text:
\section*{Introduction}

Let ${\mathcal M}_2$ be the space of complex $(2\times 2)$-matrices,
$\widehat I\in {\mathcal M}_2$ the unit matrix, and
$$
\widehat \sigma _1=\left( \begin{matrix}  0 & 1\\ 1 & 0 \end{matrix} \right)
\, ,\
\widehat \sigma _2=\left( \begin{matrix}  0 & -i\\ i & 0 \end{matrix} \right)
\, ,\
\widehat \sigma _3=\left( \begin{matrix}  1 & 0\\ 0 & -1 \end{matrix} \right)
$$
the Pauli matrices. Consider the generalized Dirac operator
$$
\widehat {\mathcal D}=\sum\limits
_{j=1}^2(h_{j1}\widehat {\sigma }_1+h_{j2}\widehat {\sigma }_2)\biggl( -i\,
\frac {\partial}{\partial x_j}\biggr) \, ,
$$
acting in $L^2({\bf R}^2;{\bf C}^2)$, with the domain
$D(\widehat {\mathcal D})=H^1({\bf R}^2;{\bf C}^2)$. The functions $h_{jl}\in
L^{\infty}({\bf R}^2;{\bf R})$ are assumed to be periodic with a (common) period
lattice $\Lambda \subset {\bf R}^2$, and $0<\varepsilon \leq h_{11}(x)h_{22}
(x)-h_{12}(x)h_{21}(x)$ for a.e. $x\in {\bf R}^2$. We denote by
${\mathbb L}_{\Lambda}({\bf R}^2)$ the set of all $\Lambda $-periodic functions
$W\in L^2_{\rm loc}({\bf R}^2;{\bf C})$ such that $W\varphi \in
L^2({\bf R}^2)$ for any $\varphi \in H^1({\bf R}^2)$, and for every $\varepsilon >0$
there exists a number $C_{\varepsilon}(W)\geq 0$ satisfying
$$
\| W\varphi \| _{L^2({\bf R}^2)}\leq \varepsilon \, \| \nabla \varphi \|
_{L^2({\bf R}^2;{\bf C}^2)}+C_{\varepsilon}(W)\| \varphi \| _{L^2({\bf R}^2)}
$$
for all $\varphi \in H^1({\bf R}^2)$. If $V^{(l)}\in {\mathbb L}_{\Lambda}({\bf R}^2)$,
$l=0,{\,}1,{\,}2,{\,}3$, then
$$
\widehat {\mathcal D}+\widehat V=\widehat {\mathcal D}+V^{(0)}\widehat I+
\sum\limits_{l=1}^3V^{(l)}\widehat \sigma _l  \eqno (0.1)
$$
is a closed operator in $L^2({\bf R}^2;{\bf C}^2)$ with domain
$D(\widehat {\mathcal D}+\widehat V)=D(\widehat {\mathcal D})=H^1({\bf R}^2;{\bf
C}^2)$ (a $\Lambda $-periodic matrix-valued potential $\widehat V$ has zero
bound relative to $\widehat {\mathcal D}$ if and only if $V^{(l)}\in {\mathbb L}
_{\Lambda}({\bf R}^2)$, $l=0,{\,}1,{\,}2,{\,}3$).

The following theorem is the main result of this paper.
\vskip 0.2cm

{\bf Theorem 0.1}. {\it Let $h_{jl}\in L^{\infty}({\bf R}^2;{\bf R})$, $j,{\,}l
=1,{\,}2$, be periodic functions with period lattice $\Lambda \subset {\bf R}^2$.
Suppose that there exists $\varepsilon >0$ such that $\varepsilon \leq h_{11}(x)
h_{22}(x)-h_{12}(x)h_{21}(x)$ for a.e. $x\in {\bf R}^2$. If
$V^{(l)}\in {\mathbb L}_{\Lambda}({\bf R}^2)$, $l=0,{\,}1,{\,}2,{\,}3$, then
the operator (0.1) has no egenvalues.}
\vskip 0.2cm

If under the conditions of Theorem 0.1, the operator (0.1) is selfadjoint,
then its spectrum is absolutely continuous. For the selfadjoint periodic
elliptic differential operator, the absolute continuity of the spectrum is a
consequence of the absence of eigenvalues (of infinite multiplicity); see [1]
(and also [2]). This fact is of general nature and is also valid for the
generalized Dirac operator $\widehat {\mathcal D}+\widehat V$.

The first results about the absence of eigenvalues in the spectrum of the
periodic Dirac operator were obtained in [3] -- [5]. For $n\geq 2$,
consider the $n$-dimensional periodic (with period lattice $\Lambda \subset
{\bf R}^n$) Dirac operator
$$
\sum\limits_{j=1}^n\widehat \alpha _j\bigl( -i\, \frac {\partial}{\partial x_j}-
A_j\bigr) +\widehat V+\widehat V_0\, ,\ x\in {\bf R}^n\, , \eqno (0.2)
$$
where
$$
\widehat V=V\widehat I\, ,\ \widehat V_0=m\widehat \alpha
_{n+1}\, ,\ m\in {\bf R}\, ,  \eqno (0.3)
$$
the $\widehat \alpha _j\, $, $j=1,\dots ,n+1$, are Hermitian $(M\times M)$-matrices
satisfying the anticommutation relations $\widehat \alpha _j\widehat
\alpha _l+\widehat \alpha _l\widehat \alpha _j=2\delta _{jl}\widehat I$, $\delta
_{jl}$ is the Kronecker symbol, and $\widehat I$ is the unit $(M\times M)$-matrix
($M\in 2{\bf N}$). The components $A_j$ of the vector-valued (magnetic) potential
and the scalar (electric) potential $V$ are real-valued periodic functions with
period lattice $\Lambda \subset {\bf R}^n$; let $K$ be the standard fundamental
domain of $\Lambda$. In [5,{\,}6], the absolute continuity of the spectrum of the
operator (0.2), (0.3) was proved for all $n\geq 2$ under the conditions $V\in
C({\bf R}^n)$, $A\in L^{\infty}({\bf R}^n;{\bf R}^n)$, and
$$
\| \, |A|\, \| _{L^{\infty}({\bf R}^n)}<\max\limits_{\gamma \, \in \, \Lambda
\backslash \{ 0\} }\, \frac {\pi}{|\gamma |}  \eqno (0.4)
$$
($|x|$ is the length of the vector $x\in {\bf R}^n$). In subsequent papers, the
restriction on the periodic scalar potential $V$ has been relaxed. The spectrum
of the operator (0.2), (0.3) is absolutely continuous if (at least) one of the
following conditions is satisfied:

1) $n=2$, $V\in L^q(K)$, $q>2$, and the vector-valued potential $A\in L^{\infty}
({\bf R}^2;{\bf R}^2)$ satisfies (0.4) (see [7]);

2) $n\geq 3$, $A\equiv 0$, and $\sum\limits_N|V_N|^p<+\infty $, where the $V_N$
are the Fourier coefficients of $V$, $p\in [1,q_n(q_n-1)^{-1})$, and the $q_n>n$
are some numbers, the smallest (found before) values of which were presented
for $n\geq 4$ in [8] (the numbers $q_n$ are found as the largest roots of the
algebraic equations
$$
q^4-(3n^2-4n-1)q^3+2(4n^2-6n-3)q^2-(9n^2-16n-4)q-4n(n-2)=0\, ,
$$
$q_3\simeq 11.645$, $n^{-2}q_n\to 3$ as $n\to +\infty $; for the first time
the above condition on the Fourier coefficients $V_N$ appeared in [4] for $n=3$);

3) $n=3$, $V\in L^q(K)$, $q>3$, and the vector-valued potential $A\in L^{\infty}
({\bf R}^3;{\bf R}^3)$ satisfies (0.4) (see [9]);

4) $n\geq 2$, $V\in L^2(K)$, $A\in L^{\infty}({\bf R}^n;{\bf R}^n)$, there exists
a vector $\gamma \in \Lambda \backslash \{ 0\} $ such that $\| \, |A|\, \|
_{L^{\infty}({\bf R}^n)}<\pi |\gamma |^{-1}$ and the map
$$
{\bf R}^n\ni x\to \{ [0,1]\ni t\to V(x+t\gamma )\} \in L^2([0,1])
$$
is continuous; see [10]. (In particular, under an appropriate choice of $\gamma
\in \Lambda \backslash \{ 0\} $, the latter condition is satisfied if $V$ is an
arbitrary piecewise continuous scalar potential with piecewise analytic
discontinuity surfaces; such potentials were considered in [3].)

Some other conditions on the scalar potential $V$ and a small vector-valued
potential $A$ can be found in [10].

The periodic Dirac operator with a nonsmall vector-valued potential $A$
was studied in [11] -- [13]. In [12], the absolute continuity of the
spectrum of the operator (0.2) was proved under the conditions $\widehat V
=V\widehat I$, $\widehat V_0=V_0\widehat \alpha _{n+1}\, $, where $V,V_0\in
L^q(K)$, $A\in L^q(K;{\bf R}^2)$, $q>2$, for $n=2$, and under the conditions
$V,V_0\in C({\bf R}^n;{\bf R})$, $A\in C^{2n+3}({\bf R}^n;{\bf R}^n)$ for
$n\geq 3$. For $n=2$, the proof is based on the results of the papers [14,{\,}15],
where the periodic Schr\" odinger operator
$$
\sum\limits_{j=1}^2\bigl( -i\, \frac {\partial}{\partial x_j}-A_j\bigr) ^2+V\, ,\
x\in {\bf R}^2\, .  \eqno (0.5)
$$
was treated. For the operator (0.5), the absolute continuity of the spectrum
was proved first for $V\in L^2_{\rm loc}({\bf R}^2;{\bf R})$, $A\in C({\bf R}^2;
{\bf R}^2)$ in [14], and then in [15] for the more general case where
$V\in L^q_{\rm loc}({\bf R}^2;{\bf R})$, $A\in L^{2q}_{\rm loc}({\bf R}^2;
{\bf R}^2)$, $q>1$ . For the periodic Dirac operator (0.2) with $n=2$, a similar
result (as in [12]) was obtained in [11] (it was assumed, however, that $V_0\equiv
m={\rm const}$, but the proof carries over to functions $V_0\in L^q(K)$, $q>2$,
without essential modifications). The methods used in [11] were the same as in [7].
More general conditions on $V$, $V_0$, and $A$ (for $n=2$) were obtained in [16]:
it suffices to require that the functions $V^2\ln (1+|V|)$, $V_0^2\ln (1+|V_0|)$,
and $|A|^2\ln ^q(1+|A|)$ belong to $L^1(K)$ for some $q>1$. For $n\geq 3$, the
results of [12] were based on Sobolev's paper [17], where the absolute
continuity of the spectrum was proved for the Schr\" odinger operator with a
periodic vector-valued potential $A\in C^{2n+3}({\bf R}^n;{\bf R}^n)$. The latter
condition was relaxed by Kuchment and Levendorski$\breve {\mathrm i}$ in [2]
(and also by Sobolev; see the remark at the end of the survey [18]): it suffices
to require that $A\in H^q_{\rm loc}({\bf R}^n;{\bf R}^n)$, $2q>3n-2$, which makes it
possible to relax accordingly the constraint on the vector-valued potential
$A$ also for the periodic Dirac operator (see [12,{\,}18]).

Let ${\mathfrak M}_h$, $h>0$, be the set of all even Borel signed measures
(charges) $\mu$ on ${\bf R}$ (of finite total variation) for which
$\int_{{\bf R}}e^{\, ipt}d\mu (t)=1$ for every $p\in (-h,h)$. In
[13], it was proved that the spectrum of the periodic (with period lattice
$\Lambda \subset {\bf R}^n$) Dirac operator (0.2) is absolutely continuous for
$n\geq 3$ if the following conditions are fulfilled:

1) the $(M\times M)$-matrix-valued functions $\widehat V$ and $\widehat
V_0$ are Hermitian and continuous, and $\widehat V(x)\widehat \alpha _j=\widehat
\alpha _j\widehat V(x)$, $\widehat V_0(x)\widehat \alpha
_j=- \widehat \alpha _j\widehat V_0(x)$ for all $x\in {\bf R}^n$, $j=1,\dots ,n$;

2) $A\in C({\bf R}^n;{\bf R}^n)$ and there exist a vector $\gamma \in \Lambda
\backslash \{ 0\} $ and a measure $\mu \in {\mathfrak M}_h\, $, $h>0$, such that
for every $x\in {\bf R}^n$ and every unit vector $e\in {\bf R}^n$ with
$(e,\gamma )=0$ (where $(.,.)$ is the scalar product in ${\bf R}^n$) we have
$$
\biggl| \ \int\limits_{\bf R}d\mu (t)\int\limits_0^1A(x-\xi \gamma -te)\, d\xi -
A_0\, \biggr| <\frac {\pi}{|\gamma |}\, ,  \eqno (0.6)
$$
where $A_0=v^{-1}(K)\int_KA(x)\, d^{\, n}x$, $v(K)$ is the volume of the
fundamental domain $K$.

For a periodic vector-valued potential $A\in C({\bf R}^n;{\bf R}^n)$, condition
(0.6) is fulfilled (under an appropriate choice of $\gamma \in \Lambda \backslash
\{ 0\} $ and measure $\mu \in {\mathfrak M}_h\, $, $h>0$) whenever $A\in H^q_{\rm loc}
({\bf R}^n;{\bf R}^n)$, $2q>n-2$, and also in the case where $\sum\limits_N|A_N|_{{\bf
C}^n}<+\infty $, where the $A_N$ are the Fourier coefficients of $A$ (see [13,{\,}19]).
The results of [13] were used in [20] in order to prove the absolute continuity of
the spectrum of the periodic Schr\" odinger operator
$$
\sum\limits_{j=1}^n\bigl( -i\, \frac {\partial}{\partial x_j}-A_j\bigr) ^2+V\, ,\
x\in {\bf R}^n\, ,\ n\geq 3\, ,
$$
with a vector-valued potential $A\in C^1({\bf R}^n;{\bf R}^n)$ satisfying condition
2) and with the scalar potential $V\in L^p_w(K;{\bf R})$ for which $t\, ({\rm
meas}\, \{ x\in K:|V(x)|>t\} )^{1/p}\to 0$ as $t\to +\infty $, where $p=n/2$ for
$3\leq n\leq 6$ and $p=n-3$ for $n\geq 7$, ${\rm meas}$ standing for Lebesgue
measure. The multidimensional periodic Schr\" odinger operator was studied in
many papers; the relevant facts and references can be found in [2,{\,}18] and
[21] -- [25].

Let ${\mathbb G}$ denote the set of all continuously differentiable and
monotone nonincreasing functions $g:(0,+\infty )\to (0,+\infty )$ (i.e., $g(t_1)
\geq g(t_2)$ for $0<t_1\leq t_2$) such that $\int_0^1(r{\,}g(r))^{-1}
dr < +\infty $ and $g(r/2)/g(r)\to 1$ as $r\to +0$. We write
$L^2\{ g,\Lambda \} $, $g\in {\mathbb G}$, for the Banach space of periodic
(with period lattice $\Lambda \subset {\bf R}^2$) functions $W\in L^2_{\rm loc}
({\bf R}^2;{\bf C})$ such that
$$
\| W\| ^2_{L^2\{ g,\Lambda \} }=\sup\limits_{x\, \in \, {\bf R}^2}\ \int\limits
_{y\, :\, |x-y|\, \leq \, 1}g(|x-y|)\, |W(y)|^2d^{\, 2}y<+\infty \, .
$$
If $g\in {\mathbb G}$, then $g(r)(\ln r)^{-1}\to -\infty $ as $r\to +0$; therefore,
for any $W\in L^2\{ g,\Lambda \} $ the function $W^2$ belongs to the Kato class
$K_2$ (see [26]), which implies that $W\in {\mathbb L}_{\Lambda}({\bf R}^2)$.

For $n=2$, the operator $\widehat {\mathcal D}+\widehat V$ (see (0.1)) was considered
in [27] in the case where $h_{jl}\in C^{\infty}({\bf R}^2)$, $j,l=1,2$, $V^{(l)}\in
C^{\infty}({\bf R}^2;{\bf R})$ for $l=1,2$, and $V^{(l)},\, \partial V^{(l)}/\partial
x_j\in L^{\infty}({\bf R}^2)$ for $l=0,3$ and $j=1,2$. In [28], a special case of
Theorem 0.1 was proved: under the same conditions on $h_{jl}\, $, it was assumed
that $\widehat V\in L^q_{\rm loc}({\bf R}^2;{\mathcal M}_2)$, $q>2$. The latter
result was improved in [29] (and was announced in [30]): it suffices to require that
$V^{(0)},V^{(3)}\in {\mathbb L}_{\Lambda}({\bf R}^2)$ and $V^{(1)},V^{(2)}\in L^2\{ g,
\Lambda \} \subset {\mathbb L}_{\Lambda}({\bf R}^2)$ for some $g\in {\mathbb G}$.

