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This article is cited in 1 scientific paper (total in 1 paper) On the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential L. I. Danilov Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, Izhevsk, Russia
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     § 1. IntroductionWe consider the Landau Hamiltonian
$$
\begin{equation}
\widehat H_B+V=\biggl(-i\,\frac{\partial}{\partial x_1}\biggr)^2 +\biggl(-i\,\frac{\partial}{\partial x_2}-Bx_1\biggr)^2+V
\end{equation}
\tag{1.1}
$$
on $L^2(\mathbb R^2)$ perturbed by a periodic electric potential $V\in L^2_{\mathrm{loc}}(\mathbb R^2; \mathbb R)$ with period lattice $\Lambda $. For the homogeneous magnetic field we assume that $B>0$. The coordinates in $\mathbb R^2$ are defined with respect to some orthonormal basis $e_1$, $e_2$. Let $E^1$ and $E^2$ be basis vectors of the lattice $\Lambda=\{N_1E^1+N_2E^2\colon N_1, N_2\in \mathbb Z\} $; let $E^l_j=(E^l,e_j)$, $l,j=1,2$ (the inner product and lengths of a vector from $\mathbb R^2$ are denoted by $(\cdot,\cdot)$ and $|\cdot |$, respectively). It can be assumed that $E^1_1>0$, $E^1_2=0$, and $E^2_2>0$. We let $K=\{\xi_1E^1+\xi_2E^2\colon 0\leqslant \xi_j<1,\ j=1,2\} $ denote the unit cell of the lattice $\Lambda $ (of area $v(K)=E^1_1E^2_2$, where $v(\,\cdot\,)$ is the Lebesgue measure on $\mathbb R^2$), and $\eta=(2\pi)^{-1}Bv(K)$ is the (normalized) magnetic flux of the magnetic field $B$ through the unit cell $K$. In what follows we assume that $\eta >0$ is a rational number. The operator (1.1) is a particular case of the two-dimensional magnetic Schrödinger operator
$$
\begin{equation}
\biggl(-i\,\frac{\partial}{\partial x_1}-A_1\biggr)^2 +\biggl(-i\, \frac{\partial}{\partial x_2}-A_2\biggr)^2+V,
\end{equation}
\tag{1.2}
$$
where $A\colon \mathbb R^2\to \mathbb R^2$ is the magnetic potential defining the magnetic field $B(x)=\partial A_2/\partial x_1- \partial A_1/\partial x_2$, $x\in \mathbb R^2$; $A_j=(A,e_j)$, $j=1,2$. If $A$ is a periodic magnetic potential with period lattice $\Lambda$, then the magnetic field $B(\,\cdot\,)$ is also periodic, and the magnetic flux is zero. The two-dimensional Schrödinger operator (1.2) with periodic potentials $V$ and $A$ with common period lattice has been studied extensively (see [1]–[5] and the references given there). In particular, in [4] and [5] it was shown that the spectrum of the periodic Schrödinger operator (1.2) is absolutely continuous if the functions $V$ and $|A|^2$ have bound zero in the sense of quadratic forms relative to the free Schrödinger operator $-\Delta=-{\partial}^2/{\partial x_1^2}-{\partial}^2/{\partial x_2^2}$ (in [4] and [5], a periodic variable metric and measure-derivative like potentials $V$ were also considered). For a survey on the absolute continuity of the spectrum of multidimensional periodic Schrödinger operators, see [6]–[8] (for recent results, see also [9] and [10]). The three-dimensional Landau operator with periodic electric potential was studied in [11] and [12]. Let $H^p(\mathbb R^2;\mathbb C)$ and $H^p_{\mathrm{loc}}(\mathbb R^2;\mathbb C)$, $p>0$, be Sobolev classes, and let $L^p_{\Lambda} (\mathbb R^2;{\mathfrak B})$, $p\in [1,+\infty ]$, and $C_{\Lambda}(\mathbb R^2;{\mathfrak B})$, $C^{\infty}_{\Lambda} (\mathbb R^2;{\mathfrak B})$, where ${\mathfrak B}=\mathbb R^m$ or ${\mathfrak B}=\mathbb C^m$, $m\in \mathbb N $, be the spaces of functions from $L^p_{\mathrm{loc}}(\mathbb R^2;{\mathfrak B})$, $C(\mathbb R^2;{\mathfrak B})$ and $C^{\infty}(\mathbb R^2;{\mathfrak B})$, respectively, which are periodic with period lattice $\Lambda $. For $V\in L^p_{\Lambda} (\mathbb R^2;\mathbb R)$ and $V\in C_{\Lambda}(\mathbb R^2;\mathbb R)$, we define the norms $\| V\|_{L^p(K)}\doteq \| V(\cdot |_K) \|_{L^p(K)}$ and $\| V\|_{C(K)}\doteq \| V(\,\cdot\,|_K)\|_{C(K)}$. The spectrum of the Landau Hamiltonian
$$
\begin{equation*}
\widehat H_B=\biggl(-i\,\frac{\partial}{\partial x_1}\biggr)^2 +\biggl(-i\,\frac {\partial}{\partial x_2}-Bx_1\biggr)^2
\end{equation*}
\notag
$$
consists of the eigenvalues $\lambda=(2m+1)B$, $m\in \mathbb Z_+=\mathbb N \cup \{0\} $, of infinite multiplicity (Landau levels). For a periodic point potential $V$, all Landau levels $\lambda=(2m+1)B$, $m\in \mathbb Z_+$, are also eigenvalues of the operator $\widehat H_B+V$ (of infinite multiplicity) if the period lattice $\Lambda $ is monatomic and $\mathbb Q \ni \eta >1$ (the spectrum also has an absolutely continuous component); see [13]. However, it is still unknown whether eigenvalues can lie in the spectrum of the operator (1.1) for $\eta \in \mathbb Q $ for nonconstant potentials $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$; see [14] and [15] (in [15] this problem was stated for potentials $V\in C_{\Lambda}(\mathbb R^2;\mathbb R)$). According to Theorem XIII.96 in [16], the electric potentials $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ are $-\Delta $-bounded with relative bound zero (and therefore bounded relative to the operators $\widehat H_B$). Hence the magnetic Floquet-Bloch theory applies. If $\eta \in \mathbb Q $, then (provided that the magnetic Floquet-Bloch theory is applicable) the spectrum of operator (1.1) has no singular component (see [13], [16], [17] and also [18], which is dedicated to this problem). Hence the absence of eigenvalues means the absolute continuity of the spectrum of operator (1.1). Two-dimensional Schrödinger operators with periodic electric potential $V$ and magnetic field $B$ (with common period lattice $\Lambda $) were considered in [19] in the case of a rational (normalized) magnetic flux. For basis vectors $E^1$ and $E^2$ of the lattice $\Lambda $, let $E^1_*,E^2_*\in \mathbb R^2$ be vectors such that $(E^{\mu},E^{\nu}_*)=\delta_{\mu \nu}$, $\mu,\nu=1,2$, where $\delta_{\mu \nu}$ is the Kronecker delta. The vectors $E^1_*$ and $E^2_*$ form a basis for the reciprocal lattice $\Lambda^*=\{ N_1E^1_*+N_2E^2_*\colon N_1,N_2\in \mathbb Z\} $ with unit cell $K^*=\{\xi_1E^1_*+\xi_2E^2_*\colon 0\leqslant \xi_j<1$, $ j=1,2\} $, where $v(K^*)=(v(K))^{-1}=(E^1_1E^2_2)^{-1}$. Let $W_Y$, $Y\in 2\pi \Lambda^*$, denote the Fourier coefficients of functions $W$ in $L^2_{\Lambda}(\mathbb R^2;\mathbb C)$: For functions $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ and vectors $Y\in 2\pi \Lambda^*$, we set
$$
\begin{equation*}
{\mathcal L}(\Lambda,B;V,Y)=|Y|\cdot \biggl(|V_Y|-\sum_{Y'\in 2\pi \Lambda^*\setminus \{ Y\}}\exp\biggl(-\frac{|Y-Y'|^2}{4B}\biggr)|V_{Y'}|\biggr).
