(\left|k\right|^{-1}).$ (22)

On the other hand, it follows from (16) and (17) that there exists $c_{2}$
such that distance between the centres of these intervals is greater than
$c_{2}\left|k\right|^{-1}.$ Therefore, if $\left|k\right|\geq N$, then these
intervals are pairwise disjoint and

$\mu\left({\textstyle\bigcup\limits_{i\in\left\\{1,2,...,s\right\\}}}\left(U(i,j,k,\delta_{k}+h_{k})\cup
U(i,j,-k-1,\delta_{k}+h_{k})\right)\right)=o(\left|k\right|^{-1}),$

where $\mu(E)$ is the measure of the set $E.$ If we eliminate these intervals
from $[-1,2\pi-1)$ then the remaining set consists of the pairwise disjoint
intervals $[a_{1,k},b_{1,k}],[a_{2,k},b_{2,k}],...,[a_{l,k},b_{l,k}]$ such
that

$\mu\left([a_{1,k},b_{1,k}]\cup[a_{2,k},b_{2,k}]\cup...\cup[a_{l,k},b_{l,k}]\right)=2\pi+o(\left|k\right|^{-1}).$
(23)

###### Theorem 5

Suppose that the matrix $C$ has a real eigenvalue $\mu_{j}$ of odd
multiplicity $m_{j}$ and $n$ is an even number. If $t_{0}\in[a_{v,k},b_{v,k}]$
for some $v\in\left\\{1,2,...,l\right\\}$ and $k\geq N$, then
$\mu_{k,j}(t_{0})\in\sigma(T).$ Moreover,

$\mu\left(\mu_{k,j}([-1,2\pi-1))\cap\sigma(T)\right)=\mu\left(\mu_{k,j}([-1,2\pi-1))\right)(1+o(\left|k\right|^{-1}).$
(24)

In other words, the spectrum $\sigma(T)$ contains the set

$\mu_{k,j}\left([a_{1,k},b_{1,k}]\cup[a_{2,k},b_{2,k}]\cup...\cup[a_{l,k},b_{l,k}]\right)$
(25)

for $k\geq N$ and $\sigma(T)$ contains the large part of the interval
$[0,\infty),$ in the sense that

$\lim_{\rho\rightarrow\infty}\frac{\mu([0,\rho]\backslash\sigma(T))}{\mu(\sigma(T)\cap[0,\rho])}=0.$
(26)

Proof. By the definition of the interval $[a_{v,k},b_{v,k}]$ if
$t_{0}\in[a_{v,k},b_{v,k}]$ then $t_{0}$ does not belong to any of the
intervals $U(i,j,k,\delta_{k}+h_{k})$ and $U(i,j,-k-1,\delta_{k}+h_{k}).$ It
means that $\left[t_{0}-h_{k},t_{0}+h_{k}\right]$ has no common points with
the intervals in (16) and (17). Therefore, using (18) we obtain that there
exists $h\in(0,\pi)$ such
that$\left[t_{0}-h_{k},t_{0}+h_{k}\right]\subset[-h,2\pi-h)\backslash
U_{\delta_{k}}\left(A(k,j)\right).$ Then by Theorem 3$(b)$ the conditions of
Lemma 1 holds. Thus $\mu_{k,j}(t_{0})\in\sigma(T)$ for each
$t_{0}\in[a_{v,k},b_{v,k}]$ and $v=1,2,...,l.$ It means that the set (25)
belong to the spectrum. Therefore, using (7) one can easily verify that (24)
follows from (23) and (26) follows from (24).

Now to prove the main result for Case 3 we use the following consequence of
Theorem 5.