The methods employed in the proof of Theorem 0.1 can also be used for the proof
of the absolute continuity of the spectrum of the two-dimensional periodic
Schr\" odinger operator
$$
\sum\limits_{j,l=1}^2\bigl( -i\, \frac {\partial}{\partial x_j}-A_j\bigr) G_{jl}
\bigl( -i\, \frac {\partial}{\partial x_l}-A_l\bigr) +V\, ,\
x\in {\bf R}^2\, ,  \eqno (0.7)
$$
with variable metric (see [29] -- [32]); here $G=(G_{jl})_{j,l=1,2}$ is a real
symmetric positive definite matrix-valued function (a metric) with $G,{\,}G^{-1}\in
L^{\infty}({\bf R}^2;{\mathcal M}_2)$. The functions $A$, $V$, and $G$ are periodic
with a common period lattice $\Lambda \subset {\bf R}^2$. For the first time, the
absolute continuity of the spectrum of the operator (0.7) was proved by Morame in [27]
under the conditions $A\in C^{\infty}({\bf R}^2;{\bf R}^2)$, $V\in L^{\infty}({\bf
R}^2;{\bf R})$, and $G\in C^{\infty}({\bf R}^2;{\mathcal M}_2)$, ${\rm det}\, G\equiv
1$. Later on, in the case where $G\in C^{\, m+\alpha }({\bf R}^2;{\mathcal M}_2)$,
$m\in {\bf Z}_+\, $, $\alpha \in (0,1)$, Kuchment and Levendorski$\breve {\mathrm i}$
in [2] proved the existence of periodic isothermal coordinates $y(x)\in C^{\,
m+1+\alpha }({\bf R}^2;{\bf R}^2)$ that reduce the matrix-valued function $G$ to a
scalar form; the use of such coordinates allowed them to relax the restrictions on
$A$, $V$, and $G$ by reducing the problem to the case of a constant matrix $G$.
The periodic isothermal coordinates were applied in a series of papers by Birman,
Suslina, and Shterenberg. In [33], the absolute continuity of the spectrum of the
operator (0.7) was proved for $G\in W^2_{2q,\, {\rm loc}}({\bf R}^2;{\mathcal M}_2)$,
$A\in L^{2q}(K;{\bf R}^2)$, $q>1$, and $V=V_1+\sigma \delta _{\, \Sigma}\, $, where
$V_1\in L^q(K;{\bf R})$, $\Sigma$ is a periodic system of piecewise smooth curves,
$\delta _{\, \Sigma}$ is the $\delta $-function concentrated on $\Sigma$, and
$\sigma \in L^q_{\rm loc}(\Sigma ;{\bf R})$. In the subsequent papers [34] -- [37],
Shterenberg relaxed the conditions imposed on $A$, $V$, and $G$. The following
conditions were given in [36]:
$$
{\rm det}\, G\in H^1_{\rm loc}({\bf R}^2)\, ,\ \frac {\partial}{\partial x_j}\,
{\rm det}\, G\in {\mathbb L}_{\Lambda}({\bf R}^2)\, ,\ j=1,{\,}2\, ,
\eqno (0.8)
$$ $$
|A|^2\, \widetilde g(|A|)\in L^1_{\rm loc}({\bf R}^2)\, ,  \eqno (0.9)
$$
where $\widetilde g(t)=l(t)\doteq l^q_m(t)\prod\limits_{i=1}^{m-1}l_i(t)$,
$m\in {\bf N}$, $q>1$, $l_1(t)=1+\ln (1+t)$, $l_i(t)=1+\ln l_{i-1}(t)$, $i=2,
\dots ,m\, $, $t\geq 0$, and the scalar potential $V$ is defined as the distribution
$d\mu /d^{\, 2}x$, where $\mu$ is a periodic Borel signed measure satisfying some
additional conditions (which occur also in [35]); in this situation the closure of
the quadratic form ${\mathcal V}(\varphi ,\varphi )=\int\limits_{{\bf R}^2}|\varphi |
^2d\mu $, $\varphi \in C^{\infty}_0({\bf R}^2)$, may fail to be bounded relative to the
form $\| \nabla \varphi \| _{L^2({\bf R}^2;{\bf C}^2)}^2\, $, $\varphi \in H^1({\bf
R}^2)\subset L^2({\bf R}^2)$. Finally, in [37], the restriction (0.9) on the
vector-valued potential $A$ was relaxed to $A_j\in {\mathbb L}_{\Lambda}({\bf R}^2)$,
$j=1,2$. It should be mentioned that in [36,{\,}37] Shterenberg studied also a
generalization of the operator (0.7) obtained by the incorporation of weight
functions. The results pertaining to the absolute continuity of the spectrum of the
periodic Schr\" odinger operator were applied to the study of the spectrum of the
Schr\" odinger operator in periodic waveguides (see [38,{\,}39] and also
[40,{\,}41]).

In [30] (without using periodic isothermal coordinates, and with the help of
results on the generalized two-dimensional periodic Dirac operator), it was proved
that no eigenvalues are present in the spectrum of the periodic operator (0.7) if
the following conditions are fulfilled: $G$ satisfies (0.8), $A_j\in L^2\{ g,\Lambda
\} $, $j=1,{\,}2$, for some $g\in {\mathbb G}$, and the scalar potential $V$ is
defined via a quadratic form ${\mathcal V}(\varphi ,\varphi )$, $\varphi \in H^1({\bf R}^2)$;
this form has zero bound relative to the form $\| \nabla \varphi \| ^2_{L^2({\bf R}^2;
{\bf C}^2)}\, $, $\varphi \in H^1({\bf R}^2)\subset L^2({\bf R}^2)$, and is such that
${\mathcal V}(\psi (.-\gamma ),\varphi (.-\gamma ))={\mathcal V}(\psi ,\varphi )$ for all
$\psi ,\varphi \in H^1({\bf R}^2)$, $\gamma \in \Lambda $, and ${\mathcal V}(e^{\,
i(k,x)}\psi ,e^{\, i(k,x)}\varphi )={\mathcal V}(\psi ,\varphi )$ for all $\psi ,\varphi \in
H^1({\bf R}^2)$ and $k\in {\bf R}^2$ (if the form ${\mathcal V}(\varphi ,\varphi )$ is
Hermitian, then it may fail to coincide with $\int\limits_{{\bf R}^2}|\varphi |^2d\mu $
for $\varphi \in C^{\infty}_0({\bf R}^2)$, where $\mu$ is a periodic (locally finite)
Borel signed measure [29]). In [30], the functions $A_j\, $, $j=1,2$, can be
complex-valued, and the form ${\mathcal V}$ is not assumed to be Hermitian.

If the function $[0,+\infty )\ni t\to \widetilde g(t)\in [0,+\infty )$ is monotone
nondecreasing and the function $(0,+\infty )\ni t\to \widetilde g(t^{-1})$ belongs
to ${\mathbb G}$ (in particular, this is true if $\widetilde g(.)=l(.)$), then for
every periodic vector-valued potential $A\in L^2_{\rm loc}({\bf R}^2;{\bf C}^2)$
satisfying (0.9) there exists $g\in {\mathbb G}$ such that $A_j\in L^2\{ g,\Lambda
\} $, $j=1,{\,}2$ (see [31]). Therefore, the condition imposed in [30] on the
vector-valued potential $A$ is less restrictive than (0.9) (with $\widetilde g$
as indicated). In [32] (as well as in [37]), the constraint on $A$ was relaxed
up to $A_j\in {\mathbb L}_{\Lambda}({\bf R}^2)$, $j=1,2$.

\section{Notation and the main statements}

Since the change $V^{(0)}-\lambda \to V^{(0)}$ reduces the operator $\widehat
{\mathcal D}+\widehat V-\lambda \widehat I$, where $\widehat I$ is the identity
operator on $L^2({\bf R}^2;{\bf C}^2)$ and $\lambda \in {\bf C}$, to the operator
$\widehat {\mathcal D}+\widehat V$, it follows that in the proof of Theorem 0.1 it
suffices to check the absence of the eigenvalue $\lambda =0$. Also, we may assume
that $\Lambda ={\bf Z}^2$ (an appropriate linear change of variables can be made,
preserving the form of the operator $\widehat {\mathcal D}+\widehat V$). Denote
${\mathbb L}({\bf R}^2)\doteq {\mathbb L}_{{\bf Z}^2}({\bf R}^2)$, $K=[0,1)^2$. Let
$0<q\leq p<+\infty $, let $F\geq 0$, and let $\Gamma (p,q,F)$ denote the set of
ordered collections $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} $ of
${\bf Z}^2$-periodic functions ${\mathcal F},{\mathcal G},{\mathcal H}$ in
$L^{\infty}({\bf R}^2;{\bf R})$ such that $q\leq {\mathcal G}(x)\leq p$, $q\leq
{\mathcal H}(x)\leq p$, and $|{\mathcal F}(x)|\leq F$ for a.e. $x\in {\bf R}^2$.
We put $\Gamma =\bigcup\limits_{p,q,F}\Gamma (p,q,F)$. Multiplying the generalized
Dirac operator (0.1) from the left by the (unitary) matrix-valued function
$$
(h^2_{21}(x)+h^2_{22}(x))^{-1/2}\, (h_{22}(x)\widehat I-ih_{21}(x)\widehat \sigma
_3)\, ,\ x\in {\bf R}^2,
$$
we obtain the operator
$$
\widehat {\mathcal D}+\widehat V=({\mathcal G}\widehat \sigma _1+
{\mathcal F}\widehat \sigma _2)\bigl( -i\, \frac {\partial}{\partial x_1}\bigr)
+{\mathcal H}\widehat \sigma _2\bigl( -i\, \frac {\partial}{\partial x_2}\bigr)
+\widehat V\, ,  \eqno (1.1)
$$
for which $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} \in \Gamma $ and the
(new) periodic matrix-valued potential $\widehat V$ satisfies the assumptions of
Theorem 0.1. Therefore, Theorem 0.1 is a direct consequence of the next
Theorem 1.1.
\vskip 0.2cm

{\bf Theorem 1.1}. {\it Suppose $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} \in
\Gamma $ and
$$
\widehat V=V^{(0)}\widehat I+\sum\limits_{l=1}^3V^{(l)}\widehat \sigma _l
$$
with $V^{(l)}\in {\mathbb L}({\bf R}^2)$, $l=0,{\,}1,{\,}2,{\,}3$. Then the
generalized Dirac operator (1.1), acting in $L^2({\bf R}^2;{\bf C}^2)$ and defined
on the Sobolev class $H^1({\bf R}^2;{\bf C}^2)$, is invertible (i.e., $\lambda =0$
is not an eigenvalue of this operator).}
\vskip 0.2cm

For the proof of Theorem 1.1, we apply Thomas' method (originating from [42] and
used for checking the absence of eigenvalues in the spectrum of periodic elliptic
differential operators). With the help of this method, in this section we reduce
Theorem 1.1 to Theorem 1.2.

As a preliminary, we introduce some notation and several definitions. The Fourier
coefficients of a function $\varphi \in L^1(K;{\bf C}^d)$, $d=1,{\,}2$, will be
denoted by
$$
\varphi _N=\int\limits_K\varphi (x)\, e^{-2\pi i\, (N,\, x)}d^{\, 2}x\, ,\ N\in {\bf
Z}^2.
$$
Let $\widetilde C(K)$, $\widetilde C^1(K)$, and $\widetilde H^1(K)$ be the spaces
of functions $\varphi :K\to {\bf C}$, the ${\bf Z}^2$-periodic extensions of which
belong to $C({\bf R}^2)$, $C^1({\bf R}^2)$, and the Sobolev class $H^1_{\rm loc}
({\bf R}^2)$, respectively; by $\widetilde C_0(K)$, $\widetilde C^1_0(K)$, and
$\widetilde H^1_0(K)$ we denote the corresponding subspaces of functions $\varphi$
such that $\varphi _0=\int\limits_K\varphi (x)\, d^{\, 2}x=0$; $\widetilde H^1(K;{\bf
C}^2)=(\widetilde H^1(K))^2$. In what follows, we identify functions defined on
$K$ with their ${\bf Z}^2$-periodic extensions to ${\bf R}^2$. The norms and scalar
products in ${\bf C}^d$, $L^2({\bf R}^2;{\bf C}^d)$, and $L^2(K;{\bf C}^d)$, $d=
1,{\,}2$, are standard (as a rule, we do not indicate a particular space in the
notation for its norm and scalar product). The scalar products are assumed to be
linear in the second argument; $\nabla =(\partial /\partial x_1 \, ,\partial /
\partial x_2)$, and ${\rm meas}$ is Lebesgue measure on ${\bf R}^2$.

Let $\{ {\mathcal F} ,{\mathcal G} ,{\mathcal H}\} \in \Gamma (p,q,F)$. For all
$k=(k_1,k_2)\in {\bf R}^2$ and all $\varkappa =(\varkappa _1,\varkappa _2)\in {\bf R}^2$,
we introduce the operators
$$
\widehat {\mathcal D}(k+i\varkappa )=({\mathcal G}\widehat \sigma _1+{\mathcal
F}\widehat \sigma _2)(k_1+i\varkappa _1-i\, \frac {\partial}{\partial x_1})+
{\mathcal H}\widehat \sigma _2(k_2+i\varkappa _2-i\, \frac {\partial}{\partial
x_2})\, ,
$$
acting in $L^2(K;{\bf C}^2)$, with $D(\widehat {\mathcal D}(k+i\varkappa ))=
\widetilde H^1(K;{\bf C}^2)$. Put
$$
\widehat d_{\pm}(k+i\varkappa )=({\mathcal G}\pm i{\mathcal F})(k_1+i\varkappa _1-i\,
\frac {\partial}{\partial x_1})\pm i{\mathcal H}(k_2+i\varkappa _2-i\, \frac
{\partial}{\partial x_2})\, ,
$$
$D(\widehat d_{\pm}(k+i\varkappa ))=\widetilde H^1(K)\subset L^2(K)$, $\widehat
d_{\pm}\doteq \widehat d_{\pm}(0)$;
$$
\widehat {\mathcal D}(k+i\varkappa )=\left( \begin{matrix} 0  &\widehat d_-(k+i
\varkappa ) \\ \widehat d_+(k+i\varkappa ) &0 \end{matrix} \right) \, .  \eqno (1.2)
$$
There exist numbers $c_1=c_1(p,q,F)>0$ and $c_2=c_2(p,q,F)\geq c_1$ such that for
all $k\in {\bf R}^2$ and all $\varphi \in \widetilde H^1(K)$ we have
$$
c_1\sum\limits_{j=1}^2\| \biggl( k_j-i\, \frac {\partial}{\partial x_j}\biggr)
\varphi \| ^2\leq \| \widehat d_{\pm}(k)\varphi \| ^2\leq c_2\sum\limits_{j=1}^2\|
\biggl( k_j-i\, \frac {\partial}{\partial x_j}\biggr) \varphi   \| ^2  \eqno (1.3)
$$
(see, e.g., [29]).

Relations (1.2) and (1.3) imply that the operators $\widehat {\mathcal D}(k)$,
$k\in {\bf R}^2$, are closed, and if $k\not\in 2\pi {\bf Z}^2$, then their range
$R(\widehat {\mathcal D}(k))$ coincides with the space $L^2(K;{\bf C}^2)$,
${\rm ker}\, \widehat {\mathcal D}(k)=\{ 0\} $, and the inverse operators $\widehat
{\mathcal D}^{-1}(k)$ are compact (see (1.2), (1.3), and the properties of
$\widehat d_{\pm}(k)$ presented in \S{\,}2).