\end{equation*}
\notag
$$
According to [20], the spectrum of the operator (1.1) is absolutely continuous if, given a potential $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ and a homogeneous magnetic field $B>0$ with magnetic flux $\eta \in \mathbb Q $, there exists a sequences of vectors $Y^{(j)}\in 2\pi \Lambda^*$, $j\in \mathbb N $, such that ${\mathcal L} (\Lambda,B;V,Y^{(j)})\to +\infty $ as $j\to +\infty $. It was shown in [15] that in the Banach space $(C_{\Lambda}(\mathbb R^2;\mathbb R),\| \cdot \|_{C(K)})$, there exists a dense $G_{\delta}$-set $\mathcal O$ such that, for each potential $V\in {\mathcal O}$ and any homogeneous magnetic field $B>0$ with magnetic flux $\eta \in \mathbb Q $, the spectrum of the operator (1.1) is absolutely continuous. A similar result for potentials $V$ in $L^p_{\Lambda}(\mathbb R^2;\mathbb R)$, $p>1$, was obtained in [21]. The case $p=2$ was also considered1 in [20]. It was shown in [22] that the spectrum of the operator (1.1) is absolutely continuous if $V$ is a nonconstant trigonometric polynomial (with period lattice $\Lambda $) and $\eta \in \{ Q^{-1}\colon Q\in \mathbb N\} $. According to the magnetic Floquet-Bloch theory, the operator $\widehat H_B+V$ for $\eta \in \mathbb N $ is unitarily equivalent to the direct integral of the discrete-spectrum ‘fibrewise’ operators $\widehat H_B(k)+V$ depending on the magnetic quasimomentum $k\in 2\pi K^*$. If $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$, then $\lambda $ is an eigenvalue of the operators $\widehat H_B(k)+V$ of the same multiplicity $\widetilde{\mathcal N} (\lambda)\in \mathbb N $ for almost all $k\in 2\pi K^*$ (see Lemma 2.3 below). If $\lambda $ is not an eigenvalue of the operator $\widehat H_B+V$, then $\lambda $ is also not an eigenvalue of the operators $\widehat H_B(k)+V$ (that is, $\widetilde{\mathcal N} (\lambda)=0$) for almost all $k\in 2\pi K^*$ (see Theorem XIII.85 in [16]). In the present paper we prove the following results. Theorem 1.1. If $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ and $\eta \in \mathbb N $, then $\widetilde{\mathcal N} (\lambda)\leqslant \eta $ for the operator (1.1) for all $\lambda \in \mathbb R $. Theorem 1.2. If $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)\setminus C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$ and $\eta \in \mathbb N $, then $\widetilde{\mathcal N} (\lambda)\leqslant \eta -1$ for the operator (1.1) for all $\lambda \in \mathbb R $. Corollary 1.1. For any periodic electric potential $V \in L^2_{\Lambda}(\mathbb R^2;\mathbb R)\setminus C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$ and any homogeneous magnetic field $B > 0$ with magnetic flux $\eta \in \{Q^{-1}\colon Q\in \mathbb N\} $ the spectrum of the operator (1.1) is absolutely continuous. Corollary 1.1 is used in the proof of the following theorem. Theorem 1.3. For any nonconstant periodic electric potential $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ and any homogeneous magnetic field $B>0$ with magnetic flux $\eta \in \{Q^{-1}\colon Q\in \mathbb N\} $, the numbers $\lambda \in \mathbb R \setminus \bigcup_{m=1}^{+\infty}\{(2m+1)B+V_0\} $ are not eigenvalues of operator (1.1). Theorem 1.3 is the main result of the paper. In § 2 we consider the magnetic Floquet-Bloch theory for operator (1.1) with $\eta \in \mathbb N $. In § 3 we study the ‘fibrewise’ operators $\widehat H_B(k)+V$ with complex values of the magnetic quasimomentum. At the end of that section we give a sketch of the proofs of Theorems 1.1, 1.2 and 1.3. Some results from [22] required in the proof (as well as some corollaries and generalizations thereof) are collected in § 4. In § 5 we prove Theorem 1.1, and in § 6 we verify Theorem 1.2. In § 7 we establish some auxiliary results on analytic functions with values in a Hilbert space. These results are used in § 8, in the proof of Theorem 1.3. § 2. Floquet-Bloch theoryFor an account of the magnetic Floquet-Bloch theory for the Landau Hamiltonian (under the Lorentz gauge condition) perturbed by a periodic electric potential, for ${\eta \in \mathbb N}$, see the first part of [13]. In this and the subsequent sections we use the notation and results from [22] and [20]. Let $\eta=PQ^{-1}$, where $P,Q\in \mathbb N $ are coprime integers. If $Q>1$, then the lattice $\Lambda $ can be replaced by the ‘enlarged’ lattice with basis vectors $QE^1$ and $E^2$. For notational convenience we denote, as before, the vector $QE^1$ by $E^1$ and the new lattice by $\Lambda $. So, for the new lattice, we have $\eta=P$ (and now for the proof of Corollary 1.1 and Theorem 1.3 it suffices to consider the case $\eta=1$). Let $\mathcal H_B$ be the space of functions $\varphi \in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb C)$ such that, for almost all $x\in \mathbb R^2$,
$$
\begin{equation*}
\varphi (x+E^{\mu})=\exp(iBE^{\mu}_1x_2)\varphi (x), \qquad \mu=1,2.
\end{equation*}
\notag
$$
Next, we set $\mathcal H^p_B\doteq \mathcal H_B \cap H^p_{\mathrm{loc}}(\mathbb R^2;\mathbb C)$, $p>0$, and $\mathcal H^{\infty}_B\doteq \mathcal H_B \cap C^{\infty}(\mathbb R^2;\mathbb C)$. Note that $\mathcal H^2_B\subset C(\mathbb R^2;\mathbb C)$. We equip the space $\mathcal H_B$ with the inner product
$$
\begin{equation*}
(\psi,\varphi)_B=\int_K{\overline {\psi}}\varphi\,dx, \qquad \psi,\varphi \in \mathcal H_B;
\end{equation*}
\notag
$$
the space $\mathcal H_B$ is a Hilbert space. Let $\| \cdot \|_B$ be the norm generated by the inner product. Next, let ${\widehat I}_B$ be the identity operator on $\mathcal H_B $, and $0_B$ be the identically zero function on $\mathcal H_B $. We set
$$
\begin{equation*}
\mathcal H '\doteq \int_{2\pi K^*}^{\oplus}\mathcal H_B\,\frac{dk}{(2\pi)^2v(K^*)}.
\end{equation*}
\notag
$$
Let
$$
\begin{equation*}
\widehat H_B(k)=\biggl(k_1-i\,\frac{\partial}{\partial x_1}\biggr)^2 +\biggl(k_2-i\,\frac {\partial}{\partial x_2}-Bx_1\biggr)^2, \qquad k\in \mathbb R^2,
\end{equation*}
\notag
$$
be self-adjoint operators on $\mathcal H_B$ with domain $D(\widehat H_B(k))=\mathcal H^2_B$. Then there exists a unitary operator $\widehat U$ from $L^2(\mathbb R^2;\mathbb C)$ onto the space $\mathcal H '$ such that
$$
\begin{equation*}
\widehat U\widehat H_B\widehat U^{-1} =\int_{2\pi K^*}^{\oplus}\widehat H_B(k)\,\frac {dk}{(2\pi)^2v(K^*)},
\end{equation*}
\notag
$$
where the $\widehat H_B(k)$, $k\in 2\pi K^*$, are ‘fibrewise’ operators in the direct integral. The vector $k\in 2\pi K^*$ is known as the magnetic quasimomentum. The operator $\widehat U$ is initially defined on the set of functions $\Phi \colon \mathbb R^2\to \mathbb C $ from the Schwartz space ${\mathcal S}(\mathbb R^2)$ as the operator associating with $\Phi $ the function
$$
\begin{equation*}
\begin{aligned} \, &2\pi K^*\times \mathbb R^2\ni (k,x)\mapsto \widehat U(\Phi)(k;x) = \sum_{\mu_1,\mu_2 \in\mathbb Z}\exp\biggl(-i\frac B2E^2_1E^2_2\mu_2(\mu_2-1)\biggr) \\ &\,\times \exp\bigl(-iB(\mu_1E^1_1+\mu_2E^2_1)x_2\bigr) \exp\bigl(-i(k,x+\mu_1E^1+\mu_2E^2)\bigr) \Phi (x+\mu_1E^1+\mu_2E^2) \end{aligned}
\end{equation*}
\notag
$$
such that $\widehat U(\Phi)(k;\,\cdot\,)\in \mathcal H^{\infty}_B$ for all $k\in 2\pi K^*$; afterwards, this operator extends to a unitary operator from $L^2(\mathbb R^2;\mathbb C)$ to $\mathcal H '$. The potential $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ is $\widehat H_B$-bounded with relative bound zero, and so this operator (as an operator of multiplication in $\mathcal H_B$) is $\widehat H_B(k)$-bounded with relative bound zero, for all $k\in \mathbb R^2$. Hence the $\widehat H_B(k)+V$, $k\in \mathbb R^2$, are self-adjoint operators in $\mathcal H_B$ (see Theorem X.12 in [23]), and $D(\widehat H_B(k)+V)=D(\widehat H_B(k))=\mathcal H^2_B$. In addition (see [13] and [22]),
$$
\begin{equation}
\widehat U(\widehat H_B+V)\widehat U^{-1} =\int_{2\pi K^*}^{\oplus}(\widehat H_B(k)+V)\,\frac{dk}{(2\pi)^2v(K^*)}
\end{equation}
\tag{2.1}
$$
and so $D(\widehat H_B+V)$ is the set of functions $\Phi \in H^2_{\mathrm {loc}}(\mathbb R^2;\mathbb C)$ such that $\widehat U(\Phi)(k;\,\cdot\,)\in \mathcal H^2_B$ for almost all $k\in 2\pi K^*$, and
$$
\begin{equation*}
\int_{2\pi K^*}\| \widehat H_B(k)\widehat U(\Phi)(k;\,\cdot\,)\|^2_B\,dk<+\infty.