###### Corollary 1

Suppose that the matrix $C$ has a real eigenvalue $\mu_{j}$ of odd
multiplicity $m_{j}$ and $n$ is an even number. Then there exist $H>0$ and
$\gamma_{k}=o(k^{n-2})$ such that the gaps of $\sigma(T)$ lying in
$[H,\infty)$ are contained in the union of the interval $S(2k,i,j)$ and
$S(2k+1,i,j)$ for $i=1,2,...,s$ and $\left|k\right|>N,$ where

$S(l,i,j)=\left(\left(\pi l\right)^{n}+\frac{\mu_{i}+\mu_{j}}{2}\left(\pi
l\right)^{n-2}-\gamma_{l},\left(\pi
l\right)^{n}+\frac{\mu_{i}+\mu_{j}}{2}\left(\pi
l\right)^{n-2}+\gamma_{l}\right).$

Proof. By definition, the set
$[a_{1,k},b_{1,k}]\cup[a_{2,k},b_{2,k}]\cup...\cup[a_{l,k},b_{l,k}]$ is

$[-1,2\pi-1)\backslash\left({\textstyle\bigcup\limits_{i=1}^{s}}\left(U(i,j,k,\delta_{k}+h_{k})\cup
U(i,j,-k-1,\delta_{k}+h_{k})\right)\right).$

On the other hand, by Theorem 5 the image (25) of this set belong to
$\sigma(T).$ Therefore, using (7) one can easily conclude that there exists
$H>0$ such that the gaps of $\sigma(T)$ lying in $[H,\infty)$ is contained in

$\mu_{k,j}\left({\textstyle\bigcup\limits_{i=1}^{s}}\left(U(i,j,k,\delta_{k}+h_{k})\cup
U(i,j,-k-1,\delta_{k}+h_{k})\right)\right)$

for $\left|k\right|>N.$ Moreover, using (7), (19) and (22) one can easily
verify that there exists $\gamma_{k}=o(k^{n-2})$ such that

$\mu_{k,j}\left({\textstyle\bigcup\limits_{i=1}^{s}}U(i,j,k,\delta_{k}+h_{k})\right)\subset
S(2k,i,j)$

and

$\mu_{k,j}\left({\textstyle\bigcup\limits_{i=1}^{s}}U(i,j,-k-1,\delta_{k}+h_{k})\right)\subset
S(2k+1,i,j).$

These inclusions give the proof of the corollary.

Now we are ready to prove the main result of this paper for the Case 3.

###### Theorem 6

If the matrix $C$ has at least three real eigenvalues
$\mu_{j_{1}},\mu_{j_{2}},\mu_{j_{3}}$ of odd multiplicity such that

$\min_{i_{1},i_{2},i_{3}}\left(diam(\\{\mu_{j_{1}}+\mu_{i_{1}},\mu_{j_{2}}+\mu_{i_{2}},\mu_{j_{3}}+\mu_{i_{3}}\\})\right)\neq
0,$ (27)

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

$diam(E)=\sup_{x,y\in E}\mid x-y\mid,$

then there exists a number $H$ such that $[H,\infty)\subset\sigma(L)$.

Proof. By Corollary 1 the gaps lie in each of the following three sets

${\textstyle\bigcup\limits_{i=1,2,...,s;\left|k\right|>N}}\left(S(2k,i,j_{u})\cup
S(2k+1,i,j_{u})\right)$

for $u=1,2,3.$ Therefore, to prove the theorem it is enough to show that these
sets have no common points. If they have a common point $x,$ then using the
definitions of these set we obtain that there exist $\left|k\right|\geq N;$
$l\in\left\\{2k,2k+1\right\\}$ and $i_{u}\in\left\\{1,2,...,s\right\\}$ such
that

$\mid x-(\pi l)^{n}-\frac{\mu_{j_{u}}+\mu_{i_{u}}}{2}\left(\pi
l\right)^{n-2}\mid<\beta_{l}$

for all $u=1,2,3$, where $\beta_{l}=o(l^{n-2}).$ This inequality implies that
$\mu_{j_{1}}+\mu_{i_{1}}=\mu_{j_{2}}+\mu_{i_{2}}=\mu_{j_{3}}+\mu_{i_{3}}$
which contradicts (27). The theorem is proved.