The generalized Dirac operator $\widehat {\mathcal D}+\widehat V$ of the form (1.1)
is unitarily equivalent to the direct integral
$$
{\int\limits_{2\pi K}}^{\oplus} \, (\widehat {\mathcal D}(k)+\widehat V)\, \frac
{d^2k}{(2\pi )^2}\, ,  \eqno (1.4)
$$
acting in
$$
{\int\limits_{2\pi K}}^{\oplus} \, L^2(K;{\bf C}^2)\, \frac {d^2k}{(2\pi )^2}
$$
(the vector $k=(k_1,k_2)\in 2\pi K\subset {\bf R}^2$ is called the {\it
quasimomentum}). The unitary equivalence mentioned above is established via the
Gel$^{\prime }$fand transformation [43,{\,}44] (for periodic Dirac operators, see also
[9,{\,}12]). The matrix-valued potential $\widehat V$, viewed as acting in $L^2(K;{\bf
C}^2)$, has zero bound relative to the operators $\widehat {\mathcal D}(k)$, $k\in {\bf
R}^2$; therefore, the operators $\widehat V\widehat {\mathcal D}^{-1}(k)$ are compact
for all $k\in {\bf R}^2\backslash 2\pi {\bf Z}^2$. Fix a vector $k^0\in
{\bf R}^2\backslash 2\pi {\bf Z}^2$. Since
$$
(\widehat {\mathcal D}(k)+\widehat V)\widehat {\mathcal D}^{-1}(k^0)=
\widehat I+\widehat S(k)\, ,\ k\in {\bf R}^2,
$$
where $\widehat I$ is the identity operator in $L^2(K;{\bf C}^2)$, and the operator
$$
\widehat S(k)=(({\mathcal G}\widehat \sigma _1+{\mathcal F}\widehat \sigma _2)
(k_1-k^0_1)+{\mathcal H}\widehat \sigma _2(k_2-k^0_2)+\widehat V)
\widehat {\mathcal D}^{-1}(k^0)
$$
is compact, the representation of the operator (1.1) in the form of a direct
integral (1.4) and the analytic Fredholm theorem imply that if $\lambda =0$
is an eigenvalue of $\widehat {\mathcal D}+\widehat V$, then $\lambda =0$ is an
eigenvalue of each of the operators $\widehat {\mathcal D}(k+i\varkappa )+\widehat
V$ (with the domain $D(\widehat {\mathcal D}(k+i\varkappa )+\widehat V)=\widetilde
H^1(K;{\bf C}^2)\subset L^2(K;{\bf C}^2)$) for all $k+i\varkappa \in {\bf C}^2$
(see [1] and [44,{\,}{\S{\,}XIII.16}]). Consequently, for the proof of Theorem 1.1
it suffices to find a complex vector $k+i\varkappa \in {\bf C}^2$ such that ${\rm ker}\,
(\widehat {\mathcal D}(k+i\varkappa )+\widehat V)=\{ 0\} $. Thus, Theorem 1.1 is
implied by the following statement.
\vskip 0.2cm

{\bf Theorem 1.2}. {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\} \in
\Gamma (p,q,F)$,
$$
\widehat V(.)=V^{(0)}(.)\widehat I+\sum\limits_{l=1}^3V^{(l)}(.)\widehat
\sigma _l\, ,
$$
$V^{(l)}\in {\mathbb L}({\bf R}^2)$, $l=0,{\,}1,{\,}2,{\,}3$. Then there are vectors
$k^{\, \prime},\, \varkappa ^{\, \prime}\in {\bf R}^2$, and a unit vector $e=(e_1,e_2)
\in {\bf R}^2$ with $e_1>0$ such that for some numbers $\widetilde \mu >0$ as large
as we wish the following is true: for all $k\in {\bf R}^2$ with $k_1=\pi $ and all
vector-valued functions $\varphi \in \widetilde H^1(K;{\bf C}^2)$ we have
$$
\| (\widehat {\mathcal D}(k+k^{\, \prime}+i(\widetilde \mu e+\varkappa ^{\,
\prime}))+\widehat V)\varphi \| \geq e^{-c\, \widetilde \mu }\, \| \varphi \| \, ,
$$
where $c=c(p,q,F)>0$.}
\vskip 0.2cm

The proof of Theorem 1.2 is presented in {\S}{\,}5. It is based on Theorems 3.1 and
5.1, proved in {\S}{\,}3 and {\S}{\,}6, respectively. In {\S}{\,}2, for the
operators $\widehat d_{\pm}(k)\, $ we list the properties needed for what follows.
In {\S}{\,}3 we prove that the Dirac operator $\widehat {\mathcal D}(0)+\widehat V$
with a matrix-valued potential $\widehat V$ of a special form is similar to the
Dirac operator $\widehat {\mathcal D}(k+i\varkappa )$ for some vectors $k,\, \varkappa \in
{\bf R}^2$. In {\S}{\,}4 we collect the auxiliary statements to be used either in
the proof of Theorem 1.2, or (mainly) in the proof of Theorem 5.1. The estimates
proved in Theorem 5.1 for the Dirac operator $\widehat {\mathcal D}(k)+\widehat V$
with a matrix-valued potential of a special form are used in the proof of Theorem 1.2.
In {\S}{\,}6, Theorem 5.1 is deduced from Theorem 6.1, which is proved in the same
section.

\section{Properties of the operators $\widehat d_{\pm}(k)$}

In this section we present the properties of $\widehat d_{\pm}(k)$, $k\in
{\bf R}^2$, which we considered in detail in [29] and the proofs of which can be
found in [28,{\,}29]. In the case where $k=0$, we use the abbreviation $\widehat
d_{\pm}\doteq \widehat d_{\pm}(0)$. The results of this section will be employed
substantially in what follows.

Estimates (1.3) imply that the operators $\widehat d_{\pm}(k)$, $k\in {\bf R}^2$,
are closed. If $k\notin 2\pi {\bf Z}^2$, then ${\rm ker}\, \widehat d_{\pm}(k)=\{
0\} $ and $R(\widehat d_{\pm}(k))=L^2(K)$; ${\rm ker}\, \widehat d_+={\rm ker}\,
\widehat d_-$ is the one-dimensional subspace of constant functions in $L^2(K)$,
and the subspaces $R(\widehat d_{\pm})$ are closed subspaces in $L^2(K)$ for which
${\rm dim\, coker}\, \widehat d_{\pm}=1$. For all $\varphi \in \widetilde H^1(K)$
we have
$$
\overline {\widehat d_+\varphi }=-\widehat d_-\overline \varphi  \eqno (2.1)
$$
(the bar stands for complex conjugation).

On the set $\Gamma $ we consider the metric
$$
\rho _{\infty}(\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\} ,\{ {\mathcal F}^{\,
\prime}, {\mathcal G}^{\, \prime}, {\mathcal H}^{\, \prime}\} )=
$$ $$
=\max \, \{ \| {\mathcal F}-{\mathcal F}^{\, \prime}\| _{L^{\infty}(K)}, \|
{\mathcal G}-{\mathcal G}^{\, \prime}\| _{L^{\infty}(K)}, \| {\mathcal H}-
{\mathcal H}^{\, \prime}\| _{L^{\infty}(K)}\} \, .
$$
Let $\widehat P({\mathcal L})$ denote the orthogonal projection in $L^2(K)$ onto
the subspace ${\mathcal L}$; the set of orthogonal projections is endowed with the
operator topology (induced by the operator norm).
\vskip 0.2cm

{\bf Lemma 2.1} ([28]). {\it The functions $(\Gamma (p,q,F),\rho _{\infty})\ni \{
{\mathcal F}, {\mathcal G}, {\mathcal H}\} \to \widehat P({\rm coker}\, \widehat
d_{\pm})$ are uniformly continuous.}
\vskip 0.2cm

This lemma and the convexity of the sets $\Gamma (p,q,F)$ imply that there
exist continuous functions
$$
(\Gamma ,\rho _{\infty})\ni \{ {\mathcal F}, {\mathcal G}, {\mathcal H}\} \to
\chi _{\pm}\in \{ \chi \in L^2(K):\| \chi \| =1\} \cap {\rm coker}\, \widehat
d_{\pm}\subset L^2(K)\, ,
$$
and we may assume that $\chi _-=\overline {\chi _+}\, $ (see (2.1)).
\vskip 0.2cm

{\bf Lemma 2.2} ([29]). {\it For any $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}
\} \in \Gamma $ for a.e. $x\in K$ we have $\chi _+(x)\neq 0$.}
\vskip 0.2cm

{\bf Lemma 2.3} ([29]). {\it Let $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\}
\in \Gamma (p,q,F)$, and let $g(.)$ be a positive, monotone nonincreasing, and
continuously differentiable function on $(0,1]$ such that $g(r/2)/g(r)\to 1$
as $r\to +0$. Then for all $x\in {\bf R}^2$ and all $\Phi \in \widetilde H^1(K)$
we have
$$
\int\limits_{y\, :\, |x-y|\, \leq \, 1}g(|x-y|)\, |\nabla \Phi (y)|_{{\bf C}^2}
^2\, d^{\, 2}y \leq c_3 \int\limits_{y\, :\, |x-y|\, \leq \, 1}g(|x-y|)\,
|\widehat d_+ \Phi (y)|^2d^{\, 2}y\, ,
$$
where $c_3=c_3(p,q,F;g)>0$ (the integrals may take the value $+\infty $).}
\vskip 0.2cm

We denote by $\widetilde H^1_0\{ g,{\bf Z}^2\} $, $g\in {\mathbb G}$, the Banach
space of functions $\Phi \in \widetilde H^1_0(K)$ such that
$$
\| \Phi \| _{\widetilde H^1_0\{ g,{\bf Z}^2\} }\doteq \| \, |\nabla \Phi (.)|
\, \| _{L^2\{ g,{\bf Z}^2\} }<+\infty \, .
$$
By Lemma 2.3, $\widetilde H^1_0\{ g,{\bf Z}^2\} =\{ \Phi \in \widetilde H^1_0(K):
\widehat d_+\Phi \in L^2\{ g,{\bf Z}^2\} \} $. Since $r^{\varepsilon}g(r)\to 0$ as
$r\to +0$ for any $\varepsilon >0$, the space $\widetilde C^1_0(K)$ is embedded
continuously in $\widetilde H^1_0\{ g,{\bf Z}^2\} $. On the other hand,
$\widetilde H^1_0\{ g,{\bf Z}^2\} \subset \widetilde C_0(K)$, and for all
$\Phi \in \widetilde H^1_0\{ g,{\bf Z}^2\} $ we have
$$
\| \Phi \| _{L^{\infty}(K)}\leq c_4\, \| \Phi \| _{\widetilde H^1_0\{ g,{\bf Z}
^2\} }\, ,  \eqno (2.2)
$$
with $c_4=c_4(g)>0$ (see [29]). The following Lemma 2.4 is an immediate consequence
of Lemma 2.3 and estimate (2.2).
\vskip 0.2cm

{\bf Lemma 2.4} ([29]). {\it If $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\}
\in \Gamma (p,q,F)$ and $g\in {\mathbb G}$, then there is a number $c_5=c_5(p,q,
F;g)>0$ such that for any $\Phi \in \widetilde H^1_0(K)$ with $\widehat d_+\Phi
\in L^2\{ g,{\bf Z}^2\} $ we have $\Phi \in \widetilde C_0(K)$ and
$$
\| \Phi \| _{L^{\infty}(K)}\leq c_5\, \| \widehat d_+ \Phi \| _{L^2\{ g,{\bf Z}
^2\} }\, .
$$}
\vskip 0.2cm

The next lemma follows from the definition of the set ${\mathbb L}({\bf R}^2)$
(see, e.g., [22,{\,}36]).
\vskip 0.2cm

{\bf Lemma 2.5}. {\it Suppose $W\in {\mathbb L}({\bf R}^2)$. Then for any
$\varepsilon >0$ there exists a number $C_{\varepsilon}(W)\geq 0$ such that for all
$k\in {\bf R}^2$ and all $\varphi \in \widetilde H^1(K)$ we have $W\varphi \in L^2(K)$ and
$$
\| W\varphi \| _{L^2(K)}\leq \varepsilon \, \| (k-i\nabla )\varphi \| _{L^2(K;{\bf C}^2)}+
C_{\varepsilon }(W)\, \| \varphi \| _{L^2(K)}\, .
$$}

{\bf Theorem 2.1} ([29]). {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\}
\in \Gamma $, $g\in {\mathbb G}$. Then for any $\Phi \in \widetilde H^1_0(K)$ with
$\widehat d_+\Phi \in L^2\{ g,{\bf Z}^2\} $, and for any $\psi \in \widetilde H^1(K)$,
we have $\Phi \in \widetilde C_0(K)$, $e^{\, i\Phi }\psi \in \widetilde H^1(K)$,
$(\partial \Phi / \partial x_j)\psi \in L^2(K)$, $j=1,{\,}2$, and
$$
\widehat d_+(e^{\, i\Phi }\psi )=e^{\, i\Phi }(i(\widehat d_+\Phi )\psi +\widehat
d_+\psi )\, .  \eqno (2.3)
$$}

Theorem 2.1 is a consequence of Lemmas 2.3, 2.4, and 2.5. For the proof of identity
(2.3), we use the fact that the operator $\widehat d_+$ is closed. First,
identity (2.3) is established for $\psi \in \widetilde C^1(K)$, and then, with
the use of Lemma 2.5, in the general case, for $\psi \in \widetilde H^1(K)$.
\vskip 0.2cm

{\bf Theorem 2.2} ([28]). {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\}
\in \Gamma $. Then there exist unique real-valued functions $\Phi ,\, \Psi \in
\widetilde H^1_0(K)$ and a vector $\widetilde \varkappa \in {\bf R}^2$ such that
$$
i\widehat d_+(\Phi -i\Psi )=-({\mathcal G}+i{\mathcal F})\widetilde {\varkappa}_1-
i{\mathcal H}(\widetilde {\varkappa}_2+i)\, .  \eqno (2.4)
$$
Moreover, (2.4) implies that $\Phi ,\, \Psi \in \widetilde C_0(K)$ and $\widetilde
{\varkappa}_1>0$.}
\vskip 0.2cm

{\bf Lemma 2.6} ([29]). {\it Suppose $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} \in
\Gamma $. Then $\chi _+=c_6\, ({\mathcal G}{\mathcal H})^{-1}(\widehat d_+\Psi -
{\mathcal H})$, where $\Psi \in \widetilde H^1_0(K)\cap \widetilde C(K)$ is the
function defined in Theorem 2.2, and $c_6=c_6\, ({\mathcal F},{\mathcal G},{\mathcal
H})\in {\bf C}\backslash \{ 0\} $.}
\vskip 0.2cm

{\bf Lemma 2.7} ([29]). {\it Suppose $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} \in
\Gamma $ and $\Psi \in \widetilde H^1_0(K)$ is the function defined in Theorem 2.2.
Then
$$
\biggl( \frac {\partial \Psi }{\partial x_1}\biggr) ^2+\biggl( \frac {\partial
\Psi }{\partial x_2}-1\biggr) ^2>0
$$
for a.e. $x\in K$.}

Lemma 2.7 is a consequence of Lemmas 2.2 and 2.6.
\vskip 0.2cm

{\bf Lemma 2.8} ([29]). {\it Under the conditions of Lemma 2.7, for all $\lambda
\in {\bf R}$ we have
$$
{\rm meas}\, \{ x\in K:\Psi (x)-x_2=\lambda \} =0\, .
$$}

Lemma 2.8 follows from Lemma 2.7, because $\Psi \in H^1_{\rm loc}({\bf R}^2)$ and
if $\Psi (x)-x_2=\lambda \ (= {\rm const})$ on some set $M\subseteq K$ with
${\rm meas}\, M>0$, then $\partial \Psi / \partial x_1=0$ and $\partial \Psi /
\partial x_2=1$ for a.e. $x\in M$, which contradicts Lemma 2.7. Lemma 2.8 will be
used in the proof of Theorem 1.2. Instead of Lemma 2.8, we could apply Theorem 2.3
stated below (this method of argument was chosen in [28] for the proof of the
absence of eigenvalues in the spectrum of the generalized two-dimensional
periodic Dirac operator $\widehat {\mathcal D}+\widehat V$ with matrix-valued
potential $\widehat V\in L^q_{\rm loc}({\bf R}^2;{\mathcal M}_2)$, $q>2$).
\vskip 0.2cm

{\bf Theorem 2.3} ([28]). {\it Suppose $\{ {\mathcal F},{\mathcal G},{\mathcal H}\}
\in \Gamma $ and identity (2.4) is fulfilled for real-valued functions $\Phi ,\,
\Psi \in \widetilde H^1_0(K)\cap \widetilde C(K)$ and a vector $\widetilde \varkappa =
(\widetilde \varkappa _1\, ,\widetilde \varkappa _2)\in {\bf R}^2$. Then ${\bf R}^2\ni
x\to {\mathcal Z}(x)=\Phi (x)-i\Psi (x)+\widetilde {\varkappa}_1x_1+(\widetilde
{\varkappa}_2+i)x_2\in {\bf C}$ is a continuous bijective map (with continuous
inverse).}
\vskip 0.2cm

Under the conditions of Theorem 2.3, ${\mathcal Z}(.)$ is a periodic map: for any
$x\in {\bf R}^2$ and $n\in {\bf Z}^2$, we have ${\mathcal Z}(x+n)={\mathcal Z}(x)
+\widetilde {\varkappa}_1n_1+(\widetilde {\varkappa}_2+i)n_2$, and $\widetilde {\varkappa}_1>0$.

\section{Similarity between the Dirac operator $\widehat {\mathcal D}(0)+\widehat V$
with a special matrix-valued potential and the Dirac operator $\widehat {\mathcal D}
(k+i\varkappa )$}

We put
$$
\widetilde H^1_0\{ {\mathbb G}\} =\bigcup\limits_{g\in {\mathbb G}}
\widetilde H^1_0\{ g,{\bf Z}^2\} \, .
$$
Lemma 2.3 and Theorem 2.1 imply that for any $\Phi \in \widetilde H^1_0\{ {\mathbb G}
\} $ the operators of multiplication by $e^{\, \Phi}$ and by $e^{\, \widehat {\sigma}
_3\Phi }$ act within the space $\widetilde H^1(K;{\bf C}^2)$.