\end{equation*}
\notag
$$
In what follows we will also consider the operators $\widehat H_B(k+i\varkappa)$ (and $\widehat H_B(k+i\varkappa) +V$) for complex values of the magnetic quasimomentum $k+i\varkappa \in \mathbb C^2$, $k,\varkappa \in \mathbb R^2$. The resolvent of the operator $\widehat H_B(k+i\varkappa)+V$, $k+i\varkappa \in \mathbb C^2$, is compact, and therefore the spectrum of this operator is discrete. Let $\lambda_j(k)$, $k\in \mathbb R^2$, $j\in \mathbb N $, be the eigenvalues of the operator $\widehat H_B(k)+V$ (counting multiplicities) arranged in increasing order. The functions $\mathbb R^2\ni k \mapsto \lambda_j(k)$ are continuous and analytic except at points of intersection (see [13] and [16]), and $\lambda_j(k+ \gamma)=\lambda_j(k)$ for all $\gamma \in 2\pi {\Lambda}^*$. The operator $\widehat H_B+V$ is unitary equivalent to the direct integral (2.1), and so $\lambda \in \mathbb R $ is an eigenvalue of the operator $\widehat H_B+V$ if and only if $\lambda $ is an eigenvalue of the operators $\widehat H_B(k)+V$ for all $k$ in some subset of positive Lebesgue measure of the cell $2\pi K^*$ (see Theorem XIII.85 in [16]). Hence, by the analytic Fredholm theorem $\lambda $ is an eigenvalue of each of the operators $\widehat H_B(k+i\varkappa)+V$ for all $k+i\varkappa \in \mathbb C^2$. Thus, we have the following result. Theorem 2.1 (see [13]). Let $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ (and $\eta=P\in \mathbb N $). Then $\lambda \in \mathbb R $ is an eigenvalue of the operator $\widehat H_B+V$ if and only if $\lambda $ is an eigenvalue of each operator $\widehat H_B(k+i\varkappa)+V$ for all $k+i\varkappa \in \mathbb C^2$. For each $k+i\varkappa \in \mathbb C^2$ the spectrum of the operator $\widehat H_B(k+i\varkappa)$ is discrete and coincides with the Landau levels; in addition, each eigenvalue $\lambda=(2m+1)B$, $m\in \mathbb Z_+$ is $P$-fold degenerate (see [13]). We let $\mathcal H (\lambda;k+i\varkappa)$ denote the subspace of eigenfunctions with eigenvalue $\lambda \in \mathbb C $ of the operator $\widehat H_B(k+i\varkappa)+V$ (if $\lambda $ is not an eigenvalue, then ${\mathcal H (\lambda;k+i\varkappa)=\{0_B\}}$); $\widetilde{\mathcal N} (\lambda;k+i\varkappa)\doteq \dim\mathcal H (\lambda;k+i\varkappa)$ is the dimension of $\mathcal H (\lambda;k+i\varkappa)$, and $\widetilde{\mathcal N} (\lambda; k+ \gamma + i\varkappa)=\widetilde{\mathcal N} (\lambda;k+i\varkappa)$ for all $\gamma \in 2\pi {\Lambda}^*$. Let $U^{(1)}_{r}(z)=\{z'\in \mathbb C \colon |z'-z|<r\} $, $z\in \mathbb C $, and $U^{(2)}_{r}(z)= \{z'\in \mathbb C^2\colon |z'_1-z_1|^2+|z_2'-z_2|^2<r^2\} $, $z\in \mathbb C^2$, $r>0$. Given a vector $e\in S^1\doteq \{x\in \mathbb R^2\colon |x|=1\} $, we denote by $e^{\perp}$ the vector obtained from $e$ by rotating it counterclockwise through an angle of ${\pi}/2$. Lemma 2.1. For each $\lambda \in \mathbb R $ there exists $\widetilde{\mathcal N} (\lambda)\in \mathbb Z_+$ such that $\widetilde{\mathcal N} (\lambda;k)\geqslant \widetilde{\mathcal N} (\lambda)$ for all $k\in \mathbb R^2$, and for each $e\in S^1$ there exists a discrete set2 ${\mathcal P}_{\perp}(\lambda;e)\subset \mathbb R $ such that for each $\tau^{\perp}\in \mathbb R \setminus {\mathcal P}_{\perp}(\lambda;e)$ and some discrete set ${\mathcal P}(\lambda;e,\tau^{\perp})\subset \mathbb R$, $\widetilde{\mathcal N} (\lambda; \tau^{\perp}e^{\perp} + \tau e)=\widetilde{\mathcal N} (\lambda)$ for all $\tau \in \mathbb R \setminus {\mathcal P}(\lambda;e,\tau^{\perp})$. Proof. Let $k\in \mathbb R^2$ and $e\in S^1$. If for some $\tau_0\in \mathbb R $ and $j_1,j_2\in \mathbb N $ we have $j_1\leqslant j_2$ and $\lambda_{j_1-1}(k+\tau_0e)<\lambda_{j_1}(k+\tau_0e) =\lambda_{j_2}(k+\tau_0e)<\lambda_{j_2+1}(k+\tau_0e)$ (the first inequality must hold for $j_1\geqslant 2$), then there exists a permutation $j\mapsto \sigma (j)$ of the integers $j_1, j_1+1, \dots, j_2$ such that, for all $j\in \{j_1, j_1+1, \dots, j_2\} $, the functions
$$
\begin{equation*}
\mathbb R \ni \tau \mapsto \widetilde {\lambda}_j(k+\tau e)= \begin{cases} \lambda_j(k+\tau e) &\text{for} \ \tau \leqslant \tau_0, \\ \lambda_{\sigma (j)}(k+\tau e) &\text{for} \ \tau > \tau_0, \end{cases}
\end{equation*}
\notag
$$
extend as analytic functions to a complex neighbourhood $U^{(1)}_{\varepsilon}(\tau_0)$, $\varepsilon >0$ of $\tau_0$ (in which they are eigenvalues of the operators $\widehat H_B(k+ze)+V$); see Theorem XII.13 in [16]. Hence (by the properties of analytic functions), for each $\lambda \in \mathbb R$ there exist a number $\widetilde{\mathcal N} (\lambda;k,e) \in \mathbb Z_+$ and a discrete set ${\mathcal P}'(\lambda;k,e) \subset \mathbb R $ such that $\widetilde{\mathcal N} (\lambda;k+\tau e)\geqslant \widetilde{\mathcal N} (\lambda;k,e)$ for all $\tau \in \mathbb R $ and $\widetilde{\mathcal N} (\lambda;k+\tau e)=\widetilde{\mathcal N} (\lambda;k,e)$ for $\tau \in \mathbb R \setminus {\mathcal P}'(\lambda;k,e)$. Next we set
$$
\begin{equation*}
\widetilde{\mathcal N}_-(\lambda;e)=\min_{k\in\mathbb R^2}\widetilde{\mathcal N}(\lambda;k,e), \qquad e\in S^1.
\end{equation*}
\notag
$$
For an arbitrary vector $\widetilde e \in S^1$ we choose a vector $k(\widetilde e)\in \mathbb R^2$ such that
$$
\begin{equation}
\widetilde{\mathcal N} (\lambda;k(\widetilde e),\widetilde e)=\widetilde{\mathcal N}_-(\lambda;\widetilde e).