###### Remark 1

Using (6) and Theorem 1 we obtain

(a) If $m$ is an odd number, then the matrix $C$ has a real eigenvalues of odd
multiplicity.

(b) If $m$ is an even number, then the number of real eigenvalues of odd
multiplicity of the matrix $C$ is an even number.

##  3 Appendix

In this section we give the proof of Theorem 2. For this we prove that if
$\lambda(t,\varepsilon)$ is an eigenvalue of the operator
$T_{t}(\varepsilon,C)$ satisfying the inequality

$|\lambda(t,\varepsilon)-\mu_{k,j}(t)|\leq|\lambda(t,\varepsilon)-\mu_{d,u}(t)|$
(28)

for all $\left(d,u\right)\neq(k,j)$, then

$\lambda(t,\varepsilon)\in U_{\varepsilon_{k}}(\mu_{k,j}(t)),$ (29)

where $\left|k\right|\geq N$ and $N$ is defined in Theorem 2. We use the
following notations and formulas. Let $A(k,t)$ be $\left\\{k\right\\},$ if $n$
is an odd number. When $n$ is an even number, then let $A(k,t)$ be
$\left\\{k\right\\},$ $\left\\{\pm k\right\\}$ and $\left\\{k,-k-1\right\\}$
respectively, if $t\in\left([1,\pi-1]\cup[\pi+1,2\pi-1)\right),$ $t\in[-1,1)$
and $t\in(\pi-1,\pi+1).$ Using the obvious inequality
$\left|a^{n}-b^{n}\right|\geq\left|a-b\right|\left|a^{n-1}+b^{n-1}\right|$ for
$a>0$ and $b>0,$ one can easily verify that if $\left|k\right|\geq N$ and
$d\notin A(k,t),$ then

$\left|\left(2\pi k+t\right)^{n}-\left(2\pi
d+t\right)^{n}\right|\geq\pi\left|\left|k\right|-\left|d\right|\right|\left(\left|k\right|^{n-1}+\left|d\right|^{n-1}\right).$

Therefore it follows from (7) and (28) we obtain

$\left|\lambda(t,\varepsilon)-\left(2\pi
d+t\right)^{n}\right|>\left|\left|k\right|-\left|d\right|\right|\left(\left|k\right|^{n-1}+\left|d\right|^{n-1}\right)$
(30)

for all $d\notin A(k,t),$ $t\in[-1,2\pi-1)$ and $\varepsilon\in[0,1].$

The eigenfunctions $\Phi_{d,u,l,t}^{\ast}(x)$ and associated functions
$\Phi_{d,u,l,q,t}^{\ast}(x)$ of $\left(T_{t}(C)\right)^{\ast}$ corresponding
to the eigenvalue $\overline{\mu_{d,u}(t)}$ are

$\Phi_{d,u,l,t}^{\ast}(x)=v_{u,l}^{\ast}e^{i\left(2\pi d+t\right)x},\text{
}\Phi_{d,u,l,q,t}^{\ast}(x)=v_{u,l,q}^{\ast}e^{i\left(2\pi d+t\right)x},$ (31)

where $v_{u,l}^{\ast}$ and $v_{u,l,q}^{\ast}$ are the eigenvector and
associated vector of $C^{\ast}$ corresponding to $\overline{\mu_{u}},$
$l=1,2,...,l_{u\text{ }}$ and $q=1,2,...,r_{u,l}-1$ (see (8) and (10)). In the
other words,

$(\left(T_{t}(C)\right)^{\ast}-\overline{\mu_{d,u}(t)}I)\Phi_{d,u,l,t}^{\ast}=0$
(32)

and

$\text{
}(\left(T_{t}(C)\right)^{\ast}-\overline{\mu_{d,u}(t)}I)\Phi_{d,u,l,q,t}^{\ast}=\Phi_{d,u,l,q-1,t}^{\ast},$
(33)

where $\Phi_{d,u,l,0,t}^{\ast}(x)=\Phi_{d,u,l,t}^{\ast}(x)$. Let
$\Psi_{\lambda(t,\varepsilon)}$ be a normalized eigenfunction of
$T_{t}(\varepsilon,C)$ corresponding to the eigenvalue
$\lambda(t,\varepsilon).$ Multiplying both sides of