{\bf Theorem 3.1}. {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H} \} \in
\Gamma (p,q,F)$ and $g\in {\mathbb G}$. Then for any two functions ${\mathcal C}_1,\,
{\mathcal C}_2\in L^2\{ g,{\bf Z}^2\} $ there exist unique vectors $k,\, \varkappa \in
{\bf R}^2$ and functions $\Phi ,\, \Psi \in \widetilde H^1_0\{ {\mathbb G}\} \subset
\widetilde H^1_0(K)\cap \widetilde C(K)$ such that for some $\mu \in {\bf C}\backslash
\{ 0\} $ (consequently, for all $\mu \in {\bf C}$) we have
$$
e^{\, \mu \widehat \sigma _3\Psi }e^{-i\mu \Phi }\, \widehat {\mathcal D}(\mu (k+
i\varkappa ))\, e^{\, i\mu \Phi }e^{\, \mu \widehat \sigma _3\Psi }=\widehat
{\mathcal D}(0)+\mu({\mathcal C}_1\widehat \sigma _1+{\mathcal C}_2\widehat
\sigma _2)\, .  \eqno (3.1)
$$
Moreover, $\Phi ,\, \Psi \in \widetilde H^1_0\{ g,{\bf Z}^2\} $ and
$$
\max \{ \| \Phi \| _{L^{\infty}(K)}, \| \Psi \| _{L^{\infty}(K)}\} \leq
c^{\, \prime}_1\, (\| {\mathcal  C}_1\| _{L^2\{ g,{\bf Z}^2\} }+\| {\mathcal C}_2\|
_{L^2\{ g,{\bf Z}^2\} })\, ,  \eqno (3.2)
$$ $$
|k|^2+|\varkappa |^2\leq c^{\, \prime}_2\, (\| {\mathcal C}_1\|
^2_{L^2(K)}+\| {\mathcal C}_2\| ^2_{L^2(K)})\, ,  \eqno (3.3)
$$
where $c^{\, \prime}_1=c^{\, \prime}_1(p,q,F;g)>0$, $c^{\, \prime}_2=c^{\, \prime}
_2(p,q,F)>0$. If ${\mathcal C}_1\pm i\, {\mathcal C}_2\in R(\widehat d_{\pm})$,
then $k=\varkappa =0$. If ${\mathcal C}_1$ and ${\mathcal C}_2$ are real-valued,
then $\varkappa =0$ and $\Phi $ and $\Psi $ are also real-valued.}
\vskip 0.2cm

{\it Proof}. By Theorem 2.1 and relations (1.2) and (2.1), vectors $k,\, \varkappa
\in {\bf R}^2$ and functions $\Phi ,\, \Psi \in \widetilde H^1_0\{ {\mathbb G}
\} \subset \widetilde H^1_0(K)\cap \widetilde C(K)$ satisfy (3.1) for some
$\mu \in {\bf C}\backslash \{ 0\} $ (and hence, for all $\mu \in {\bf C}$)
if and only if, for both signs $+$ and $-{\, }$,
$$
i\widehat d_{\pm}\Phi _{\pm}={\mathcal C}^{\, \prime}_{\pm} \doteq {\mathcal
C}_{\pm}-(({\mathcal G}\pm i{\mathcal F})k_1\pm i{\mathcal H}k_2)-i(({\mathcal G}\pm i{\mathcal
F})\varkappa _1\pm i{\mathcal H}\varkappa _2)\, ,  \eqno (3.4)
$$
where $\Phi _{\pm}\doteq \Phi \mp i\Psi $, ${\mathcal C}_{\pm}\doteq {\mathcal C}_1
\pm i{\mathcal C}_2\in L^2\{ g,{\bf Z}^2\} \subset L^2(K)$ (multiplication by
$e^{\, i\mu \Phi }$ and by $e^{\, \mu \Psi }$, $\mu \in {\bf C}$, acts within the
space $\widetilde H^1(K)$). We denote $(\chi _{\pm},{\mathcal G}\pm i{\mathcal F})
=\mu^{(1)}_{\pm}$, $(\chi _{\pm},\pm i{\mathcal H})=\mu ^{(2)}_{\pm}$. Since the
functions $\chi _{\pm}$ (with $\| \chi _{\pm}\| =1$) are chosen so that $\chi _-=
\overline {\chi _+}\, $, we have $\mu ^{(1)}_-=\overline {\mu ^{(1)}_+}$, $\mu
^{(2)}_-=\overline {\mu ^{(2)}_+}$. Also, $|\mu ^{(1)}_{\pm}|\leq p+F$ and $|\mu
^{(2)}_{\pm}|\leq p$. Since the subspaces $R(\widehat d_{\pm})$ are closed in
$L^2(K)$ and ${\rm dim\, coker}\, \widehat d_{\pm}=1$, equations (3.4) can be
solved for $\Phi _{\pm}\in \widetilde H^1_0(K)$ and $k,{\,}\varkappa \in {\bf
R}^2$ if and only if
$$
(\chi _{\pm},{\mathcal C}_{\pm})=\mu ^{(1)}_{\pm}\, (k_1+i\varkappa _1)+\mu ^{(2)}
_{\pm}\, (k_2+i\varkappa _2)\, .  \eqno (3.5)
$$

{\bf Lemma 3.1}. {\it We have $|\, {\rm Im}\, \mu ^{(1)} _+\overline
{\mu ^{(2)}_+}\, |\geq c_0=c_0(p,q,F)>0$.}
\vskip 0.2cm

{\it Proof}. Let ${\mathcal K}(x)=({\mathcal G}(x){\mathcal H}(x))
^{-1/2}$, $x\in K$. For any vector $\tau =(\tau _1,\tau _2)\in {\bf R}^2$ and any
function $\Omega _+\in \widetilde H^1(K)$, we can write
$$
\| {\mathcal K}(i\widehat d_+\Omega _++({\mathcal G}+ i{\mathcal
F})\tau _1+ i{\mathcal H}\tau _2)\| ^2=
$$ $$
=\biggl\| {\mathcal K}{\mathcal G}(\tau _1+\frac {\partial \Omega _+}
{\partial x_1})\biggr\| ^2+\biggl\| {\mathcal K}\bigl( {\mathcal F}(\tau _1+\frac
{\partial \Omega _+}{\partial x_1})+{\mathcal H}(\tau _2+\frac
{\partial \Omega _+}{\partial x_2})\bigr) \biggr\| ^2\, ,
$$
whence
$$
\| i\widehat d_+\Omega _++({\mathcal G}+ i{\mathcal
F})\tau _1+ i{\mathcal H}\tau _2\| ^2\geq  \eqno (3.6)
$$ $$
\geq c_0\, \sum\limits_{j=1}^2\, \biggl\| \tau _j+\frac {\partial \Omega
_+}{\partial x_j}\biggr\| ^2 =c_0\, \biggl( |\tau |^2+
\sum\limits_{j=1}^2\, \biggl\| \frac {\partial \Omega _+} {\partial
x_j}\biggr\| ^2\, \biggr) \, ,
$$
where $c_0=c_0(p,q,F)>0$. Since the function $\Omega _+\in \widetilde H^1(K)$
in (3.6) is arbitrary, we obtain
$$
c_0\, |\tau |^2\leq \min\limits_{\Omega _+\in \widetilde H^1(K)}
\| i\widehat d_+\Omega _++({\mathcal G}+ i{\mathcal F})\, \tau _1+
i{\mathcal H}\, \tau _2\| ^2=
$$ $$
=|(\chi _+,{\mathcal G}+ i{\mathcal F})\, \tau _1+(\chi _+, i{\mathcal
H})\, \tau _2|^2=|\mu ^{(1)}_+\tau _1+\mu ^{(2)}_+\tau _2
|^2
$$
(in particular, this implies that $|\mu ^{(j)}_{\pm}|\geq \sqrt {c_0}$, $j=
1,{\,}2$). Consequently,
$$
c_0\leq \sqrt {c_0}\, |\mu ^{(2)}_+|\leq \min\limits_{t\in {\bf R}}|(\mu
^{(1)}_++\mu ^{(2)}_+t)\overline {\mu ^{(2)}_+}|=|\, {\rm Im}\, \mu ^{(1)}_+
\overline {\mu ^{(2)}_+}\, |\, .
$$
Lemma 3.1 is proved.  \hfill $\square$
\vskip 0.2cm

Since
$$
{\rm det}\, \left( \begin{matrix} \mu ^{(1)}_+  &\mu ^{(2)}_+\\  \mu ^{(1)}_-
&\mu ^{(2)}_-\end{matrix} \right) = 2i\, {\rm Im}\, \mu ^{(1)} _+\overline {\mu
^{(2)}_+}\, ,
$$
Lemma 3.1 shows that there exist unique vectors $k,\, \varkappa \in {\bf
R}^2$ satisfying (3.5):
$$
k_1+i\varkappa _1=\bigl( 2i\, {\rm Im}\, \mu ^{(1)}_+\overline {\mu ^{(2)}_+}\bigr)
^{-1}(\mu ^{(2)}_-(\chi _+,{\mathcal C}_+)-\mu ^{(2)}_+(\chi _-,{\mathcal C}_-))
\, ,  \eqno (3.7)
$$ $$
k_2+i\varkappa _2=\bigl( 2i\, {\rm Im}\, \mu ^{(1)}_+\overline {\mu ^{(2)}_+}\bigr)
^{-1}(-\mu ^{(1)}_-(\chi _+,{\mathcal C}_+)+\mu ^{(1)}_+(\chi _-,{\mathcal C}_-))
\, .  \eqno (3.8)
$$
Relations (3.7) and (3.8) imply (3.3). For the vectors $k,\, \varkappa \in {\bf R}^2$
chosen as above, we have ${\mathcal C}_{\pm}^{\, \prime}\in R(\widehat d_{\pm})$,
therefore, we can find (unique) functions $\Phi _{\pm}\in \widetilde H^1_0(K)$ such
that $i\widehat d_{\pm}\Phi _{\pm}={\mathcal C}_{\pm}^{\, \prime}\, $. On the other
hand, ${\mathcal C}_{\pm}^{\, \prime}\in L^2\{ g,{\bf Z}^2\} $, so that, by Lemmas
2.3 and 2.4, $\Phi _{\pm}\in \widetilde H^1_0\{ g,{\bf Z}^2\} \subset \widetilde
C_0(K)$ and
$$
\| \Phi _{\pm}\| _{L^{\infty}(K)}\leq c_5\, \| {\mathcal C}^{\, \prime}_{\pm}\|
_{L^2\{ g,{\bf Z}^2\} }\, .  \eqno (3.9)
$$
Also, we have $\Phi ,\, \Psi \in \widetilde H^1_0\{ g,{\bf Z}^2\}
\subset \widetilde H^1_0(K)\cap \widetilde C(K)$, and (3.2) is implied by (3.3),
(3.4), and (3.9). If ${\mathcal C}_{\pm}\in R(\widehat d_{\pm})$, then
$(\chi _{\pm},{\mathcal C}_{\pm})=0$, whence $k=\varkappa =0$. If ${\mathcal
C}_1$ and ${\mathcal C}_2$ are real-valued, then ${\mathcal C}_-=\overline {{\mathcal
C}_+}$ and $(\chi _-,{\mathcal C}_-)=\overline {(\chi _+,{\mathcal C}_+)}$, and from
(3.7), (3.8) it follows that $\varkappa =0$ and $i\widehat d_{\pm}\Phi _{\pm}=
{\mathcal C}_{\pm}-(({\mathcal G}\pm i{\mathcal F})k_1\pm i{\mathcal H}k_2)$ (see
(3.4)). By complex conjugation and (2.1), we obtain $i\widehat d_{\pm}\overline
{\Phi _{\mp}}={\mathcal C}_{\pm}-(({\mathcal G}\pm i{\mathcal F})k_1\pm i{\mathcal
H}k_2)=i\widehat d_{\pm}\Phi _{\pm}$, whence $\widehat d_+({\rm Im}\, \Phi -i\, {\rm
Im}\, \Psi )=0$. Consequently, $\Phi $ and $\Psi $ are real-valued. This proves
Theorem 3.1.            \hfill $\square$
\vskip 0.2cm

If under the assumptions of Theorem 3.1 we put ${\mathcal C}_1=i{\mathcal H}$,
${\mathcal C}_2\equiv 0$, then ${\mathcal C}_1\, ,{\mathcal C}_2\in L^{\infty}(K)
\subset L^2\{ g,{\bf Z}^2\} $ for any $g\in {\mathbb G}$. Therefore, the next
theorem, which we need for what follows, is a consequence of Theorem 3.1.
\vskip 0.2cm

{\bf Theorem 3.2}. {\it Suppose $\{ {\mathcal F},{\mathcal G},{\mathcal H}\} \in
\Gamma (p,q,F)$. Then the following objects exist and are unique: a vector
$\widetilde {\varkappa}\in {\bf R}^2$ and real-valued functions $\Phi ,\, \Psi \in
\widetilde H^1_0\{ {\mathbb G}\} \subset \widetilde H^1_0(K)\cap \widetilde C(K)$
such that for all $k,\, \varkappa \in {\bf R}^2$ and all $\mu \in {\bf R}$ we have
$$
e^{\, i\mu \widehat {\sigma}_3\Psi }e^{\, \mu \Phi }\widehat {\mathcal D}
(k+i\varkappa +i\mu \widetilde {\varkappa})e^{-\mu \Phi }e^{\, i\mu \widehat {\sigma}_3
\Psi }=\widehat {\mathcal D}(k+i\varkappa )+i\mu {\mathcal H}\widehat {\sigma}_1\, .
\eqno (3.10)
$$
Moreover, $\Phi ,\, \Psi \in \widetilde H^1_0\{ g,{\bf Z}^2\} $ for every $g\in
{\mathbb G}$, and $\max \{ \| \Phi \| _{L^{\infty}(K)}\, ,\, \| \Psi \| _{L^{\infty}
(K)}\} \leq c^*_1$, $|\widetilde {\varkappa}|\leq c^*_2$, where $c^*_1=c^*_1\, (p,q,F)
>0$ and $c^*_2=c^*_2\, (p,q,F)>0$.}
\vskip 0.2cm

The functions $\Phi $, $\Psi $ and the vector $\widetilde \varkappa $ defined in
Theorem 3.2 coincide with the corresponding objects in Theorem 2.2, because they
satisfy condition (2.4), which is implied by (3.10).
\vskip 0.2cm

{\bf Lemma 3.2}. {\it For the vector $\widetilde \varkappa =(\widetilde \varkappa
_1\, ,\widetilde \varkappa _2)\in {\bf R}^2$ defined in Theorem 3.2 we have
$\widetilde \varkappa _1\geq c_3^*=c_3^*(p,q,F)>0$.}
\vskip 0.2cm

{\it Proof}. Identity (2.4) yields
$$
\mu ^{(1)}_+\widetilde \varkappa _1+\mu ^{(2)}_+(\widetilde \varkappa _2+i)=0\, .
\eqno (3.11)
$$
Since $|\mu ^{(1)}_+|\leq p+F$ and $|\mu ^{(2)}_+|\geq \sqrt {c_0}$ (see the proof
of Lemma 3.1), from (3.11) we deduce that
$$
|\widetilde \varkappa _1|\geq \frac {\sqrt {c_0}}{p+F}\, |\, {\rm Im}\ \frac {\mu
^{(1)}_+}{\mu ^{(2)}_+}\ \widetilde \varkappa _1\, |=\frac {\sqrt {c_0}}{p+F}\doteq
c_3^*\, ,
$$
and $\widetilde \varkappa _1>0$ by Theorem 2.2.           \hfill $\square$

\section{Auxiliary statements}

Let $k\in {\bf R}^2$, $\mu \in {\bf R}$. For every $N\in {\bf Z}^2$, we denote
$$
G^{\pm}_N(k;\mu )=\bigl( (k_1+2\pi N_1)^2+(
k_2+2\pi N_2\pm \mu )^2 \bigr) ^{1/2}\, ,
$$ $$
G_N(k;\mu )=\min \, \{ G^-_N(k;\mu ), G_N^+(k;\mu )\}
$$
(here and in the sequel, we agree that the statements and formulas involving
$\pm$ and $\mp \, $ are understood independently for the upper and the lower
combination of signs). If $k_1=\pi $, then $G_N(k;\mu )\geq \pi $. For $\varphi \in
\widetilde H^1(K)$, put
$$
\| \varphi \| _*=\biggl( \, \sum\limits_{N\in {\bf Z}^2}G^2_N(k;\mu )\, |\varphi
_N|^2 \biggr) ^{1/2}\, ,
$$ $$
\| \varphi \| _{*,\pm}=\biggl( \, \sum\limits_{N\in {\bf Z}^2}(G^{\pm}_N(k;\mu ))^2
\, |\varphi _N|^2 \biggr) ^{1/2}\, .
$$
For $a\geq 2\pi $, we introduce the finite sets
$$
T^{\pm}(a)=\{ N\in {\bf Z}^2: G^{\pm}_N(k;\mu )\leq a\} \, .
$$
In the above notation, the dependence on the vector $k\in {\bf R}^2$ and the
number $\mu \in {\bf R}$, which will be specified in advance, is not indicated
explicitly. Let $\# \, {\mathcal O}$ denote the number of elements of a finite
set ${\mathcal O}$. We have
$$
1\leq \# \, T^{\pm}(a) <6\pi a^2\, . \eqno (4.1)
$$
\vskip 0.2cm

{\bf Lemma 4.1}. {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H}\} \in
\Gamma (p,q,F)$. Then for all vectors $k\in {\bf R}^2$, all numbers
$\mu \in {\bf R}$, and all functions $\varphi \in \widetilde H^1(K)$ we have
$$
c_1\, \| \varphi \| ^2_{*,\pm}\leq \| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi \|
^2\leq c_2\, \| \varphi \| ^2_{*,\pm}\, ,
$$
where $c_1=c_1(p,q,F)>0$ and $c_2=c_2(p,q,F)\geq c_1$.}
\vskip 0.2cm

This lemma is a consequence of estimates (1.3) (with the same constants $c_1$
and $c_2$).