\end{equation}
\tag{2.2}
$$
Let $e\in S^1$ be any vector not parallel to $\widetilde e$. Then the set of numbers $\tau^{\perp} \in \mathbb R $ such that
$$
\begin{equation*}
\tau^{\perp}e^{\perp}+\tau e\in \bigl\{k(\widetilde e)+\widetilde {\tau}\widetilde e\colon \widetilde {\tau}\in {\mathcal P}'(\lambda;k(\widetilde e),\widetilde e)\bigr\}
\end{equation*}
\notag
$$
for some $\tau \in \mathbb R $ is a discrete set ${\mathcal P}'(\lambda;k(\widetilde e),\widetilde e,e)\subset \mathbb R $. In addition, for each $\tau^{\perp}\in \mathbb R \setminus {\mathcal P}'(\lambda;k(\widetilde e), \widetilde e,e)$ there exists $\tau '\in \mathbb R $ such that $\tau^{\perp}e^{\perp}+\tau 'e= k(\widetilde e)+\widetilde {\tau}'\widetilde e$ for some $\widetilde {\tau}'\in \mathbb R \setminus {\mathcal P}'(\lambda;k(\widetilde e),\widetilde e)$. Therefore,
$$
\begin{equation}
\begin{aligned} \, \notag &\widetilde{\mathcal N}_-(\lambda;e)\leqslant \widetilde{\mathcal N} (\lambda;\tau^{\perp}e^{\perp},e)\leqslant \widetilde{\mathcal N} (\lambda;\tau^{\perp} e^{\perp}+\tau 'e) \\ &\qquad =\widetilde{\mathcal N} (\lambda;k(\widetilde e)+\widetilde {\tau}'\widetilde e)= \widetilde{\mathcal N} (\lambda;k(\widetilde e),\widetilde e)=\widetilde{\mathcal N}_-(\lambda;\widetilde e). \end{aligned}
\end{equation}
\tag{2.3}
$$
Hence $\widetilde{\mathcal N}_-(\lambda;e)\leqslant \widetilde{\mathcal N}_-(\lambda;\widetilde e)$. Swapping $\widetilde e$ and $e$ we obtain the inequality ${\widetilde{\mathcal N}_-(\lambda;\widetilde e)\leqslant \widetilde{\mathcal N}_-(\lambda;e)}$. Therefore, $\widetilde{\mathcal N}_- (\lambda;e)=\widetilde{\mathcal N}_-(\lambda;\widetilde e)$. Let $\widetilde{\mathcal N} (\lambda)$ be the common value of all numbers $\widetilde{\mathcal N}_- (\lambda;e)$, $e\in S^1$. For all $k\in \mathbb R^2$ and $e\in S^1$ we have
$$
\begin{equation*}
\widetilde{\mathcal N} (\lambda;k)\geqslant \widetilde{\mathcal N} (\lambda;k,e)\geqslant \widetilde{\mathcal N}_-(\lambda;e)=\widetilde{\mathcal N} (\lambda).
\end{equation*}
\notag
$$
If, given a vector $e\in S^1$, we choose any vector $\widetilde e\in S^1$ not parallel to it and a vector $k(\widetilde e)\in \mathbb R^2$ satisfying (2.2) (so that $\widetilde{\mathcal N} (\lambda;k(\widetilde e), \widetilde e)=\widetilde{\mathcal N} (\lambda)$), then, considering the discrete set ${\mathcal P}_{\perp}(\lambda;e)\doteq {\mathcal P}'(\lambda;k(\widetilde e),\widetilde e,e)$, we find from (2.3) that for all $\tau ^{\perp}\in \mathbb R \setminus {\mathcal P}_{\perp}(\lambda;e)$, there exists a discrete set ${\mathcal P} (\lambda;e,\tau^{\perp})\subset \mathbb R $ such that
$$
\begin{equation*}
\widetilde{\mathcal N} (\lambda)\leqslant \widetilde{\mathcal N} (\lambda;\tau^{\perp}e^{\perp}+\tau e)=\widetilde{\mathcal N} (\lambda;\tau^{\perp}e^{\perp}, e)\leqslant \widetilde{\mathcal N} (\lambda;k(\widetilde e),\widetilde e)=\widetilde{\mathcal N} (\lambda)
\end{equation*}
\notag
$$
for all $\tau \in \mathbb R \setminus {\mathcal P} (\lambda;e,\tau^{\perp})$. Hence $\widetilde{\mathcal N} (\lambda;\tau^{\perp}e^{\perp}+\tau e)=\widetilde{\mathcal N} (\lambda)$. This proves the lemma. Lemma 2.2. For all $\lambda \in \mathbb C $ and $k+i\varkappa \in \mathbb C^2$ there exists a positive number $\varepsilon $ such that, for all $k'+i{\varkappa}'\in U^{(2)}_{\varepsilon}(k+i\varkappa)$ Proof. Assume the contrary. Then there exist a sequence of vectors $k_j+i\varkappa_j\to k+i\varkappa $ as $j\to +\infty $ and functions $\Phi_j\in \mathcal H (\lambda;k_j+i\varkappa_j)$, $\| \Phi_j\|_B=1$, orthogonal to all functions $\Phi \in \mathcal H (\lambda;k+i\varkappa)$. Note that the operators $-i\,{\partial}/{\partial x_1}$ and ${-i\,{\partial}/{\partial x_2}-Bx_1}$ are $\widehat H_B(k)$-bounded with relative bound zero. Hence the sets $\{\Phi_j\colon j\in \mathbb N\}$, $\{-i\,{\partial \Phi_j}/{\partial x_1}\colon j\in \mathbb N\} $ and $\{(-i\,{\partial}/{\partial x_2}-Bx_1) \Phi_j\colon j\in \mathbb N\} $ are precompact in $\mathcal H_B$. Passing to a subsequence if required, we may assume that $\Phi_j\to \Phi_0\in \mathcal H_B$ and that the sequences $-i\,{\partial \Phi_j}/{\partial x_1}$ and $(-i\,{\partial}/{\partial x_2}-Bx_1) \Phi_j$ also converge as $j\to +\infty $. In addition, $\| \Phi_0\|_B=1$ and the function $\Phi_0$ is orthogonal to all $\Phi \in \mathcal H (\lambda;k+i\varkappa)$. On the other hand, the sequence $(\widehat H_B(k+i\varkappa)+V-\lambda) \Phi_j=(\widehat H_B(k+i\varkappa) -\widehat H_B(k_j+i\varkappa_j)) \Phi_j$, $j\in \mathbb N $, converges to $0_B$ as $j\to +\infty $, and therefore, since the operator $\widehat H_B(k+i\varkappa)+V-\lambda $ is closed, it follows that $\Phi_0\in \mathcal H (\lambda;k+i\varkappa)$. Hence $(\Phi_0,\Phi_0)_B=0$. This contradiction proves the lemma. The next result is a direct consequence of Lemmas 2.1 and 2.2. Lemma 2.3. For each $\lambda \in \mathbb R $ there exists $\widetilde{\mathcal N} (\lambda)\in \mathbb Z_+$ such that $\widetilde{\mathcal N}(\lambda;k)\geqslant \widetilde{\mathcal N} (\lambda)$ for all $k\in \mathbb R^2$, the set $\mathbb M (\lambda)\doteq \{k\in \mathbb R^2\colon \widetilde{\mathcal N} (\lambda;k)=\widetilde{\mathcal N} (\lambda)\} $ is open and $v(\mathbb R^2\setminus \mathbb M (\lambda))=0$. § 3. Properties of the operators $\widehat H_B(k+i\varkappa)$Let $C^{\omega}({\mathcal O};\mathbb C)$, where $\mathcal O$ is a domain in $\mathbb R^2$, be the set of functions $f\colon {\mathcal O}\to \mathbb C $ such that both $\operatorname{Re} f$ and $\operatorname{Im} f$ are real analytic. In a neighbourhood of each point $x^{(0)}\in {\mathcal O}\subseteq \mathbb R^2$ each function $f\in C^{\omega}({\mathcal O};\mathbb C)$ expands in an absolutely and uniformly convergent series
$$
\begin{equation*}
\sum_{m, n=0}^{+\infty}f_{m,n}(x^{(0)})(x_1-x^{(0)}_1)^m(x_2-x^{(0)}_2)^n, \qquad f_{m,n}(x^{(0)})\in \mathbb C.
\end{equation*}
\notag
$$
Given a vector $k\in \mathbb R^2$, we let $\mathcal H^{(m)}_B(k)$, $m\in \mathbb Z_+$, denote the subspace of eigenfunctions with eigenvalue $\lambda = (2m + 1)B$ of the operators $\widehat H_B(k)$. Note that $\dim\mathcal H^{(m)}_B(k)=P$ and $\mathcal H^{(m)}_B(k)\subset C^{\omega}(\mathbb R^2;\mathbb C)$ (see [13] and also [22]). Consider the operators
$$
\begin{equation*}
\widehat Z_{\mp}(k)=\biggl(k_1-i\,\frac{\partial}{\partial x_1}\biggr) \pm i\biggl(k_2-i\,\frac{\partial}{\partial x_2}-Bx_1\biggr)
\end{equation*}
\notag
$$
on $\mathcal H_B$, where $D(\widehat Z_{\mp}(k))=\mathcal H^1_B$ and $\widehat Z_{\mp}^*(k)=\widehat Z_{\pm}(k)$. We have
$$
\begin{equation*}
\widehat H_B(k)=\widehat Z_+(k)\widehat Z_-(k)+B=\widehat Z_-(k)\widehat Z_+(k)-B.
\end{equation*}
\notag
$$
Given functions $\psi (k)\in \mathcal H^{(0)}_B(k)$, we set
$$
\begin{equation*}
\psi^{(m)}(k)=\frac{(2B)^{-m/2}}{\sqrt {m!}}\widehat Z_+^m(k)\psi (k), \qquad m\in \mathbb Z_+.