$T_{t}(\varepsilon,C)\Psi_{\lambda(t,\varepsilon)}=\lambda(t,\varepsilon)\Psi_{\lambda(t,\varepsilon)}$
(34)

by $\Phi_{d,u,l,t}^{\ast}(x),$ using
$T_{t}(\varepsilon,C)=T_{t}(C)+(T_{t}(\varepsilon,C)-T_{t}(C))$ and (32), we
get

$(\lambda(t,\varepsilon)-\mu_{d,u}(t))(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,t}^{\ast})=(T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,t}^{\ast}).$

Similarly, multiplying (34) by $\Phi_{d,u,l,1,t}^{\ast}$ and using (33) for
$q=1$ we obtain

$\left(\lambda(t,\varepsilon)-\mu_{d,u}(t)\right)(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,1,t}^{\ast})=$
$(T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,1,t}^{\ast})+(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,t}^{\ast}).$

Now, using the last two equalities, one can easily verify that

$(\lambda(t,\varepsilon)-\mu_{d,u})^{2}(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,1,t}^{\ast})=$
$(\lambda(t,\varepsilon)-\mu_{d,u})((T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,1,t}^{\ast})+$
$((T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,t}^{\ast}).$

In this way one can deduce the formulas

$(\lambda(t,\varepsilon)-\mu_{d,u}(t))^{q+1}(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,q,t}^{\ast})=$
(35)
$\sum_{p=0}^{q}(\lambda(t,\varepsilon)-\mu_{d,u})^{p}((T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,p,t}^{\ast}).$

To prove (29) we estimate the terms of (35).

###### Lemma 2

If $n$ is an even number, then there exists $d\in A(k,t)$ such that

$\left|(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,q,t}^{\ast})\right|\geq
c_{3}$ (36)

for some $u,l,q$ and

$\left|((T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,p,t}^{\ast})\right|\leq
c_{4}\left(\frac{1}{|k|}+q_{k}\right)\left|k\right|^{n-2}$ (37)

for all $u,l,p$, where $\left|k\right|\geq N$, $N$ and $q_{k}$ are defined in
Theorem 2. If $n$ is an odd number, then (36) and (37) hold for $d=k$ and
$q_{k}=0.$

Proof. To prove the lemma we use the following formula

$\left(\lambda(t,\varepsilon)-\left(2\pi
d+t\right)^{n}\right)\left(\Psi_{\lambda(t,\varepsilon)},\varphi_{d,s,t}\right)={\textstyle\sum\limits_{\nu=2}^{n}}(-i)^{n-v}(P_{\nu}\Psi_{\lambda(t,\varepsilon)}^{\left(n-\nu\right)},\varphi_{d,s,t})$
(38)

which can be obtained from (34) by multiplying by
$\varphi_{d,s,t}(x)=:e_{s}e^{i\left(2\pi d+t\right)x}$ and using the equality
$T_{t}\left(0\right)\varphi_{d,s,t}=\left(2\pi d+t\right)^{n}\varphi_{d,s,t},$
where $e_{1},e_{2},...,e_{m}$ is a standard basis of $\mathbb{C}^{m},$
$T_{t}(0)$ is the operator generated by the expression $(-i)^{n}y^{(n)}$ and
the boundary conditions (3). Using (38), (30) and Bessel inequality for the
orthonormal system