For a set ${\mathcal O}\subseteq {\bf Z}^2$, denote ${\mathcal L}({\mathcal
O})=\{ \psi \in L^2(K):\psi _N=0$ for $N\in {\bf Z}^2\backslash {\mathcal O}\}$,
${\mathcal L}({\bf Z}^2)=L^2(K)$, ${\mathcal L}(\emptyset )=\{ 0\} $. Let $\widehat
P^{\, {\mathcal O}}=\widehat P({\mathcal L}({\mathcal O}))$ be the orthogonal
projection in $L^2(K)$ that takes a function $\varphi \in L^2(K)$ to the function
$$
\widehat P^{\, {\mathcal O}}\varphi = \sum\limits_{N\in {\mathcal
O}}\varphi _Ne^{\, 2\pi i\, (N,\, x)}\, .
$$
\vskip 0.2cm

{\bf Lemma 4.2}. {\it If $W\in L^2(K)$, then for any finite set ${\mathcal O}\subset
{\bf Z}^2$ the operator $W\widehat P^{\, \mathcal O}$ is bounded on $L^2(K)$, and
$$
\| W\widehat P^{\, \mathcal O}\| \leq f_W(\# \, {\mathcal O})\, ,
$$
where $f_W:{\bf Z}_+\to [0,+\infty )$ is a monotone nondecreasing function
satisfying $f_W(N)=o\, (\sqrt N)$ as $N\to +\infty $.}
\vskip 0.2cm

{\it Proof}. For $b\geq 0$, we introduce the following functions:
$$
K\ni x\to W_b(x)=
\left\{
\begin{array}{ll}
W(x) & \text {if}\ |W(x)|>b, \\
0\ & \text {otherwise},
\end{array}
\right. \eqno (4.2)
$$
$\widetilde W_b(x)=W(x)-W_b(x)$, $x\in K$. If $\varphi \in L^2(K)$ and ${\mathcal
O}\subset {\bf Z}^2$ is a finite set, then
$$
\| W_b\widehat P^{\, \mathcal O}\varphi \| = \biggl( \, \sum\limits_{N\in {\bf Z}^2}
\, \biggl| \, \sum\limits_{M\in {\bf Z}^2}(W_b)_M(\widehat P^{\, \mathcal O}\varphi )
_{N-M}\biggr| ^2 \,
\biggr) ^{1/2}\leq
$$ $$
\leq \biggl( \, \sum\limits_{N\in {\bf Z}^2}\biggl( \, \sum\limits_{M\, :\, N-M
\in {\mathcal O}}|(W_b)_M|^2\biggr) \biggl( \, \sum\limits_{M\, :\, N-M\in
{\mathcal O}}|\varphi _{N-M}|^2\biggr) \biggr) ^{1/2}\leq
$$ $$
\leq \biggl( \, \sum\limits_{M\in {\bf Z}^2}\biggl( \, \sum\limits_{N\, :\, N-M
\in {\mathcal O}}1\biggr) |(W_b)_M|^2\biggr) ^{1/2}\| \varphi \| = (\# \, {\mathcal O})
^{1/2}\, \| W_b \| _{L^2(K)}\| \varphi \| \, ,
$$
whence
$$
\| W\widehat P^{\, \mathcal O}\varphi \| \leq \| \widetilde W_b\widehat P^{\,
\mathcal O}\varphi \| +\| W_b\widehat P^{\, \mathcal O}\varphi \| \leq (b+(\# \, {\mathcal
O})^{1/2}\, \| W_b\| _{L^2(K)})\| \varphi \| \, .
$$
Put
$$
f_W(N)=\inf\limits_{b\geq 0}\ (b+\sqrt N\, \| W_b\| _{L^2(K)})\, ,\ N\in {\bf
Z}_+={\bf N}\cup \{ 0\} \, .
$$
Then $\| W\widehat P^{\, \mathcal O}\| \leq f_W(\# \, {\mathcal O})$, the function
$f_W$ is monotone nondecreasing, and for any $\varepsilon >0$ we can find a number
$b(\varepsilon )>0$ such that $\| W_{b(\varepsilon )}\| _{L^2(K)}<\varepsilon $,
so that $f_W(N)/{\sqrt N}\to 0$ as $N\to +\infty $.       \hfill $\square$
\vskip 0.2cm

Let $W\in {\mathbb L}({\bf R}^2)$; we put
$$
h_W(t)=\inf\limits_{\varepsilon >0}\ (\varepsilon +t^{-1}\, C_{\varepsilon}(W))\, ,\
t>0\, ,
$$
where $C_{\varepsilon}(W)$ is as in Lemma 2.5. The function $h_W$ is monotone
nonincreasing, and $h_W(t)\to 0$ as $t\to +\infty $.
\vskip 0.2cm

{\bf Lemma 4.3}. {\it Suppose $W\in {\mathbb L}({\bf R}^2)$, $\mu
\geq 4\pi $. Then for all $k\in {\bf R}^2$ with $k_1=\pi $ and all
$\varphi \in {\mathcal L}(T^{\pm}(\mu /2))$ we have
$$
\| W\varphi \| \leq c_7\, \| \varphi \| _{*,\pm} = c_7\, \| \varphi \| _*\, ,
\eqno (4.3)
$$
where $c_7=c_7\, (W)>0$. If $2\pi \leq a\leq \mu /2$, then
$$
\| W\varphi \| \leq h_W(a)\| \varphi \| _*  \eqno (4.4)
$$
for all $k\in {\bf R}^2$ and all $\varphi \in {\mathcal L}(T^{\pm}(\mu /2)\backslash
T^{\pm}(a))$.}

{\it Proof}. By Lemma 2.5 (with $\varepsilon =1$), for all $\mu
\in {\bf R}$, all $k\in {\bf R}^2$, and all $\varphi \in \widetilde
H^1(K)$ we have
$$
\| W\varphi \| \leq \| \varphi \| _{*,\, \pm }+C_1(W)\| \varphi \| \, .  \eqno (4.5)
$$
On the other hand, if $\varphi \in {\mathcal L}(T^{\pm}(\mu /2))$, $\mu \geq
4\pi $, then $\| \varphi \| _{*,\, \pm }=\| \varphi \| _*\, $, and $\| \varphi \|
_*\geq \pi \, \| \varphi \| $ whenever $k_1=\pi $. Therefore, (4.5) implies
(4.3) with $c_7=1+\pi ^{-1}C_1(W)$. Now, suppose that $2\pi \leq a\leq \mu
/2$ and $\varphi \in {\mathcal L}(T^{\pm}(\mu /2)\backslash T^{\pm}(a))$. Then, for
any $k\in {\bf R}^2$, we have $\| \varphi \| _{*,\, \pm }\geq a\, \| \varphi \| $,
and by Lemma 2.5 we obtain
$$
\| W\varphi \| \leq (\varepsilon +a^{-1}\, C_{\varepsilon}(W))\| \varphi \| _*
$$
for any $\varepsilon >0$, which yields (4.4).          \hfill $\square$
\vskip 0.2cm

{\bf Lemma 4.4}. {\it Suppose $W\in {\mathbb L}({\bf R}^2)$, $\mu
\geq 4\pi $. Then for all $k\in {\bf R}^2$ and all $\varphi \in
\widetilde H^1(K)\cap {\mathcal L}({\bf Z}^2\, \backslash (T^+(\mu /2)\cup T^-
(\mu /2)))$ we have
$$
\| W\varphi \| \leq 3h_W(\mu )\, \| \varphi \| _*\, .
$$}

This lemma follows from Lemma 2.5, since, under the assumptions of Lemma 4.4,
we have $\| \varphi \| \leq 2\mu ^{-1}\| \varphi \| _*$ and $\| (k-i\nabla )
\varphi \| _{L^2(K;{\bf C}^2)}\leq 3\, \| \varphi \| _*\, $.
\vskip 0.2cm

{\bf Lemma 4.5}. {\it Suppose $W\in L^2(K)$, $\mu >4\pi $, and $2\pi \leq
a<a^{\, \prime}\leq \mu /2$. Then for all $\varphi \in {\mathcal L}({\bf
Z}^2\backslash T^{\pm}(a^{\, \prime}))$ and all $\psi \in {\mathcal L}(T^{\pm}(a))$
we have
$$
|(\varphi ,W\psi )|\leq \sqrt {6\pi }\, a\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\,
:\, 2\pi |N|\, >\, a^{\, \prime}-a}|W_N|^2\, \biggr) ^{1/2}\, \| \varphi \|
_{L^2(K)}\, \| \psi \| _{L^2(K)}\, .
$$}

{\it Proof}. Indeed,
$$
|(\varphi ,W\psi )|\leq \sum\limits_{M\in {\bf Z}^2}|\varphi _M|\, \sum\limits_{N\in
{\bf Z}^2}|W_N\psi _{M-N}|\leq
$$ $$
\leq \| \varphi \| _{L^2(K)} \biggl( \, \sum\limits_{M\in {\bf Z}^2\backslash
T^{\pm}(a^{\, \prime})} \biggl( \, \sum\limits_{N\in {\bf Z}^2}|W_N \psi _{M-N}|
\, \biggr) ^2 \, \biggr) ^{1/2}\leq
$$ $$
\leq \biggl( \, \sum\limits_{N\in {\bf Z}^2} \, \biggl( \, \sum\limits_{M\, :\,
M\notin \, T^{\pm}(a^{\, \prime}),\, M-N\, \in \, T^{\pm}(a)}1\, \biggr) \,
|W_N|^2\biggr) ^{1/2}\, \| \varphi \| _{L^2(K)} \, \| \psi \| _{L^2(K)} \leq
$$ $$
\leq \sqrt {6\pi }\, a\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\, a^{\,
\prime}-a}|W_N|^2\biggr) ^{1/2}\, \| \varphi \| _{L^2(K)} \, \| \psi \|
_{L^2(K)}
$$
(we have used estimate (4.1)). \hfill $\square$
\vskip 0.2cm

{\bf Lemma 4.6}. {\it For $W\in {\mathbb L}({\bf R}^2)$, let $W_b\, $, $b\geq 0$,
be the functions defined in (4.2). There exists a monotone nonincreasing function
$\widetilde h_W:[0,+\infty )\to [0,+\infty )$ such that $\widetilde h_W(t)\to
0$ as $t\to +\infty $ and for all $\mu \in {\bf R}$, all $k\in {\bf R}^2$ with
$k_1=\pi $, all $\varphi \in \widetilde H^1(K)$, and all $b\geq 0$ we have
$$
\| W_b\varphi \| \leq \widetilde h_W(b)\, \| \varphi \| _{*,\pm}\, .  \eqno (4.6)
$$}

{\it Proof}. By Lemma 2.5, for any $\varepsilon >0$ there exists a number
$C_{\varepsilon}(W)\geq 0$ such that
$$
\| W\psi \| \leq \varepsilon \, \| \psi \| _{*,\pm}+C_{\varepsilon}(W)\| \psi \|
\eqno (4.7)
$$
for all $\mu \in {\bf R}$, $k\in {\bf R}^2$, and $\psi \in \widetilde H^1(K)$.
We define
$$
\widetilde h_W(b)=\inf\limits_{\varepsilon >0}\, \min\limits_{a\geq 2\pi }\, \biggl(
\sqrt {\frac 6{\pi }}\, a\, \| W_b\| _{L^2(K)}+\varepsilon +a^{-1}C_{\varepsilon}(W)
\biggr) \, ,\ b\geq 0\, .
$$
Since the function $[0,+\infty )\ni b\to \| W_b\| _{L^2(K)}$ is monotone
nonincreasing, and $\| W_b\| _{L^2(K)}\to 0$ as $b\to +\infty $, the function
$\widetilde h_W$ is also monotone nonincreasing, and $\widetilde h_W(b)\to 0$ as
$b\to +\infty $. On the other hand, using (4.1) and (4.7), we see that if $\mu
\in {\bf R}$, $k\in {\bf R}^2$, $k_1=\pi $, $\varphi \in \widetilde H^1(K)$ (in
which case $\pi \| \varphi \| \leq \| \varphi \| _{*,\pm}$), $b\geq 0$, $\varepsilon
>0$ and $a\geq 2\pi $, then
$$
\| W_b\varphi \| \leq \| W_b\widehat P^{\, T^{\pm}(a)}\varphi \| +\| W_b\widehat P^{\,
{\bf Z}^2\backslash T^{\pm}(a)}\varphi \| \leq
$$ $$
\leq \| W_b\| _{L^2(K)}\| \widehat P^{\, T^{\pm}(a)}\varphi \| _{L^{\infty}(K)}+
\| W\widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi \| \leq
$$ $$
\leq \sqrt {6\pi }\, a\, \| W_b\| _{L^2(K)}\| \widehat P^{\, T^{\pm}(a)}\varphi \| +
\varepsilon \, \| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi \| _{*,\pm}+
C_{\varepsilon}(W)\| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi \| \leq
$$ $$
\leq \biggl( \sqrt {\frac 6{\pi }}\, a\, \| W_b\| _{L^2(K)}+\varepsilon +a^{-1}C
_{\varepsilon}(W)\biggr) \, \| \varphi \| _{*,\pm}\, .
$$
These inequalities and the definition of $\widetilde h_W$ imply (4.6).
\hfill $\square$

\section{Proof of Theorem 1.2}

For an arbitrary set ${\bf M}^{\, \prime}\subseteq {\bf N}$ we put
$$
{\mathcal Q}\, ({\bf M}^{\, \prime})=\overline {\lim\limits_{N\to +\infty }}\ \
\frac {\# \{ n\in {\bf M}^{\, \prime}:n\leq N\} }{N} \, .
$$
In this and the next sections we use the symbol $\sum\limits_{+,-}$ to denote
the sum of two terms obtained from expressions with indices $\pm$ and $\mp$ when
fixing the upper or the lower combination of signs.
\vskip 0.2cm

{\bf Theorem 5.1}. {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H} \} \in
\Gamma (p,q,F)$, $\widetilde V^{(l)}\in {\mathbb L}({\bf R}^2)$, $l=0,3$, and
$\Psi $ is a real-valued function of class $\widetilde C(K)$ such that
$$
{\rm meas}\, \{ x\in K:\Psi (x)-x_2=\lambda \} =0   \eqno (5.1)
$$
for any $\lambda \in {\bf R}$. Then there exists a number $a_0=a_0(p,q,F;\widetilde
V^{(0)},\widetilde V^{(3)})\geq 2\pi $ such that for any $a\geq a_0$ there is a set
${\bf M}\subset {\bf N}$, depending also on ${\mathcal F}$, ${\mathcal G}$,
${\mathcal H}$, $\widetilde V^{(0)}$, $\widetilde V^{(3)}$, and $\Psi $, for
which ${\mathcal Q}\, ({\bf N}\backslash {\bf M})=0$ and for all
$\mu \in \pi {\bf M}$, all $k\in {\bf R}^2$ with $k_1=\pi $, and all
$$
\varphi =\left( \begin{matrix} \varphi _+\\ \varphi _- \end{matrix} \right) \in \widetilde
H^1(K;{\bf C}^2)\, ,  \eqno (5.2)
$$
we have the estimate
$$
\| (\widehat {\mathcal D}(k)+i\mu {\mathcal H}\widehat \sigma _1 +e^{\, 2i\mu
\widehat \sigma _3\Psi }(\widetilde V^{(0)}\widehat I+\widetilde V^{(3)}
\widehat \sigma _3))\varphi \| ^2\geq  \eqno (5.3)
$$ $$
\geq \frac {c_1}6\, \sum\limits_{+,-}\| \widehat P^{\, T^{\pm}(a)}\varphi _{\pm}\|
^2_*+c_8 \sum\limits_{+,-}\| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi
_{\pm}\| ^2_{*,\pm}\, ,
$$
where $c_8=c_8(p,q,F;\widetilde V^{(0)},\widetilde V^{(3)})\in (0,\frac 16\, c_1]
$.}
\vskip 0.2cm

The proof of this theorem is postponed until {\S}{\,}6.