\end{equation*}
\notag
$$
Note that
$$
\begin{equation}
\begin{gathered} \, \widehat Z_+(k)\psi^{(m)}(k)=\sqrt {2B(m+1)}\, \psi^{(m+1)}(k), \qquad m\in \mathbb Z_+, \\ \widehat Z_-(k)\psi^{(m)}(k)=\sqrt {2Bm}\, \psi^{(m-1)}(k), \qquad m\in \mathbb N, \end{gathered}
\end{equation}
\tag{3.1}
$$
and $\mathcal H^{(0)}_B(k)=\{\Phi \in \mathcal H^1_B\colon \widehat Z_-(k)\Phi=0_B\} $. If $\psi_j(k)=\psi_j^{(0)}(k)$, $j=1,\dots,P$, is an orthonormal basis of the space $\mathcal H^{(0)}_B(k)$, then $\psi_j^{(m)}(k)$, $j=1,\dots,P$, is an orthonormal basis of the space $\mathcal H^{(m)}_B(k)$, $m\in \mathbb N $, and $\psi_j^{(m)}(k)$, $j=1,\dots,P$, $m\in \mathbb Z_+$, is an orthonormal basis of $\mathcal H_B$. Let $\widehat P^{(m)}(k)$, $m\in \mathbb Z_+$, be an orthogonal projection of $\mathcal H_B$ onto the subspace $\mathcal H^{(m)}_B(k)$. For all $n\in \mathbb N $ and $k\in \mathbb R^2$ we have
$$
\begin{equation*}
\mathcal H^n_B=\biggl\{\Phi \in \mathcal H_B\colon \sum_{m=0}^{+\infty}m^n\| \widehat P^{(m)}(k)\Phi \|^2_B <+\infty \biggr\}.
\end{equation*}
\notag
$$
Given vectors $k,k'\in \mathbb R^2$ in $\mathcal H_B$, consider the unitary operator
$$
\begin{equation*}
\mathcal H_B \ni \Phi \mapsto \widehat U(k,k')\Phi=\exp(-i(k_1'-k_1)x_1) \Phi \biggl(\cdot -\frac{k_2'-k_2}B\, e_1+\frac{k_1'-k_1}B\, e_2\biggr).
\end{equation*}
\notag
$$
The following result is verified directly. Lemma 3.1. The sets $\mathcal H^n_B$, $n\in \mathbb Z_+$, are invariant under the operator $\widehat U(k,k')$, and By Lemma 3.1, the operators $\widehat U(k,k')$ map the subspaces $\mathcal H^{(m)}_B(k)$ onto the subspaces $\mathcal H^{(m)}_B(k')$, $m\in \mathbb Z_+$. The domains $D(\exp(z\widehat Z_{\mp}(k)))$ of the operators $\exp(z\widehat Z_{\mp}(k))$, $z\in \mathbb C $, consist of the functions $\Phi \in \mathcal H^{\infty}_B$ with convergent series
$$
\begin{equation*}
\exp(z\widehat Z_{\mp}(k))\Phi \doteq \sum_{m=0}^{+\infty}\frac{z^m}{m!}\widehat Z_{\mp}^m (k)\Phi.
\end{equation*}
\notag
$$
Let $\mathcal H_B(+0;k)$ be the set of functions $\Phi \in \mathcal H_B$ such that, for some $\alpha >0$,
$$
\begin{equation*}
\sup_{m\in\mathbb Z_+} e^{m\alpha}\| \widehat P^{(m)}(k)\Phi \|_B<+\infty.
\end{equation*}
\notag
$$
Lemma 3.2 (see [22]). For all $k'\in \mathbb R^2$ and $z\in \mathbb C $, $\mathcal H_B(+0;k)\subset D(\exp(z\widehat Z_{\mp}(k')))$, and the set $\mathcal H_B(+0;k)$ is invariant under the operators $\widehat Z_{\mp}(k')$ and $\exp(z\widehat Z_{\mp}(k'))$, $k'\in \mathbb R^2$, and under multiplication by the functions $\exp(i(Y,x))$, $Y\in 2\pi \Lambda^*$. For these operators acting on functions in $\mathcal H_B(+0;k)$ (for all $z, z',z''\in \mathbb C $ and $Y\in 2\pi \Lambda^*$), the following equations are fulfilled: By Lemma 3.2, for all $z\in \mathbb C $ we can introduce the operators
$$
\begin{equation*}
\widehat U_z(k)=\exp\biggl(-\frac{|z|^2}{4B}\biggr) \exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr) \exp\biggl(\frac{\overline z}{2B}\widehat Z_-(k)\biggr)
\end{equation*}
\notag
$$
on $\mathcal H_B(+0;k)$. They extend from $\mathcal H_B(+0;k)$ to the space $\mathcal H_B$ as unitary operators. In addition, we have $\widehat U_z(k)\mathcal H^n_B=\mathcal H^n_B$, $n\in \mathbb N $, and if, for a vector $k'\in \mathbb R^2$, we set $z=k_1'-k_1+i(k_2'-k_2)$, then the operators $\widehat U_z(k)$ satisfy analogues of equalities (3.2) and (3.3) (for $\widehat U(k,k')$ replaced by $\widehat U_z(k)$). Hence the operators $\widehat U_z(k)$ also map the subspaces $\mathcal H^{(m)}_B(k)$ onto the subspaces $\mathcal H^{(m)}_B(k')$, $m\in \mathbb Z_+$, and for all $k'\in \mathbb R^2$ the functions
$$
\begin{equation*}
\begin{gathered} \, \widetilde {\psi}^{(m)}_j(k')=\frac{(2B)^{-m/2}}{\sqrt {m!}} \exp\biggl(-\frac{|z|^2}{4B}\biggr)(\widehat Z_+(k)+\overline z)^m \exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi_j(k), \\ j=1,\dots,P, \end{gathered}
\end{equation*}
\notag
$$
form an orthonormal basis of $\mathcal H^{(m)}_B(k')$, $m\in \mathbb Z_+$. In particular, the following result holds. Lemma 3.3. If $\psi (k)\in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $, $k\in \mathbb R^2$, then for each $k'\in \mathbb R^2$ the function $\widetilde \psi (k')\doteq \exp(-(z/(2B))\widehat Z_+(k))\psi (k)$, where $z=k_1'-k_1+i(k_2'-k_2)$, belongs to $\mathcal H^{(0)}_B(k')\setminus \{0_B\} $. That the functions $\mathbb C \ni z\mapsto \widetilde \psi (k')= \exp(-(z/(2B))\widehat Z_+(k))\psi (k)$, $k\in \mathbb R^2$, are analytic will be important in our further analysis. Given a vector $k\in \mathbb R^2$ and a function $\psi (k)\in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $, we can also define the function $\psi (k')\doteq \exp(-|z|^2/(4B))\widetilde \psi (k')$, where $z=k_1'-k_1+i(k_2'-k_2)$; $\| \psi (k')\|_B=\| \psi (k)\|_B$. The next result follows from equalities (3.4). Lemma 3.4 (see [22]). For all vectors $k\in\mathbb R^2$ and $Y\in 2\pi \Lambda^*$, there exists a unitary operator $\widehat U^{(Y)}(k)\colon \mathcal H^{(0)}_B(k)\to \mathcal H^{(0)}_B(k)$ such that, for all $\psi\in\mathcal H^{(0)}_B(k)$, For $k\in \mathbb R^2$ consider the operators
$$
\begin{equation*}
\widehat H_{\mp}(k;\zeta)\doteq \widehat Z_+(k)\widehat Z_-(k)+\zeta \widehat Z_{\mp}(k)+B+V, \qquad \zeta \in \mathbb C.
\end{equation*}
\notag
$$
Note that $D(\widehat H_{\mp}(k;\zeta))=\mathcal H^2_B$ and $\widehat H_{\mp}^*(k;\zeta)=\widehat H_{\pm}(k; \overline {\zeta})$. We have
$$
\begin{equation}
\widehat H_{\mp}(k;\zeta)=\widehat H_B\biggl(k+\frac{\zeta}2\, e_1\pm i \frac\zeta2\, e_2\biggr)+V.