$\left\\{\varphi_{d,s,t}(x)=:e_{s}e^{i\left(2\pi
d+t\right)x}:d\in\mathbb{Z},\text{ }s=1,2,...,m\right\\}$ (39)

we obtain

$\sum\limits_{d\in\left(\mathbb{Z}\backslash A(k,t)\right),\text{
}s=1,2,...,m}\left|\left(\Psi_{\lambda(t,\varepsilon)},\varphi_{d,s,t}\right)\right|^{2}\leq\frac{\left\|{\textstyle\sum\limits_{\nu=2}^{n}}(-i)^{n-v}P_{\nu}\Psi_{\lambda(t,\varepsilon)}^{\left(n-\nu\right)}\right\|^{2}}{k^{2n-2}}.$

On the other hand, in [9] (see Lemma 2 of [9]) we proved that there exists
$c_{5}$ such that

$\left|\Psi_{\lambda(t,\varepsilon)}^{(\nu)}(x)\right|\leq
c_{5}\left|k\right|^{\nu}$ (40)

for all $x\in[0,1],t\in[-1,2\pi-1)$ and $\varepsilon\in[0,1].$ Therefore we
have

$\sum\limits_{d\in\left(\mathbb{Z}\backslash
A(k,t)\right),s=1,2,...,m}\left|\left(\Psi_{\lambda(t,\varepsilon)},\varphi_{d,s,t}\right)\right|^{2}\leq\frac{c_{6}}{k^{2}}.$

Then by the Parsevals equality

$\sum\limits_{d\in
A(k,t),s=1,2,...,m}\left|\left(\Psi_{\lambda(t,\varepsilon)},\varphi_{d,s,t}\right)\right|^{2}\geq\frac{1}{2}.$

Since the system of the root vectors of the matrix $C^{\ast}$ is a basis of
$\mathbb{C}^{m}$, (31) and the last inequality imply that (36) holds for some
$u,l,q$ and $d\in A(k,t).$ If $n$ is an odd number, then
$A(k,t)=\left\\{k\right\\}$ and hence (36) holds for $d=k.$

Now we prove (37). By the definitions of $T_{t}(\varepsilon,C)$ and $T_{t}(C)$
we have

$\displaystyle((T_{t}(\varepsilon,C)-T_{t}(C))\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,p,t}^{\ast})$
$\displaystyle=((P_{2}-C)\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\Phi_{d,u,l,p,t}^{\ast})+$
(41)
$\displaystyle{\textstyle\sum\limits_{v=3}^{n}}(P_{v}\Psi_{\lambda(t,\varepsilon)}^{\left(n-\nu\right)},\Phi_{d,u,l,p,t}^{\ast}).$

Using (40) and (31) we obtain that

$\left|{\textstyle\sum\limits_{v=3}^{n}}(P_{v}\Psi_{\lambda(t,\varepsilon)}^{\left(n-\nu\right)},\Phi_{d,u,l,p,t}^{\ast})\right|\leq
c_{7}\left|k\right|^{n-3}.$

By (31) to estimate the first term in the right side of (41) it is enough to
prove that

$\left|((P_{2}-C)\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{d,i,t})\right|\leq
c_{8}q_{k}\left|k\right|^{n-2}$

for $i=1,2,...,m.$ Using the decomposition

$\Psi_{\lambda(t,\varepsilon)}^{(n-2)}=\sum\limits_{l\in\mathbb{Z},\text{
}s=1,2,...,m}\left(\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{l,s,t}\right)\varphi_{l,s,t}$

of $\Psi_{\lambda(t,\varepsilon)}^{(n-2)}$ by the orthonormal basis (39) we
obtain.