We pass to the proof of Theorem 1.2, in which Theorems 3.1, 3.2, and 5.1 will play
an important part. For $l=1,2$ and $b\geq 0$ (as in (4.2)), we introduce the
functions
$$
{\bf R}^2\ni x\to V^{(l)}_b(x)=
\left\{
\begin{array}{ll}
V^{(l)}(x) & \text {if}\ |V^{(l)}(x)|>b, \\
0\ & \text {otherwise}.
\end{array}
\right.
$$
Since $V^{(l)}\in {\mathbb L}({\bf R}^2)$, $l=1,2$, Lemma 4.6 allows us to choose
a number $b=b(c_1;V^{(1)},V^{(2)})\geq 0$ so that for all $\mu \in {\bf R}$, all
$k\in {\bf R}^2$ with $k_1=\pi $, and all $\psi \in \widetilde H^1(K)$ we have
the inequalities
$$
\| V^{(l)}_b\psi \| ^2\leq \frac {c_1}{192}\, \| \psi \| ^2_{*,\pm}\, ,\ l=1,2\,
,  \eqno (5.4)
$$
for both signs $+$ and $-\, $. For $l=1,2$ we have $\| V^{(l)}-V^{(l)}_b\|
_{L^{\infty}({\bf R}^2)}\leq b<+\infty $; therefore, by Theorem 3.1, there exist
vectors $k^{\, \prime}, {\varkappa}^{\, \prime}\in {\bf R}^2$ and functions $\Phi
^{\, \prime}, \Psi ^{\, \prime}\in \widetilde H^1_0(K)\cap \widetilde C(K)$ with the
following properties: the operators of multiplication by the functions $e^{\, \pm
i\Phi ^{\, \prime}}$ and by the matrix-valued functions $e^{\, \pm \widehat \sigma
_3\Psi ^{\, \prime}}$ act within the space $\widetilde H^1(K;{\bf C}^2)$, for every
$k,\varkappa \in {\bf R}^2$ we have
$$
e^{\, \widehat \sigma _3\Psi ^{\, \prime}}e^{-i\Phi ^{\, \prime}}\, (\widehat
{\mathcal D}(k+k^{\, \prime}+i(\varkappa +\varkappa ^{\, \prime}))+\widehat V)\, e^{\,
i\Phi ^{\, \prime}}e^{\, \widehat \sigma _3\Psi ^{\, \prime}}=  \eqno (5.5)
$$ $$
=\widehat {\mathcal D}(k+i\varkappa )+\widetilde V^{(0)}\widehat I+\sum\limits_{l=1}
^2V^{(l)}_b\widehat \sigma _l+\widetilde V^{(3)}\widehat \sigma _3\, ,
$$
where
$$
\widetilde V^{(0)}=V^{(0)} \cosh 2\Psi ^{\, \prime} +V^{(3)} \sinh 2\Psi ^{\, \prime}
\, ,\ \widetilde V^{(3)}=V^{(0)} \sinh 2\Psi ^{\, \prime} +V^{(3)} \cosh 2\Psi ^{\,
\prime}\, ,
$$
and
$$
\max \{ \| \Phi ^{\, \prime}\| _{L^{\infty}(K)}, \| \Psi ^{\, \prime}\|
_{L^{\infty}(K)}\} \leq c^{\, \prime \prime}_1\, b\, ,  \eqno (5.6)
$$
where $c^{\, \prime \prime}_1=c^{\, \prime \prime}_1(p,q,F)>0$. Inequality (5.6)
shows that $\widetilde V^{(0)},\widetilde V^{(3)}\in {\mathbb L}({\bf R}^2)$. Let
$\Phi , \Psi \in \widetilde H^1_0(K)\cap \widetilde C(K)$ and $\widetilde \varkappa
\in {\bf R}^2$ be the vector-valued functions and the vector defined in Theorem
3.2 (for the functions ${\mathcal F}$, ${\mathcal G}$, ${\mathcal H}$).
Multiplications by $e^{\, \mu \Phi }$ and by $e^{\, i\mu \widehat
{\sigma}_3\Psi }$, $\mu \in {\bf R}$, also act within the space
$\widetilde H^1(K;{\bf C}^2)$. From (3.10) we obtain
$$
e^{\, i\mu \widehat {\sigma}_3\Psi }e^{\, \mu \Phi }\, \bigl( \widehat {\mathcal
D}(k+i\mu \widetilde {\varkappa})+\widetilde V^{(0)}\widehat I+\sum\limits_{l=1}
^2V^{(l)}_b\widehat \sigma _l+\widetilde V^{(3)}\widehat \sigma _3\bigr) \,
e^{-\mu \Phi }e^{\, i\mu \widehat {\sigma}_3\Psi }=  \eqno (5.7)
$$ $$
=\widehat {\mathcal D}(k)+i\mu {\mathcal H}\widehat {\sigma}_1+\sum\limits_{l=1}
^2V^{(l)}_b\widehat \sigma _l+e^{\, 2i\mu \widehat {\sigma}_3\Psi }\, (\widetilde
V^{(0)}\widehat I+\widetilde V^{(3)}\widehat \sigma _3)
$$
for all $\mu \in {\bf R}$ and all $k\in {\bf R}^2$. By Lemma 2.8, $\Psi $
satisfies (5.1). Let $a_0$ and $c_8$ be the numbers defined in Theorem 5.1 (for the
functions $\widetilde V^{(0)}$, $\widetilde V^{(3)}$, and $\Psi $). We put
$\varepsilon =\frac 1{16}\, \sqrt {2c_8}$ and choose a number $a\geq a_0$ so that
$\varepsilon a\geq C_{\varepsilon}(V^{(l)})$, $l=1,2$, where $C_{\varepsilon}(.)$ is
as in (4.7). By Theorem 5.1, there exists a set ${\bf M}\subset {\bf N}$, depending
on ${\mathcal F}$, ${\mathcal G}$, ${\mathcal H}$, on the matrix-valued potential
$\widehat V$, and also on the choice of $b$ and $a$, such that ${\mathcal Q}({\bf N}
\backslash {\bf M})=0$ and estimate (5.3) is valid for all $\mu \in \pi {\bf M}$, all
$k\in {\bf R}^2$ with $k_1=\pi $, and all vector-valued functions (5.2). Using (4.7),
(5.4), and the estimates
$$
\| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi _{\pm}\| \leq \frac 1a\
\| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi _{\pm}\| _{*,\pm}\, ,
$$
we obtain the inequalities
$$
\| \sum\limits_{l=1}^2V^{(l)}_b\widehat \sigma _l\, \varphi \| ^2\leq
2\sum\limits_{l=1}^2\| V^{(l)}_b \varphi \| ^2\leq
$$ $$
\leq 4\sum\limits_{l=1}^2\sum\limits_{+,-}\| V^{(l)}_b\widehat P^{\, T^{\pm}(a)}
\varphi _{\pm}\| ^2+4\sum\limits_{l=1}^2\sum\limits_{+,-}\| V^{(l)}\widehat P^{\,
{\bf Z}^2\backslash T^{\pm}(a)}\varphi _{\pm}\| ^2 \leq
$$ $$
\leq \frac {c_1}{24}\, \sum\limits_{+,-}\| \widehat P^{\, T^{\pm}(a)}\varphi _{\pm}
\| ^2_{*,\pm}+
$$ $$
+8\sum\limits_{l=1}^2\sum\limits_{+,-}\biggl( \varepsilon ^2\| \widehat
P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi _{\pm}\| ^2_{*,\pm}+C_{\varepsilon}^2(V^
{(l)})\| \widehat P^{\, {\bf Z}^2\backslash T^{\pm}(a)}\varphi _{\pm}\| ^2\biggr)
\leq
$$ $$
\leq \frac {c_1}{24}\, \sum\limits_{+,-}\| \widehat P^{\, T^{\pm}(a)}\varphi _{\pm}
\| ^2_{*,\pm}+\frac {c_8}4\, \sum\limits_{+,-}\| \widehat P^{\, {\bf Z}^2
\backslash T^{\pm}(a)}\varphi _{\pm}\| ^2_{*,\pm}\, .
$$
Therefore, (5.3) implies that, again for all $\mu \in \pi {\bf M}$, all
$k\in {\bf R}^2$ with $k_1=\pi $, and all $\varphi $ as in (5.2),
$$
\| (\widehat {\mathcal D}(k)+i\mu {\mathcal H}\widehat {\sigma}_1+\sum\limits
_{l=1}^2V^{(l)}_b\widehat \sigma _l+e^{\, 2i\mu \widehat {\sigma}_3\Psi }\,
(\widetilde V^{(0)}\widehat I+\widetilde V^{(3)}\widehat \sigma _3))\varphi \| ^2
\geq  \eqno (5.8)
$$ $$
\geq \frac 12\, \| (\widehat {\mathcal D}(k)+i\mu {\mathcal H}\widehat {\sigma}_1
+e^{\, 2i\mu \widehat {\sigma}_3\Psi }\, (\widetilde V^{(0)}\widehat I+\widetilde
V^{(3)}\widehat \sigma _3))\varphi \| ^2- \| \sum\limits_{l=1}^2V^{(l)}_b\widehat
\sigma _l\, \varphi \| ^2 \geq
$$ $$
\geq \frac {c_1}{24}\, \sum\limits_{+,-}\| \widehat P^{\, T^{\pm}(a)}\varphi _{\pm}
\| ^2_{*,\pm}+\frac {c_8}4\, \sum\limits_{+,-}\| \widehat P^{\, {\bf Z}^2
\backslash T^{\pm}(a)}\varphi _{\pm}\| ^2_{*,\pm}\geq
$$ $$
\geq \frac {c_8}4\, \sum\limits_{+,-}\| \varphi _{\pm}\| ^2_{*,\pm}\geq
\frac {c_8}4\, \sum\limits_{N\in {\bf Z}^2}G^2_N(k;\mu )|\varphi _N|^2\geq
\frac {\pi ^2}4\, c_8\, \| \varphi \| ^2\, .
$$
Now we use (5.5) with $\varkappa =\mu \widetilde \varkappa $ and also (5.6) -- (5.8)
and the estimate $\| \Phi \| _{L^{\infty}(K)}\leq c_1^*$ (see Theorem 3.2) to obtain
the inequality
$$
\| (\widehat {\mathcal D}(k+k^{\, \prime}+i(\mu \widetilde \varkappa +\varkappa ^{\,
\prime}))+\widehat V)\varphi \| \geq \frac {\pi}2\, \sqrt {c_8}\, e^{-4c_1^{\,
\prime \prime}b}\, e^{-2c_1^*\mu }\, \| \varphi \| \, ,
$$
which is valid for all $\mu \in \pi {\bf M}$, all $k\in {\bf R}^2$ with $k_1=\pi $,
and all $\varphi \in \widetilde H^1(K;{\bf C}^2)$. To complete the proof of Theorem
1.2 it remains to put $e=\widetilde \varkappa /|\widetilde \varkappa |$ and $c=
3c_1^*/c_3^*$ (see Lemma 3.2), and it suffices to choose numbers $\widetilde \mu
\doteq |\widetilde \varkappa |\mu \in \pi |\widetilde \varkappa |\, {\bf M}$ for
which
$$
4c_1^{\, \prime \prime}\, b-\ln \bigl( \, \frac {\pi}2\, \sqrt {c_8}\, \bigr)
\leq c_1^*\, \frac {\widetilde \mu }{|\widetilde \varkappa |}\, .
$$
Theorem 1.2 is proved.    \hfill $\square$
\vskip 0.2cm

{\it Remark}. Under the conditions of Theorem 1.2, if $V^{(l)}\in L^2\{ g,{\bf
Z}^2\} $, $l=1,2$, for some $g\in {\mathbb G}$, then the proof of Theorem 1.2
simplifies, because Theorem 3.1 provides an identity similar to (5.5) but without
the term  $\sum\limits_{l=1}^2V^{(l)}_b\widehat \sigma _l$ on the right-hand side
(see also [29]).

\section{Proof of Theorem 5.1}

Under the conditions of Theorem 5.1, denote $V^{\, (\pm )}=\widetilde V^{(0)}\pm
\widetilde V^{(3)}$. Since
$$
\widehat {\mathcal D}(k)+i\mu {\mathcal H}\widehat \sigma _1+e^{\, 2i\mu \widehat
\sigma _3\Psi }\, (\widetilde V^{(0)}\widehat I+\widetilde V^{(3)}\widehat \sigma
_3)=\left( \begin{matrix} e^{\, 2i\mu \Psi }\, V^{\, (+)} &\widehat
d_-(k)+i\mu {\mathcal H}\\ \widehat d_+(k)+i\mu {\mathcal H} &e^{-2i\mu \Psi }\,
V^{\, (-)}\end{matrix} \right) \, ,
$$
Theorem 5.1 is equivalent to the following statement.
\vskip 0.2cm

{\bf Theorem 6.1}. {\it Suppose $\{ {\mathcal F}, {\mathcal G}, {\mathcal H} \} \in
\Gamma (p,q,F)$, $V^{\, (\pm )}\in {\mathbb L}({\bf R}^2)$, and $\Psi $ is a
real-valued function of class $\widetilde C(K)$ satisfying (5.1) for all $\lambda
\in {\bf R}$. Then there exists a number $a_0^{\, \prime}=a_0^{\, \prime}(p,q,F;V^{\,
(+)},V^{\, (-)})\geq 2\pi $ with the following property: for every $a_1\geq a_0^{\,
\prime}$ we can find a set ${\bf M}\subset {\bf N}$, depending also on ${\mathcal F}$,
${\mathcal G}$, ${\mathcal H}$, $V^{\, (+)}$, $V^{\, (-)}$, and $\Psi $, such that
${\mathcal Q}\, ({\bf N}\backslash {\bf M})=0$ and for all $\mu \in \pi {\bf M}$, all
$k\in {\bf R}^2$ with $k_1=\pi $, and all $\varphi _{\pm}\in \widetilde H^1(K)$ we
have
$$
\| (\widehat d_+(k)+i\mu {\mathcal H})\varphi _++e^{-2i\mu \Psi }\, V^{\, (-)}
\varphi _- \| ^2+\| (\widehat d_-(k)+i\mu {\mathcal H})\varphi _-+e^{\, 2i\mu \Psi }\,
V^{\, (+)}\varphi _+ \| ^2\geq
$$ $$
\geq \frac {c_1}{6}\, \bigl( \| \widehat P^{\, T^+(a_1)}\varphi _+\| ^2_*+
\| \widehat P^{\, T^-(a_1)}\varphi _-\| ^2_*\bigr) +
$$ $$
+c^{\, \prime}_8\, \bigl( \| \widehat P^{\, {\bf Z}^2\backslash T^+(a_1)}\varphi _+
\| ^2_{*,+}+\| \widehat P^{\, {\bf Z}^2\backslash T^-(a_1)}\varphi _-\| ^2_{*,-}
\bigr) \, ,
$$
where $c_8^{\, \prime}=c_8^{\, \prime}(p,q,F;V^{\, (+)},V^{\,
(-)})\in (0,\frac 16\, c_1]$.}
\vskip 0.2cm

The next lemma is a version of the Wiener theorem (see, e.g., [45, Theorem
XI.114 and the remark after it]).
\vskip 0.2cm

{\bf Lemma 6.1}. {\it Suppose $W\in L^1(K)$, and $\Psi $ is a real-valued function
of class $\widetilde C(K)$ satisfying (5.1) for all $\lambda \in {\bf R}$. Then
$$
\lim\limits_{N\to +\infty }\, \frac 1N \sum\limits_{\nu =1}^N\, \biggl| \,
\int\limits_Ke^{\, 2\pi i\nu \, (\Psi - x_2)}\, W\, d^2x\, \biggr| ^2=0\, .
$$}
\vskip 0.2cm

{\bf Corollary}. {\it Under the conditions of Lemma 6.1, denote
$$
{\bf M}_{\pm}(W,\Psi ;\theta )=\biggl\{ \nu \in {\bf N}:\biggl| \, \int\limits
_Ke^{\, \pm 2\pi i\nu \, (\Psi -x_2)}\, W\, d^2x\biggr| \geq \theta \biggr\}
\, ,\ \theta >0\, .
$$
Then ${\mathcal Q}\, ({\bf M}_{\pm}(W,\Psi ;\theta ))=0$ (for any $\theta >0$).}
\vskip 0.2cm