\end{equation}
\tag{3.5}
$$
If $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$, then it follows from (3.5) and Theorem 2.1 that the operators $\widehat H_{\mp}(k;\zeta)$ also have the eigenvalue $\lambda $ for all $k\in \mathbb R^2$ and $\zeta \in \mathbb C $. For all $\zeta \in \mathbb C $ the operators $\widehat Z_+(k)\widehat Z_-(k)+\zeta \widehat Z_{\mp}(k)$ have the eigenvalues $\lambda=2Bm$, $m\in \mathbb Z_+$. In addition, all these eigenvalues are $P$-fold degenerate, and the functions $\exp(\pm (\zeta /(2B))\widehat Z_{\mp}(k)) \psi^{(m)}_j(k)$, $j=1,\dots,P$, where the functions $\psi^{(m)}_j(k)$, $j=1,\dots,P$, form an orthonormal basis of $\mathcal H^{(m)}_B(k)$, and are linearly independent eigenfunctions of the operators $\widehat Z_+(k)\widehat Z_-(k)+\zeta \widehat Z_{\mp}(k)$ corresponding to the eigenvalues $\lambda=2Bm$. The proofs of Theorems 1.1–1.3 depend substantially on Theorems 3.1 and 3.2 that follow. Theorem 3.1 (see [20]). For all vectors $k\in\mathbb R^2$, all functions $\Phi \in\mathcal H^2_B$ with ${\widehat P^{(0)}(k)\Phi=0_B}$ and all $\zeta \in \mathbb C $, the following estimate3 holds: Lemma 3.5. For all $k\in \mathbb R^2$, all $m\in \mathbb Z_+$ and all functions $\Phi \in \mathcal H^{(m)}_B(k)$, where $C_1=C_1(\Lambda,B)>0$. According to [20], there exists a vector $k\in 2\pi K^*$ such that estimate (3.6) holds for all $m\in \mathbb Z_+$ and $\Phi \in \mathcal H^{(m)}_B(k)$. For all $k'\in \mathbb R^2$ and $\Phi \in \mathcal H^{(m)}_B(k)$ we have the inclusion $\widehat U(k,k')\Phi \in \mathcal H^{(m)}_B(k')$ and, in addition, $\| \widehat U(k,k')\Phi \|_B=\| \Phi \|_B$, $\| \widehat U(k,k')\Phi \| _{L^{\infty}(K)}=\| \Phi \|_{L^{\infty}(K)}$. This verifies estimate (3.6) for all $k\in \mathbb R^2$. Theorem 3.2. Let $W\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$. Then for each $\varepsilon >0$ there exists $C_2=C_2(\Lambda, B,W;\varepsilon)>0$ such that for all $\zeta \in \mathbb C $ satisfying $|\zeta |\geqslant C_2$, all $k\in \mathbb R^2$ and all functions $\Phi \in \mathcal H^2_B$ such that $\widehat P^{(0)}(k)\Phi=0_B$, The proof of Theorem 3.2 was presented in [20] for fixed $k\in 2\pi K^*$ satisfying estimate (3.6) in Lemma 3.5 (so that $C_2$ depends on $C_1$). This estimate holds for all $k\in \mathbb R^2$, and therefore the conclusion of Theorem 3.2 also holds for all $k\in \mathbb R^2$. For $\lambda \in \mathbb R $ and $k\in \mathbb R^2$ we set
$$
\begin{equation*}
\mathcal H_{\mp}(\lambda,k;\zeta) =\operatorname{Ker}(\widehat H_{\mp}(k;\zeta)-\lambda) =\mathcal H \biggl(\lambda;k+\frac\zeta2\, e_1 \pm i\frac\zeta2\, e_2\biggr), \qquad \zeta \in \mathbb C.
\end{equation*}
\notag
$$
First we present a sketch of the proofs of Theorems 1.1–1.3. For a fixed vector $k\in \mathbb R^2$ we assume that $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ and the operator $\widehat H_B+V$ has an eigenvalue $\lambda \in \mathbb R $. By Lemma 4.1 there exists a set ${\mathfrak M}_-(\lambda;k)\subset \mathbb C $ such that $\mathbb C \setminus {\mathfrak M}_-(\lambda;k)$ is a discrete set, and for all $\zeta \in {\mathfrak M}_-(\lambda;k)$ the subspaces $\mathcal H_-(\lambda,k;\zeta)$ have the same dimension $\dim\mathcal H_-(\lambda,k;\zeta)=\mathcal N [\lambda;k]$. We have $\dim\mathcal H^{(0)}_B(k)=P$ and also $\| \Phi -\widehat P^{(0)}(k)\Phi \|_B\to 0$ as $\zeta \to \infty $ uniformly for all $\Phi \in \mathcal H_-(\lambda,k;\zeta)$ such that $\| \Phi \|_B=1$ (see Theorems 3.1 and 3.2). Hence $\mathcal N [\lambda;k]\leqslant P=\eta $ (see Theorem 4.1), and by Lemma 5.2 we also have the estimate $\widetilde {\mathcal N}(\lambda)\leqslant P=\eta $, which proves Theorem 1.1. In the proof of Theorem 1.2 we assume that $\widetilde {\mathcal N}(\lambda)=P=\eta $. In this case $\mathcal N [\lambda;k]= \widetilde {\mathcal N}(\lambda)=P$ (for all $k\in \mathbb R^2$; see Lemma 5.3). From Theorem 4.4 and inequalities (4.1) it follows that for any function $\psi \in \mathcal H^{(0)}_B(k) \setminus \{0_B\}$ one can (uniquely) define a meromorphic function $\zeta \mapsto \Phi (k,\psi;\zeta)\in \mathcal H_-(\lambda,k;\zeta)\subset \mathcal H_B$ (see (6.2)) with a finite number of poles $\zeta_j(k,\psi) \in \mathbb C $ of finite orders $\mu_j(k,\psi)$, $j=1,\dots,n(k,\psi)$, such that $\widehat P^{(0)}(k) \Phi (k,\psi;\zeta) = \psi $ outside these poles. In addition, we have $|\zeta_j(k,\psi)|\leqslant C_2'$ by Lemma 6.2 (where the positive number $C_2' $ is independent of $k$ and $\psi $). Multiplying the function $\Phi (k,\psi;\zeta)$ by the polynomial ${\mathfrak P}(k,\psi;\zeta)$ (see (6.3)) we obtain a function (6.4), which is also a polynomial of degree $m(k,\psi)=\sum_{j=1}^{n(k,\psi)}\mu_j(k,\psi)$ with coefficients ${\mathcal F}_{\nu}(k,\psi)\in \mathcal H^2_B$, $\nu=1,\dots,m(k,\psi)$, satisfying conditions (6.5). But at the end of § 6 we show that these conditions can only be satisfied (for all $k\in \mathbb R^2$ and $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\}$) for $V\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$. This proves Theorem 1.2. In the proof of Theorem 1.3 we assume that $\widetilde {\mathcal N}(\lambda)=P=1$ (in this case, for the functions $\Phi (k,\psi;\zeta)$ the numbers $\zeta_j(k,\psi)=\zeta_j(k)$, $\mu_j(k,\psi)=\mu_j(k)$, $n(k,\psi)=n(k)$ and $m(k,\psi)=m(k)$ are independent of $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $). By Theorem 1.2 it can be assumed that $V\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$. Next, let ${\mathfrak B}_j(k,\psi)$ (see (8.2)), $j\in \mathbb N $, be obtained by expanding the function $\Phi (k,\psi;\zeta)$ in (8.1) in a series in the powers $\zeta^{-j}$. The functions ${\mathfrak B}_j(k,\psi)$, $j=1,\dots, m(k)$, are linearly independent, and the functions ${\mathfrak B}_j(k,\psi)$, for $j>m(k)$, are linear combinations of them. An explicit expression of the function ${\mathfrak B}_{m(k)+1}(k,\psi)$ in terms of the functions ${\mathfrak B}_j(k,\psi)$, $j=1,\dots,m(k)$, is obtained from (8.3). With numbers $z\in \mathbb C $ (for a fixed vector $k\in \mathbb