$((P_{2}-C)\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{d,i,t})=\sum\limits_{l\in\left(\mathbb{Z}\backslash\left\\{d\right\\}\right),\text{
}s=1,2,...,m}p_{2,i,s,d-l}\left(\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{l,s,t}\right).$
(42)

The right-hand side of (42) is the sum of

$S_{1}=:\sum_{l\in A(k,t)\backslash\left\\{d\right\\};\text{
}s=1,2,...,m}p_{2,i,s,d-l}\left(\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{l,s,t}\right)$

and

$S_{2}=:\sum\limits_{l\in\left(\mathbb{Z}\backslash A(k,t)\right);\text{
}s=1,2,...,m}p_{2,i,s,d-l}\left(\Psi_{\lambda(t,\varepsilon)}^{(n-2)},\varphi_{l,s,t}\right).$

First, let us estimate $S_{1}.$ It follows from the definition of $A(k,t)$
that the set $A(k,t)\backslash\left\\{d\right\\}$ for $d\in A(k,t)$ consists
of at most one number. Moreover, it follows from the definition of $A(k,t)$
that $(d-l)\in\left\\{\pm 2k,\pm(2k+1),\right\\}$ for all $d\in A(k,t)$ and
$l\in\left(A(k,t)\backslash\left\\{d\right\\}\right).$ Therefore, using (40)
and the definition of $q_{k}$ we obtain

$\left|S_{1}\right|\leq c_{9}q_{k}\left|k\right|^{n-2}.$ (43)

It remains to estimate $S_{2}.$ Using the Schwards inequality for the space
$l_{2}$ and the integration by parts formula and the last relation in (2) we
obtain

$\left|S_{2}\right|^{2}\leq c_{10}\sum\limits_{l\in\left(\mathbb{Z}\backslash
A(k,t)\right);\text{
}s=1,2,...,m}|l|^{2n-4}\left|\left(\Psi_{\lambda(t,\varepsilon)},\varphi_{l,s,t}\right)\right|^{2}.$
(44)

Now, first use (38) and then (30) in (44) to conclude that

$\left|S_{2}\right|^{2}\leq\sum\limits_{\begin{subarray}{c}l\in\left(\mathbb{Z}\backslash
A(k,t)\right),\\\
s=1,2,...,m\end{subarray}}\frac{c_{11}|l|^{2n-4}}{\left|\left|k\right|-\left|l\right|\right|^{2}\left(\left|k\right|^{n-1}+\left|l\right|^{n-1}\right)^{2}}\left|{\textstyle\sum\limits_{v=2}^{n}}(-i)^{n-v}(P_{v}\Psi_{\lambda(t,\varepsilon)}^{\left(n-v\right)},\varphi_{l,s,t})\right|^{2}.$

Finally, using (40) in the last inequality we get

$\left|S_{2}\right|^{2}\leq\sum\limits_{l\in\left(\mathbb{Z}\backslash
A(k,t)\right),\text{
}s=1,2,...,m}\frac{c_{12}|l|^{2n-4}|k|^{2n-4}}{\left|\left|k\right|-\left|l\right|\right|^{2}\left(\left|k\right|^{n-1}+\left|l\right|^{n-1}\right)^{2}}$

from which by direct calculations we obtain

$\left|S_{2}\right|^{2}\leq c_{13}|k|^{2n-6},\text{ }\left|S_{2}\right|\leq
c_{14}|k|^{n-3}.$ (45)

Thus (37) follows from (43) and (45). It is clear that if $n$ is an odd
number, then $A(k,t)\backslash\left\\{d\right\\}$ is an empty set for $d\in
A(k,t).$ Therefore $S_{1}=0$ and in (37) the term $q_{k}$ does not appear.

Now we are ready to prove Theorem 2. Dividing (35) by
$(\Psi_{\lambda(t,\varepsilon)},\Phi_{d,u,l,q,t}^{\ast}),$ and using (36) and
(37) we obtain

$(\lambda(t,\varepsilon)-\mu_{d,u}(t))^{q+1}=\sum_{p=0}^{q}(\lambda(t,\varepsilon)-\mu_{d,u}(t))^{p}\left|k\right|^{n-2}O\left(\frac{1}{|k|}+q_{k}\right),$

where $q+1\leq r$ and $r$ is defined in Theorem 2. From the last equality we
obtain that $\lambda(t,\varepsilon)\in U_{\varepsilon_{k}}(\mu_{d,u}(t)).$
This inclusion with (28) implies (29) which gives the proof of Theorem 2.

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