{\it Proof of Theorem 6.1}. Let $f_{V^{(\pm )}}$, $h_{V^{(\pm )}}$, and $c_7\,
(V^{(\pm )})$ be as in Lemmas 4.2 and 4.3. We denote
$$
c^{\, \prime}_7=\max \, \{ c_7(V^{\, (+)}),c_7(V^{\, (-)})\} \, ,\ c_8^{\,
\prime}=\frac 16\, c_1^2(c_1+4(c_7^{\, \prime})^2)^{-1}
$$
(then $c_8^{\, \prime}\in (0,\frac 16\, c_1]$). Suppose $\mu \geq \mu _0>0$, where
$\mu _0$ is a sufficiently large number to be chosen later. To start with, we
assume that $\mu _0\geq 4\pi $. Let $\varphi _{\pm}\in \widetilde H^1(K)$, and let
$k\in {\bf R}^2$. For $a\in [2\pi ,\mu /2]$, we denote
$$
\varphi ^{(a)}_{\pm}=\widehat P^{\, T^{\, \pm}(a)}\varphi _{\pm}\, ,\ \widetilde
\varphi ^{\, (a)}_{\pm}=\widehat P^{\, {\bf Z}^2\backslash T^{\, \pm}(a)}\varphi
_{\pm}\, ,
$$
where $T^{\, \pm}(a)=\{ N\in {\bf Z}^2: G^{\pm}_N(k;\mu )\leq a\} $ (the functions
$G^{\pm}_N(k;\mu )$ and $G_N(k;\mu )$, $N\in {\bf Z}^2$, and also the norms
$\| .\| _*$ and $\| .\| _{*,\pm}\, $, were defined at the beginning of {\S}{\,}4).
Lemma 4.1 yields the estimates
$$
c_1\, \| \varphi ^{(a)}_{\pm}\| ^2_*=
c_1\, \| \varphi ^{(a)}_{\pm}\| ^2_{*,\pm}\leq \| (\widehat d_{\pm}(k)+i\mu
{\mathcal H})\varphi ^{(a)}_{\pm}\| ^2\leq
c_2\, \| \varphi ^{(a)}_{\pm}\| ^2_{*,\pm}=c_2\, \| \varphi ^{(a)}_{\pm}\| ^2_*\, ,
\eqno (6.1)
$$ $$
c_1\, \| \widetilde \varphi ^{\, (a)}_{\pm}\| ^2_{*,\pm}\leq \| (\widehat d_{\pm}(k)
+i\mu {\mathcal H})\widetilde \varphi ^{(a)}_{\pm}\| ^2\leq c_2\, \| \widetilde
\varphi ^{\, (a)}_{\pm}\| ^2_{*,\pm}\, .  \eqno (6.2)
$$
We choose a number  $a_0^{\, \prime}\geq 2\pi $ for which
$$
\max\ \{ h^2_{V^{(+)}}(a_0^{\, \prime}),h^2_{V^{(-)}}(a_0^{\, \prime})\}
\leq \frac 16\, c_8^{\, \prime}  \eqno (6.3)
$$
(this can be done because $h_{V^{(\pm)}}(t)\to 0$ as $t\to +\infty $). Let $a_1
\geq a_0^{\, \prime}\, $, and let $\delta = \min \, \{ \frac 1{32}\,
c_1, \frac 32\, c_8^{\, \prime}\, \} $. We denote by $J$ the smallest integer with
$c_2^2\leq J\delta ^2$ and choose numbers $a_2,\dots , a_{J+1}$ so that
$a_{j+1}>a_j$ ($j=1,\dots ,J$) and each of the (four) functions ${\mathcal P}=
{\mathcal G}^2+{\mathcal F}^2$, ${\mathcal P}=({\mathcal G}\pm i{\mathcal F})
{\mathcal H}$, and ${\mathcal P}={\mathcal H}^2$ in the space $L^{\infty}(K)
\subset L^2(K)$ satisfies the inequalities
$$
a_j\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\, a_{j+1}-a_j}|
{\mathcal P}_N|^2\biggr) ^{1/2}\leq \frac {\delta}{4\, \sqrt {6\pi }}\, ,
\eqno (6.4)
$$
where the ${\mathcal P}_N$ are the Fourier coefficients of ${\mathcal P}$
(the numbers $a_0^{\, \prime}$, $\delta$, and $J$ depend on $p$, $q$, $F$, $V^{(+)}$
and $V^{(-)}$, and the numbers $a_2,\dots ,a_{J+1}$ depend on $a_1$ and the
functions ${\mathcal F}$, ${\mathcal G}$, ${\mathcal H}$, $V^{(+)}$, and $V^{(-)}$).
Now we specify the choice of $\mu _0\, $. We assume that $\mu _0\geq 2a_{J+1}$,
$$
\max\ \{ h^2_{V^{(+)}}(\mu _0),h^2_{V^{(-)}}(\mu _0)\} \leq \frac 1{54}\,
c_8^{\, \prime}\, , \eqno (6.5)
$$
and, for all $\mu \geq \mu _0$ (and all $k\in {\bf R}^2$),
$$
\mu ^{-2}\, \max\  \{ f^2_{V^{(-)}}(\# \, T^{+}(\mu /2)),f^2_{V^{(+)}}
(\# \, T^{-}(\mu /2))\} \leq \frac 1{24}\, c_8^{\, \prime}  \eqno (6.6)
$$
(by Lemma 4.2 and the estimate $\# \, T^{\pm }(\mu /2)< \frac {3\pi }2
\, \mu ^2$ (see (4.1) with $a=\mu /2\geq \mu _0/2\geq 2\pi $), condition (6.6)
can indeed be ensured if $\mu _0$ is chosen sufficiently large). Since
$$
\sum\limits_{j=1}^J\| \widetilde \varphi _{\pm}^{(a_j)}-\widetilde \varphi
_{\pm}^{(a_{j+1})}\| ^2_{*,\pm}=\| \widetilde \varphi _{\pm}^{(a_1)}-\widetilde
\varphi _{\pm}^{(a_{J+1})}\| ^2_{*,\pm}=\| \widetilde \varphi _{\pm}^{(a_1)}-\widetilde
\varphi _{\pm}^{(a_{J+1})}\| ^2_*\leq \| \varphi _{\pm}\| ^2_*\, ,
$$
it follows that, depending on the functions $\varphi _{\pm}\in \widetilde H^1(K)$,
we can find indices $j_{\pm}\in \{ 1,\dots ,J\} $ such that
$$
\| \widetilde \varphi _{\pm}^{(a_{j_{\pm}})}-\widetilde \varphi _{\pm}^{(a_{j_{\pm}+1})}\|
^2_{*,\pm}\leq J^{-1}\, \| \varphi _{\pm}\| ^2_*\, .
$$
Then, by Lemma 4.1 and (6.2), we have
$$
\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})(\widetilde \varphi _{\pm}^{(a_{j_{\pm}})}
-\widetilde \varphi _{\pm}^{(a_{j_{\pm}+1})})\| \leq  \eqno (6.7)
$$ $$
\leq \sqrt {c_2}\, \| \widetilde \varphi _{\pm}^{(a_{j_{\pm}})}-\widetilde
\varphi _{\pm}^{(a_{j_{\pm}+1})}\| _{*,\pm}\leq \biggl( \frac {c_2}J\biggr) ^{1/2}
\| \varphi _{\pm}\| _*\, .
$$
The above choice of the indices $j_{\pm}$ is important for the proof of the next
lemma.
\vskip 0.2cm

{\bf Lemma 6.2}. {\it We have
$$
|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde \varphi _{\pm}^{(a_{j_{\pm}})},
(\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})})|\leq
2\delta\, \| \varphi _{\pm}\| _{*,\pm}\, \| \varphi _{\pm} ^{(a_{j_{\pm}})}\| _*\ .
\eqno (6.8)
$$}

{\it Proof}. From (6.7) we deduce that
$$
|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde \varphi _{\pm}^{(a_{j_{\pm}})},
(\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})})|\leq
\eqno (6.9)
$$ $$
\leq |((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde
\varphi _{\pm}^{(a_{j_{\pm}+1})}, (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}^{(a_{j_{\pm}})})|+
$$ $$
+|((\widehat d_{\pm}(k)+i\mu {\mathcal H})(\widetilde \varphi _{\pm}^{(a_{j_{\pm}})}
-\widetilde \varphi _{\pm}^{(a_{j_{\pm}+1})}), (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}^{(a_{j_{\pm}})})|\leq
$$ $$
\leq |((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde
\varphi _{\pm}^{(a_{j_{\pm}+1})}, (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}^{(a_{j_{\pm}})})|+
$$ $$
+\biggl( \frac {c_2}J\biggr) ^{1/2}\| \varphi _{\pm}
\| _*\, \| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}\| \, .
$$
Also, we have $\| \varphi _{\pm}\| _*\leq \| \varphi _{\pm}\| _{*,\pm}$ and
$$
\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi ^{(a_{j_{\pm}})}_{\pm}\| \leq
\sqrt c_2\, \| \varphi ^{(a_{j_{\pm}})}_{\pm}\| _{*,\pm}=\sqrt c_2\,
\| \varphi ^{(a_{j_{\pm}})}_{\pm}\| _*
$$
by (6.1). Since
$$
\widehat d_{\pm}(k)+i\mu {\mathcal H}=
({\mathcal G}\pm i{\mathcal F})\bigl( k_1-i\frac {\partial}{\partial x_1}\bigr)
+i{\mathcal H}\bigl( \mu \pm \bigl( k_2-i\frac {\partial}{\partial x_2}\bigr)
\bigr)\, ,
$$
we can write (see Lemma 4.5 and inequalities (6.4))
$$
|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde
\varphi _{\pm}^{(a_{j_{\pm}+1})}, (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}^{(a_{j_{\pm}})})|\leq
$$ $$
\leq \sqrt {6\pi }\, a_{j_{\pm}}\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\,
a_{j_{\pm}+1}-a_{j_{\pm}}}|({\mathcal G}^2+{\mathcal F}^2)_N|^2\biggr) ^{1/2}
\times
$$ $$
\times \ \bigl\| \bigl( k_1-i\frac {\partial}{\partial x_1}\bigr) \widetilde
\varphi _{\pm}^{\, (a_{j_{\pm}+1})}\bigr\| \cdot \bigl\| \bigl( k_1-i\frac
{\partial}{\partial x_1}\bigr) \varphi _{\pm} ^{(a_{j_{\pm}})}\bigr\| +
$$ $$
+\sqrt {6\pi }\, a_{j_{\pm}}\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\,
a_{j_{\pm}+1}-a_{j_{\pm}}}|(({\mathcal G}\mp i{\mathcal F}){\mathcal H})_N|^2
\biggr) ^{1/2}\times
$$ $$
\times \ \bigl\| \bigl( k_1-i\frac {\partial}{\partial x_1}\bigr) \widetilde
\varphi _{\pm}^{\, (a_{j_{\pm}+1})}\bigr\| \cdot \bigl\| \bigl( \mu \pm \bigl(
k_2-i\frac {\partial}{\partial x_2}\bigr) \bigr) \varphi _{\pm} ^{(a_{j_{\pm}})}
\bigr\| +
$$ $$
+\sqrt {6\pi }\, a_{j_{\pm}}\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\,
a_{j_{\pm}+1}-a_{j_{\pm}}}|(({\mathcal G}\pm i{\mathcal F}){\mathcal H})_N|^2
\biggr) ^{1/2}\times
$$ $$
\times \ \bigl\| \bigl( \mu \pm \bigl( k_2-i\frac {\partial}{\partial x_2}
\bigr) \bigr) \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}+1})}\bigr\| \cdot \bigl\|
\bigl( k_1-i\frac {\partial}{\partial x_1}\bigr) \varphi _{\pm} ^{(a_{j_{\pm}})}
\bigr\| +
$$ $$
+\sqrt {6\pi }\, a_{j_{\pm}}\, \biggl( \, \sum\limits_{N\in {\bf Z}^2\, :\, 2\pi |N|\, >\,
a_{j_{\pm}+1}-a_{j_{\pm}}}|({\mathcal H}^2)_N|^2\biggr) ^{1/2}\times
$$ $$
\times \ \bigl\| \bigl( \mu \pm \bigl( k_2-i\frac {\partial}{\partial x_2}
\bigr) \bigr) \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}+1})}\bigr\| \cdot
\bigl\| \bigl( \mu \pm \bigl( k_2-i\frac {\partial}{\partial x_2}\bigr)
\bigr) \varphi _{\pm} ^{(a_{j_{\pm}})}\bigr\| \leq
$$ $$
\leq \delta \, \| \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}+1})}\| _{*,\pm}\,
\| \varphi _{\pm} ^{(a_{j_{\pm}})}\| _{*,\pm}\leq \delta \, \| \varphi _{\pm}
\| _{*,\pm}\, \| \varphi _{\pm} ^{(a_{j_{\pm}})}\| _*\, .
$$
Now, estimate (6.8) for all $\mu \geq \mu _0$ and all $k\in {\bf R}^2$ follows
from (6.9) and the choice of $J\in {\bf N}$. Lemma 6.2 is proved.
\hfill  $\square$
\vskip 0.2cm

Now we define a set ${\bf M}\subset {\bf N}$ (for which ${\mathcal Q}\, ({\bf N}
\backslash {\bf M})=0$). We have $\# \, \{ N\in {\bf Z}^2:\pi \, |N|<a_J\} \leq
{\tau}^*\doteq 4{\pi }^{-1}a_J^2\, $. Put $\theta =\frac 13\, 2^{-6}c_1(\pi
a_1^2{\tau}^*)^{-1}$. The corollary to Lemma 6.1 implies that for all functions
${\mathcal P}^{\, \prime}_{\pm}=({\mathcal G}\pm i{\mathcal F})V^{\, (\pm )}e^{\,
2\pi i\, (N,\, x)}$ and ${\mathcal P}^{\, \prime}_{\pm}={\mathcal H}V^{\, (\pm )}
e^{\, 2\pi i\, (N,\, x)}$, where $N\in {\bf Z}^2$ and $\pi \, |N|<a_J\, $, we have
${\mathcal Q}\, ({\bf M}_{\pm}({\mathcal P}^{\, \prime}_{\pm}, \Psi ; \theta ))=0$.
Let ${\bf M}^{\, \prime}$ be the union of the sets ${\bf M}_{\pm}({\mathcal P}^{\,
\prime}_{\pm},\Psi ; \theta )$ for both signs $+$ and $-$ and for all functions
${\mathcal P}^{\, \prime}_{\pm}\, $ as above. Since there are at most $4{\tau}^*$
such sets, we have ${\mathcal Q}\, ({\bf M}^{\, \prime})=0$. We put ${\bf M}={\bf N}
\backslash ({\bf M}^{\, \prime}\cup \{ m\in {\bf N}: \pi m <\mu _0\} )$. Then
${\mathcal Q}\, ({\bf N}\backslash {\bf M})=0$, and $\mu \geq \mu _0$ whenever
$\mu \in \pi {\bf M}$. In what follows, we assume that $\mu \in \pi {\bf M}$.
\vskip 0.2cm

{\bf Lemma 6.3}. {\it For all $\mu \in \pi {\bf M}$ (and all $k\in {\bf R}^2$)
we have
$$
|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}, e^{\,
\mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)})|\leq \frac {c_1}{16}\,
\| \varphi _{\pm}^{(a_{j_{\pm}})}\| _*\, \| \varphi _{\mp}^{(a_1)}\| \, . \eqno (6.10)
$$}