R^2$) we associate the vectors $k'=k'(z)\in \mathbb R^2$ such that $z=k'_1-k_1+i(k'_2-k_2)$. Next we consider functions depending on $z\in \mathbb C $. The functions $z\mapsto {\mathfrak B}_j(k',\widetilde {\psi}(k'))$ are not analytic; in place of them we consider the analytic functions $z\mapsto {\mathfrak F}_j(k,\psi;z)\in \mathcal H_B$ (see (8.9)). For all $z\in \mathbb C $ the functions ${\mathfrak F}_j(k,\psi;z)$, $j=1,\dots, m(k')$, are linearly independent, and the functions ${\mathfrak F}_j(k,\psi;z)$, for $j>m(k')$, are linear combinations of them. For all $z\in \mathbb C $ the dimension of the linear span of the functions ${\mathfrak F}_j(k,\psi;z)$, $j\in \mathbb N $, is finite, and so by Lemma 7.2 there exist a discrete set $\widetilde M(k)\subset \mathbb C $ and a number $\widetilde m(k)\in \mathbb N $ such that $m(k')=\widetilde m(k)$ for all $z\in \mathbb C \setminus \widetilde M(k)$. The numbers $\widetilde m(k)=\widetilde m\in \mathbb N $ are independent of $k$ (see Lemma 8.10). By Lemma 7.3, for $z\in \mathbb C \setminus \widetilde M(k)$ the functions ${\mathfrak F}_{\widetilde m+1}(k,\psi;z)\in \mathcal H_B$ are linear combinations of the functions ${\mathfrak F}_j(k,\psi;z)$, $j=1,\dots,\widetilde m$, with coefficients ${\mathcal C}_j(k;z)\in \mathbb C $. In addition, the functions $z\mapsto {\mathcal C}_j(k;z)$ are meromorphic (or analytic). The numbers ${\mathcal C}_{\widetilde m}(k;z)$ satisfy equality (8.14), which (in combination with the estimates $|\zeta_j(k')|\leqslant C_2'$) implies that the function $z\mapsto {\mathcal C}_{\widetilde m}(k;z)$ is locally bounded in a neighbourhood of each point $z_0\in \mathbb C $, and therefore is analytic. By Theorem 8.1 this is possible only for $\lambda=(2\widetilde m+1)B+V_0$. This proves Theorem 1.3. § 4. Some results from [22] and generalizations thereofIn this section we recall some results from [22] required in the proof of Theorems 1.1, 1.2 and 1.3. Let $\lambda $ be an eigenvalue of the operator $\widehat H_B+V$, let $k\in \mathbb R^2$, and let $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$. Lemma 4.1 (see [22]). There exist a number $\mathcal N [\lambda;k]\in \mathbb N $ and a set ${\mathfrak M}_{\mp}(\lambda;k)\subset \mathbb C$, where $\mathbb C \setminus {\mathfrak M}_{\mp}(\lambda;k)$ is a discrete set, such that $\dim\mathcal H_{\mp}(\lambda,k;\zeta)=\mathcal N [\lambda;k]$ for all $\zeta \in {\mathfrak M}_{\mp}(\lambda;k)$ and $\dim\mathcal H_{\mp}(\lambda,k;\zeta)>\mathcal N [\lambda;k]$ for all $\zeta \in \mathbb C \setminus {\mathfrak M}_{\mp}(\lambda;k)$. In addition, ${\mathfrak M}_-(\lambda;k)=\{z\in \mathbb C \colon \overline z\in {\mathfrak M}_+(\lambda;k)\} $. Proof. From Lemma 3.5 and Theorems 3.1 and 3.2, for all $\zeta \in \mathbb C $ such that $|\zeta |\geqslant C_2'=C_2(\Lambda,B,B+V-\lambda;\frac12)>0$ and for all $\Phi \in \mathcal H_-(\lambda,k;\zeta)\setminus \{0_B\} $ we have
$$
\begin{equation}
\begin{aligned} \, \notag & \| B+V-\lambda \|_{L^2(K)}\| \widehat P^{(0)}(k)\Phi \|_B\geqslant C_1^{-1} \| B+V-\lambda \|_{L^2(K)}\| \widehat P^{(0)}(k)\Phi \|_{L^{\infty}(K)} \\ \notag &\qquad \geqslant C_1^{-1}\bigl\| (B+V-\lambda)\widehat P^{(0)}(k)\Phi \bigr\|_B= C_1^{-1}\bigl\| (\widehat H_-(k;\zeta)-\lambda)(\widehat I_B-\widehat P^{(0)}(k))\Phi\bigr \|_B \\ \notag &\qquad \geqslant \frac12 C_1^{-1}\bigl\| (\widehat Z_+(k)\widehat Z_-(k)+\zeta \widehat Z_-(k))(\widehat I_B-\widehat P^{(0)}(k))\Phi \bigr\|_B \\ &\qquad \geqslant \frac12C_1^{-1}\sqrt {\frac B2}\, |\zeta |\, \bigl\| (\widehat I_B-\widehat P^{(0)}(k))\Phi \bigr\|_B. \end{aligned}
\end{equation}
\tag{4.1}
$$
Hence $\widehat P^{(0)}(k)\Phi \neq 0_B$ (for all $\zeta \in \mathbb C $ such that $|\zeta |\geqslant C_2'$). As a result, $\dim \mathcal H_-(\lambda,k;\zeta)\leqslant \dim \mathcal H^{(0)}_B(k)=P$. But $\dim \mathcal H_-(\lambda,k; \zeta)=\mathcal N [\lambda;k]$ for all ${\zeta \in {\mathfrak M}_- (\lambda;k)}$ and $\mathbb C \setminus {\mathfrak M}_-(\lambda;k)$ is a discrete set. Therefore, $\mathcal N [\lambda;k]\leqslant P$. This proves Theorem 4.1. For ${\mathcal A} \subseteq \mathcal H_B$ we set ${\mathcal A}^{\perp}\doteq\{ \Phi \in \mathcal H_B\colon (\Phi,\chi)_B=0\text{ for all }\chi\in{\mathcal A}\} $. Theorem 4.2 (see [22]). For each $\zeta_0\in {\mathfrak M}_-(\lambda;k)$ there exists a number $r>0$ such that $U^{(1)}_{r}(\zeta_0) \subset {\mathfrak M}_-(\lambda;k)$, and for any function $\Phi_{\zeta_0}\in \mathcal H_-(\lambda,k;\zeta_0)$, there exists a (unique) analytic function $U^{(1)}_{r}(\zeta_0)\ni \zeta \mapsto \Phi (\zeta_0;\zeta)\,{\in}\, \mathcal H_B$ such that ${\Phi (\zeta_0;\zeta_0)\,{=}\,\Phi_{\zeta_0}}$, $\Phi (\zeta_0;\zeta)\in \mathcal H_-(\lambda,k;\zeta)$ and $\Phi (\zeta_0;\zeta)-\Phi_{\zeta_0}\in (\mathcal H_-(\lambda,k;\zeta_0))^{\perp}$ for all $\zeta \in U^{(1)}_{r}(\zeta_0)$. Lemma 4.2 (see [22]). For each $\zeta_0\in \mathbb C \setminus {\mathfrak M}_-(\lambda;k)$ there exist a number $r>0$ (such that $U^{(1)}_{r}(\zeta_0)\setminus \{\zeta_0\} \,{\subset}\, {\mathfrak M}_-(\lambda;k)$) and analytic functions ${U^{(1)}_{r}(\zeta_0)\,{\ni}\, \zeta \,{\mapsto}\, \Phi_{\zeta_0,j}(\zeta)\!\in\! \mathcal H_B}$, $j=1,\dots, \mathcal N [\lambda;k]$, such that for all $\zeta \in U^{(1)}_{r}(\zeta_0)\setminus \{\zeta_0\} $ the functions $\Phi_{\zeta_0,j}(\zeta)$, $j=1,\dots, \mathcal N [\lambda;k]$, form a (not necessarily orthonormal) basis of $\mathcal H_-(\lambda,k;\zeta)$. Let $\mathcal L$ be an $\mathcal N [\lambda;k]$-dimensional subspace of $\mathcal H_B$. To define a continuous function ${\mathfrak M}_-(\lambda;k)\ni \zeta \mapsto {\Delta}^-_{\mathcal L}(\zeta)\in [0,1]$, for each $\zeta \in {\mathfrak M}_-(\lambda;k)$ in $\mathcal H_-(\lambda,k;\zeta)$ we choose an orthonormal basis $E^-_{\zeta,\mu}$, $\mu=1,\dots, \mathcal N [\lambda;k]$, and consider some orthonormal basis $\Phi_{\nu}$, $\nu=1,\dots, \mathcal N [\lambda;k]$, of $\mathcal L$. We set
$$
\begin{equation*}
{\Delta}^-_{\mathcal L}(\zeta)=\bigl| \det((E^-_{\zeta,\mu},\Phi_{\nu})_B)_{\mu,\nu=1}^{\mathcal N [\lambda;k]} \bigr|.