{\it Proof}. If $\mu \in \pi {\bf M}$, then ${\mu}/{\pi}\in
{\bf N}\backslash {\bf M}_{\pm}({\mathcal P}^{\, \prime}_{\pm},\Psi ; \theta )$
for all functions ${\mathcal P}^{\, \prime}_{\pm}\, $ indicated above.
Therefore, the definition of the set ${\bf M}_{\pm}({\mathcal P}^{\, \prime}_{\pm},
\Psi ; \theta )$ implies the inequalities
$$
6\pi a_1^2\sum\limits_{N\in {\bf Z}^2\, :\, \pi |N|\, <\, a_J}\, \biggl| \,
\int\limits_Ke^{\, \mp 2i\mu \, (\Psi -x_2)}({\mathcal G}\mp i{\mathcal F})V^{\,
(\mp )}e^{\, 2\pi i\, (N,\, x)}d^2x\, \biggr| <
$$ $$
<6\pi a_1^2{\tau}^*\theta =\frac {c_1}{32}
\, ,
$$ $$
6\pi a_1^2\sum\limits_{N\in {\bf Z}^2\, :\, \pi |N|\, <\, a_J}\, \biggl| \,
\int\limits_Ke^{\, \mp 2i\mu \, (\Psi -x_2)}\, {\mathcal H}\, V^{\, (\mp )}
e^{\, 2\pi i\, (N,\, x)}d^2x\, \biggr| <
$$ $$
<6\pi a_1^2{\tau}^*\theta =\frac {c_1}{32}\, .
$$
Since $\# \, T^{\pm }(a_1)\leq 6\pi a_1^2$, we deduce the estimate
$$
|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}, e^{\,
\mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)})|\leq
$$ $$
\leq \bigl| \bigl( ({\mathcal G}\pm i{\mathcal F})\bigl( k_1-i\frac {\partial}
{\partial x_1}\bigr) \varphi _{\pm}^{(a_{j_{\pm}})}, e^{\, \mp 2i\mu \Psi }\, V^{\,
(\mp )}\varphi _{\mp}^{(a_1)}\bigr) \bigr| +
$$ $$
+\bigl| \bigl( {\mathcal H}\bigl( \mu \pm \bigl( k_2-i\frac {\partial}{\partial
x_2}\bigr) \bigr) \varphi _{\pm}^{(a_{j_{\pm}})}, e^{\, \mp 2i\mu \Psi }\, V^{\,
(\mp )}\varphi _{\mp}^{(a_1)}\bigr) \bigr| \leq
$$ $$
\leq \biggl| \ \sum\limits_{N\in \, T^{\, \pm}(a_{j_{\pm}}),\, M\in \, T^{\,
\mp}(a_1)} \ (k_1+2\pi N_1)\, \overline {(\varphi
_{\pm})_N}\, (\varphi _{\mp})_M \times
$$ $$
\times \int\limits_K e^{\, \mp 2i \mu \Psi +2\pi
i\, (M-N,\, x)}({\mathcal G}\mp i{\mathcal F})V^{\, (\mp )}d^2x\ \biggr| \, +
$$ $$
+\, \biggl| \sum\limits_{N\in \, T^{\, \pm}(a_{j_{\pm}}),\, M\in \, T^{\, \mp}
(a_1)}(\mu \pm (k_2+2\pi N_2))\overline {(\varphi _{\pm})_N}(\varphi
_{\mp})_M \times
$$ $$
\times \int\limits_K e^{\, \mp 2i \mu \Psi +2\pi i\, (M-N,\, x)}{\mathcal H}
V^{\, (\mp )}d^2x\, \biggr| \leq
$$ $$
\leq 6\pi a_1^2\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| _*\, \| \varphi _{\mp}^{(a_1)}
\| \times
$$ $$
\times \biggl( \, \sum\limits_{N^{\, \prime}\in {\bf Z}^2\, :\, \pi |N^{\,
\prime}| \, <\, a_J} \biggl| \, \int\limits^{\phantom a}_K e^{\, \mp 2i\mu \,
(\Psi -x_2)}({\mathcal G}\mp i{\mathcal F})V^{\, (\mp )}e^{\, 2\pi i\, (N^{\,
\prime},\, x)}d^2x\, \biggr| +
$$ $$
+\sum\limits_{N^{\, \prime}\in {\bf Z}^2\, :\, \pi |N^{\, \prime}|
\, <\, a_J} \biggl| \, \int\limits_K e^{\, \mp 2i\mu \, (\Psi -x_2)}\, {\mathcal H}
\, V^{\, (\mp )}e^{\, 2\pi i\, (N^{\, \prime},\, x)}d^2x\, \biggr| \, \biggr) \leq
$$ $$
\leq \frac {c_1}{16}\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| _*\, \| \varphi
_{\mp}^{(a_1)}\| \, . \phantom {\int}
$$
This proves Lemma 6.3.   \hfill  $\square$
\vskip 0.2cm

We pass to the proof of the main estimates. In the sequel it is assumed that
$k\in {\bf R}^2$ is such that $k_1=\pi $. Taking into account the inequalities
$$
\| \varphi _{\mp}^{(a_1)}\| \leq \| \varphi _{\mp}^{(a_{j_{\mp}})}\| \leq
\pi ^{-1}\, \| \varphi _{\mp}^{(a_{j_{\mp}})}\| _*\leq \| \varphi _{\mp}^{(a_{j
_{\mp}})}\| _*\, ,
$$
and using (6.8) and (6.10), we obtain
$$
\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}+e^{\,
\mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2=
$$ $$
=\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}
+(\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2\geq
$$ $$
\geq \| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}\| ^2
+\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2-
$$ $$
-2\, |((\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde
\varphi _{\pm}^{\, (a_{j_{\pm}})}, (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}^{(a_{j_{\pm}})})|-
$$ $$
-2|((\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}^{(a_{j_{\pm}})}, e^{\,
\mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)})|\geq
$$ $$
\geq c_1\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_*+\| (\widehat d_{\pm}(k)+i\mu
{\mathcal H})\widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}+e^{\, \mp 2i\mu \Psi }\,
V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2-
$$ $$
-4\delta \, \| \varphi _{\pm}\| _{*,\pm}\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| _*
-\frac {c_1}8\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| _*\, \| \varphi _{\mp}^{(a_1)}\|
\geq
$$ $$
\geq c_1\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_*+\| (\widehat d_{\pm}(k)+i\mu
{\mathcal H})\widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}+e^{\, \mp 2i\mu \Psi }\,
V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2-
$$ $$
-2\delta \, \bigl( \, \| \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}\|
^2_{*,\pm}+ 2\, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_{*,\pm}\, \bigr) -\frac
{c_1}{16}\, \bigl( \, \| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_*+\| \varphi
_{\mp}^{(a_{j_{\mp}})}\| ^2_* \, \bigr) \, .
$$
For any $\varepsilon \in (0,1)$, we have (see Lemma 4.3)
$$
\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\widetilde
\varphi _{\pm}^{\, (a_{j_{\pm}})}+e^{\, \mp 2i\mu \Psi }\,
V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2\geq
$$ $$
\geq (1-\varepsilon )\, \| (\widehat d_{\pm}(k)+i\mu
{\mathcal H})\widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}\| ^2-
(1-\varepsilon )\varepsilon ^{-1}\| V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2\geq
$$ $$
\geq (1-\varepsilon )c_1\, \| \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}\|
^2_{*,\pm}- (1-\varepsilon )\varepsilon ^{-1}(c_7(V^{(\mp )}))^2\, \| \varphi
_{\mp}^{(a_1)}\| ^2_*\, \geq
$$ $$
\geq (1-\varepsilon )c_1\, \| \widetilde \varphi _{\pm}^{\, (a_{j_{\pm}})}\|
^2_{*,\pm}- (1-\varepsilon )\varepsilon ^{-1}(c_7^{\, \prime})^2\, \| \varphi
_{\mp}^{(a_{j_{\mp}})}\| ^2_*\, .
$$
Therefore,
$$
\sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2\geq
$$ $$
\geq \bigl( c_1-(1-\varepsilon )\varepsilon ^{-1}(c_7^{\, \prime})^2-4\delta
-\frac {c_1}8\, \bigr) \sum\limits_{+,-}\| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_* +
\bigl( (1-\varepsilon )c_1-2\delta \bigr) \sum\limits_{+,-} \| \widetilde \varphi
_{\pm}^{\, (a_{j_{\pm}})}\| ^2_{*,\pm}\, .
$$
If $c_7^{\, \prime}>0$, we put $\varepsilon =4\, (c_7^{\, \prime})^2(c_1
+4\, (c_7^{\, \prime})^2)^{-1}$. Since $(1-\varepsilon )\varepsilon ^{-1} (c_7^{\,
\prime})^2=c_1/4$, $4\delta \leq c_1/8$ and $2\delta \leq \frac 12\, (1-\varepsilon )
c_1=3c_8^{\, \prime}\, $, we have
$$
\sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2\geq \eqno (6.11)
$$ $$
\geq \frac {c_1}2\,
\sum\limits_{+,-}\| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_*+3c_8^{\, \prime}
\, \sum\limits_{+,-} \| \widetilde \varphi
_{\pm}^{\, (a_{j_{\pm}})}\| ^2_{*,\pm}\, ,
$$
which is also true if $c_7^{\, \prime}=0$. Estimate (6.11) is a key point in the
proof of Theorem 6.1, and now it is not too hard to complete the proof. For
$\varphi _{\pm}\in \widetilde H^1(K)$, we have
$$
\widetilde \varphi ^{\, (a_1)}_{\pm}=\widehat P^{\, T^{\pm}(\mu /2)\backslash
T^{\pm}(a_1)}\varphi _{\pm}+\widehat P^{\, {\bf Z}^2\backslash (T^+(\mu /2)\cup
T^-(\mu /2))}\varphi _{\pm}+\widehat P^{\, T^{\mp}(\mu /2)}\varphi _{\pm}\, .
$$
Using Lemmas 4.2, 4.3, and 4.4, and conditions (6.3), (6.5), and (6.6), we obtain
$$
\| V^{\, (\mp )}\widehat P^{\, T^{\mp}(\mu /2)\backslash
T^{\mp}(a_1)}\varphi _{\mp}\| ^2\leq h^2_{V^{(\mp )}}(a_1) \, \| \widehat
P^{\, T^{\mp}(\mu /2)\backslash T^{\mp}(a_1)}\varphi _{\mp}\| ^2_*\leq
$$ $$
\leq \frac 16\, c_8^{\, \prime}\, \| \widehat
P^{\, T^{\mp}(\mu /2)\backslash T^{\mp}(a_1)}\varphi _{\mp}\| ^2_*\, ,
$$ $$
\| V^{\, (\mp )}\widehat P^{\, {\bf Z}^2\backslash (T^+(\mu /2)\cup
T^-(\mu /2))}\varphi _{\mp}\| ^2\leq 9h^2_{V^{(\mp )}}(\mu ) \, \| \widehat P^{\,
{\bf Z}^2\backslash (T^+(\mu /2)\cup T^-(\mu /2))}\varphi _{\mp}\| ^2_*\leq
$$ $$
\leq \frac 16\, c_8^{\, \prime}\, \| \widehat P^{\,
{\bf Z}^2\backslash (T^+(\mu /2)\cup T^-(\mu /2))}\varphi _{\mp}\| ^2_*\, ,
$$ $$
\| V^{\, (\mp )}\widehat P^{\, T^{\pm}(\mu /2)}\varphi _{\mp}\| ^2\leq
f^2_{V^{(\mp )}}(\# \, T^{\pm }(\mu /2)) \, \| \widehat P^{\, T^{\pm}(\mu /2)}
\varphi _{\mp}\| ^2\leq
$$ $$
\leq 4\mu ^{-2}\, f^2_{V^{(\mp )}}(\# \, T^{\pm }(\mu /2)) \, \| \widehat P^{\,
T^{\pm}(\mu /2)}\varphi _{\mp}\| ^2_{*,\mp}\leq \frac 16\, c_8^{\, \prime}\,
\| \widehat P^{\, T^{\pm}(\mu /2)}\varphi _{\mp}\| ^2_{*,\mp}\, .
$$
Therefore,
$$
\sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}\| ^2\geq  \eqno (6.12)
$$ $$
\geq \frac 12\, \sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2-
$$ $$
-3\, \sum\limits_{+,-}\| V^{\, (\mp )}\widehat P^{\, T^{\mp}(\mu /2)\backslash
T^{\mp}(a_1)}\varphi _{\mp}\| ^2-
3\, \sum\limits_{+,-}\| V^{\, (\mp )}\widehat P^{\, {\bf Z}^2\backslash
(T^+(\mu /2)\cup T^-(\mu /2))}\varphi _{\mp}\| ^2-
$$ $$
-3\, \sum\limits_{+,-}\| V^{\, (\mp )}\widehat P^{\, T^{\pm}(\mu /2)}
\varphi _{\mp}\| ^2\geq
$$ $$
\geq \frac 12\, \sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})
\varphi _{\pm}+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}^{(a_1)}\| ^2-
\frac 12\, c_8^{\, \prime}\, \sum\limits_{+,-}\|\widetilde \varphi _{\mp}^{\,
(a_1)}\| ^2_{*,\mp}\, .
$$
Finally, from (6.11), (6.12), and the condition $c_8^{\, \prime}\leq \frac 16\,
c_1$ we deduce the estimate claimed in Theorem 6.1:
$$
\sum\limits_{+,-}\| (\widehat d_{\pm}(k)+i\mu {\mathcal H})\varphi _{\pm}
+e^{\, \mp 2i\mu \Psi }\, V^{\, (\mp )}\varphi _{\mp}\| ^2\geq
$$ $$
\geq \frac {c_1}4\, \sum\limits_{+,-}\| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_{*,\pm}
+\frac 32\, c_8^{\, \prime}\, \sum\limits_{+,-} \| \widetilde \varphi _{\pm}^{\,
(a_{j_{\pm}})}\| ^2_{*,\pm}-\frac 12\, c_8^{\, \prime}\, \sum\limits_{+,-}
\| \widetilde \varphi _{\mp}^{\, (a_1)}\| ^2_{*,\mp}\geq
$$ $$
\geq \frac {c_1}6\,
\sum\limits_{+,-}\| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_{*,\pm}+
c_8^{\, \prime}\, \sum\limits_{+,-} \| \widetilde \varphi
_{\pm}^{\, (a_{j_{\pm}})}\| ^2_{*,\pm}+\frac 12\, c_8^{\, \prime}\,
\sum\limits_{+,-} \| \varphi _{\pm}^{\, (a_1)}\| ^2_{*,\pm}\geq
$$ $$
\geq \frac {c_1}6\,
\sum\limits_{+,-}\| \varphi _{\pm}^{(a_{j_{\pm}})}\| ^2_*+
c_8^{\, \prime}\, \sum\limits_{+,-} \| \widetilde \varphi
_{\pm}^{\, (a_{j_{\pm}})}\| ^2_{*,\pm} \geq
\frac {c_1}6\,
\sum\limits_{+,-}\| \varphi _{\pm}^{(a_1)}\| ^2_*+
c_8^{\, \prime}\, \sum\limits_{+,-} \| \widetilde \varphi
_{\pm}^{\, (a_1)}\| ^2_{*,\pm}\, .
$$
Theorem 6.1 is proved.   \hfill $\square$

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text:
Admission Essay on Demographics of Japan
Type of paper: Admission essay
Topic: Demography, Population, Japan
The demographic characteristics of the Japan's population include ethnicity, population density, population health, education level, religious affiliations, economic status, and other aspects of the population (Lie 22).
As of March 2009, the population of Japan was 127,076,184, making it the world's tenth most highly populated country. Japans population density was 336 individuals per square kilometer (Hughes 45).
Japan's demography shows the country to be highly population and concerning the density of inhabitants Japan stands to be 30th country in the globe. There is very high life expectancy and low birth rate. Japan's demography also shows that during high growth of population occurred in19 th and 20 th centuries which has been lowered now due to very low birth rate and roughly no immigration.
Demographically the most crowded regions and cities in Japan are Kyushu, Honshu, Yokohama, Kyoto, Osaka, Kawasaki, Kobe, and many more. Tokyo, the capital city of Japan is the largest municipal city in Japan with highest number of population. Japan has a high number of urban populations. Japan also has the highest population of elderly people across the globe.
Japanese society is linguistically and ethnically very pure with 98% of the population speaking Japanese. The other 2% consists of immigrants and new Japanese population of mainly Chinese Koreans, and Loochoos, as well as the tiny native smaller group of the Ainu on Hokkaido. The Japan's government, as an official strategy, does not recognize full citizenship of many immigrants who have lived in Japan for many generations and decades (kinyau 89)
Lie, John. Demographics of Japan. London: W.W. Havard university press, 2001.
Hughes, Japan's population. 7th ed. New York: Addison Wesley, 2006.
Kinyau, James. Japan demographics. Kenya: U.G. Lomgman Publishers, 2008.
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text:
Clari Leads the Evolution and Growth of Revenue Operations
Clari, the Revenue Leader, spent the last decade creating the new software category of Revenue Operations, purpose-built to help companies inspect and control their revenue process. This fundamentally changed the way companies think about their go-to-market, and gave rise to a brand new job function with the same title: Revenue Operations. A study just published reveals that Revenue Operations is now the #1 fastest-growing job in the United States, cementing the role as a must-have for businesses everywhere. The best leverage for this job function goes beyond RevOps, and into a new strategic framework Clari is pioneering called Revenue Collaboration & Governance.
According to LinkedIn's 2023 Jobs on the Rise report, Head of Revenue Operations tops the list of the most in-demand jobs in the U.S, with over 500,000 open Revenue Operations jobs available across the country. Clari's own product usage data affirms this growth in Revenue Operations roles, as the company now counts more than 10,000 Revenue Operations professionals using the Clari platform to run revenue, growing at a rate of more than 100% year-over-year.
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Revenue Operations is a component of a much larger movement - Revenue Collaboration & Governance. This strategic framework is designed to help all revenue-critical employees work together more effectively and provide executive teams and boards with greater control over their revenue processes. By implementing Revenue Collaboration & Governance, companies can identify and stop all areas of revenue leak and achieve unprecedented levels of revenue precision.
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