\end{equation*}
\notag
$$
Note that ${\Delta}^-_{\mathcal L}(\zeta)$ is independent of the choice of the orthonormal bases $E^-_{\zeta,\mu}$ and $\Phi_{\nu}$. We have ${\Delta}^-_{\mathcal L}(\zeta)=0$ if and only if there exists a nonzero function ${\Phi \in \mathcal H_-(\lambda,k;\zeta) \cap {\mathcal L}^{\perp}}$. For each $\mathcal N [\lambda;k]$-dimensional subspace $\mathcal L$ we either have ${\Delta}^-_{\mathcal L}(\zeta)\equiv 0$, or ${\mathcal M}_{\mathcal L}(\lambda;k)\doteq \{\zeta \in {\mathfrak M}_-(\lambda;k)\colon {\Delta}^-_{\mathcal L} (\zeta)=0\} $ is a discrete set in $\mathbb C $ (see [22], where the subspaces ${\mathcal L}=\mathcal H_-(\lambda,k;\zeta_0)$, $\zeta_0\in {\mathfrak M}_-(\lambda;k)$ were considered). Let $\widehat P_{\mathcal L}$ be the orthogonal projection of $\mathcal H_B$ onto the subspace $\mathcal L$. We let ${\mathbb L}(\lambda;k)$ denote the set of $\mathcal N [\lambda;k]$-dimensional subspaces ${\mathcal L} \in \mathcal H_B$ such that the ${\mathcal M}_{\mathcal L}(\lambda;k)$ are discrete sets in $\mathbb C $ (in this case the sets $(\mathbb C \setminus {\mathfrak M}_-(\lambda;k)) \cup {\mathcal M}_{\mathcal L}(\lambda;k)$ are also discrete). For each ${\mathcal L} \in {\mathbb L}(\lambda;k)$ and any number $\zeta \in {\mathfrak M}_-(\lambda;k)\setminus {\mathcal M}_{\mathcal L}(\lambda;k)$ the restriction of the orthogonal projection $\widehat P_{\mathcal L}$ to the subspace $\mathcal H_-(\lambda,k;\zeta)$ is a bijective linear map onto the subspace $\mathcal L$. Theorem 4.3. Let ${\mathcal L}\in {\mathbb L}(\lambda;k)$. Then for any $\psi \in {\mathcal L}$ there exists a (unique) analytic function such that $\widehat P_{\mathcal L}\Phi (\psi;\zeta)=\psi $ and $\| \Phi (\psi;\zeta)\|_B\leqslant \mathcal N [\lambda;k]({\Delta}^-_{\mathcal L}(\zeta))^{-1}\| \psi \|_B$ for all $\zeta \in {\mathfrak M}_-(\lambda;k)\setminus {\mathcal M}_{\mathcal L}(\lambda;k)$. Theorem 4.4. If ${\mathcal L}\in {\mathbb L}(\lambda;k)$, then for each function (4.2) any point $\zeta \in (\mathbb C \setminus {\mathfrak M}_-(\lambda;k)) \cup {\mathcal M}_{\mathcal L}(\lambda;k)$ is either a pole of it (of finite order), or the function extends analytically to some neighbourhood of this point. The proof of Theorem 4.3 (via Theorem 4.2) is similar to the proof of Theorem 3.4 in [22]. The proof of Theorem 4.4 (via Lemma 4.2) is the same as the proof of Lemmas 3.17 and 3.18 in [22]. In [22] the subspaces $\mathcal H_-(\lambda,k;\zeta_0)$, $\zeta_0\in {\mathfrak M}_-(\lambda;k)$ were considered in place of the subspaces ${\mathcal L}\in {\mathbb L}(\lambda;k)$. § 5. Multiplicities of eigenvalues for $k+i\varkappa \in {\mathbb C}^2$Let $\lambda \in \mathbb R $ be an eigenvalue of the operator $\widehat H_B+V$, $\widetilde{\mathcal N} (\lambda)\geqslant 1$. Lemma 5.1. If $k\in {\mathbb M}(\lambda)$, then there exists a positive number $\varepsilon $ such that $\widetilde{\mathcal N} (\lambda; k'+ i{\varkappa}')=\widetilde{\mathcal N} (\lambda;k)=\widetilde{\mathcal N} (\lambda)$ for all $k'+i {\varkappa}'\in U^{(2)}_{\varepsilon}(k)$. Proof. Let ${\gamma}_{\widetilde {\varepsilon}}$ be the circle of radius $\widetilde {\varepsilon}>0$ with centre $\lambda $ oriented counterclockwise, lying in the resolvent set of the operator $\widehat H_B (k)+V$, $k\in {\mathbb M}(\lambda)$, and encircling only one of its eigenvalues $\lambda $. For some $\delta >0$ the circle ${\gamma}_{\widetilde {\varepsilon}}$ also lies in the resolvent set of the operators $\widehat H_B(k'+i{\varkappa}')+V$ for all $k'+i{\varkappa}' \in U^{(2)}_{\delta}(k)$, the resolvent $(\widehat H_B(k'+i{\varkappa}')+V-\zeta )^{-1}$, $\zeta \in {\gamma}_{\widetilde {\varepsilon}}$, depends continuously (in the operator norm) on $k'+i{\varkappa}'$, and this continuous dependence is uniform with respect to all $\zeta \in {\gamma}_{\widetilde {\varepsilon}}$. So we have the Riesz projection
$$
\begin{equation*}
\widehat P(k;k'+i{\varkappa}')=-\frac1{2\pi i}\oint_{{\gamma}_{\widetilde {\varepsilon}}} \bigl(\widehat H_B(k'+i{\varkappa}')+V-\zeta \bigr)^{-1}\, d\zeta
\end{equation*}
\notag
$$
onto the subspace of eigenfunctions and associated functions of the operator $\widehat H_B(k'+i\varkappa')+V$ with eigenvalues $\lambda'\in U^{(1)}_{\widetilde{\varepsilon}} (\lambda)$. The operators $\widehat H_B(k')\,{+}\,V$ are self-adjoint, and by changing $\delta >0$ if necessary it can be assumed that, for $k'+i{\varkappa}'\in U^{(2)}_{\delta}(k)$, the operators $\widehat H_B(k'+i{\varkappa}')+V$ have no associated functions with eigenvalues ${\lambda}'\in U^{(1)}_{\widetilde {\varepsilon}}(\lambda)$. Hence (by the continuity of the Riesz projection) for all $k'+i{\varkappa}'\in U^{(2)}_{\delta}(k)$ we have
$$
\begin{equation}
\sum_{{\lambda}'\in U^{(1)}_{\widetilde {\varepsilon}}(\lambda)}\widetilde{\mathcal N} ({\lambda}'; k'+i{\varkappa}')=\widetilde{\mathcal N} (\lambda).
\end{equation}
\tag{5.1}
$$
If $\Phi_j(k)$, $j=1, \dots, \widetilde{\mathcal N} (\lambda)$, is an orthonormal basis of $\mathcal H (\lambda;k)$, then the functions $U^{(2)}_{\delta}(k)\,{\ni}\,k'+i{\varkappa}' \,{\mapsto}\, \widehat P(k;k'+ i{\varkappa}')\Phi_j(k)$ are analytic, $\widehat P(k;k)\Phi_j(k)=\Phi_j(k)$, and for sufficiently small $\delta >0$ their values are linearly independent for all $k'+i{\varkappa}'\in U^{(2)}_{\delta}(k)$. The set ${\mathbb M}(\lambda)$ is open in $\mathbb R^2$, and therefore there exists $\varepsilon \in (0,{\delta}/2)$ such that $\widetilde k\in {\mathbb M}(\lambda)$ for each vector $\widetilde k\in \mathbb R^2$ with $|\widetilde k-k|<\varepsilon $. For each such vector $\widetilde k$ let $\Phi_{\mu}(\widetilde k)$, $\mu=1, \dots, \widetilde{\mathcal N} (\lambda)$, be an orthonormal basis for $\mathcal H (\lambda;\widetilde k)$. For each $e\in S^1$ there exist $\delta (\widetilde k,e)>0$ and analytic functions
$$
\begin{equation*}
U^{(1)}_{\delta (\widetilde k,e)}(0)\ni z\mapsto \Phi_{\mu}(\widetilde k;z)\in \mathcal H (\lambda;k+ze)\subset \mathcal H_B, \qquad \mu= 1, \dots, \widetilde{\mathcal N} (\lambda),
\end{equation*}
\notag
$$
such that $\Phi_{\mu}(\widetilde k;0)=\Phi_{\mu}(\widetilde k)$ and whose values, for all $z\in U^{(1)}_{\delta (\widetilde k,e)}(0)$, are also linearly independent (see Theorem XIII.13 in [16]). We set $\delta_1(\widetilde k,e;\delta)=\min\{\delta (\widetilde k,e),\delta/2\} $. For all vectors $\widetilde k\in {\mathbb M}(\lambda)$ under consideration, all $e\in S^1$ and all $z\in U^{(1)}_{\delta_1(\widetilde k,e;\delta)}(0)$, we have $\widetilde k+ze\in U^{(2)}_{ \delta}(k)$. Hence by (5.1) the functions $\widehat P(k;\widetilde k+ze)\Phi_j(k)$ are linear combinations of the functions $\Phi_{\mu}(\widetilde k;z)$. Therefore, $\widehat P(k;\widetilde k+ze)\Phi_j(k)\in \mathcal H (\lambda; \widetilde k+ze)$. By the analytic Fredholm theorem the last inclusion holds for all vectors $\widetilde k\in \mathbb R^2$ such that $|\widetilde k-k|<\varepsilon $, all $e\in S^1$ and all $z\in U^{(1)} _{\delta /2}(0)$. Any vector $k'+i{\varkappa}'\in U^{(2)}_{\varepsilon}(k)$ can be written in the form $\widetilde k+ze$, where $\widetilde k\in \mathbb R^2$, $|\widetilde k-k|<\varepsilon $, $e\in S^1$ and $z\in U^{(1)}_{\delta /2}(0)$, and hence $\widehat P(k;k'+i{\varkappa}')\Phi_j(k)\in \mathcal H (\lambda;k'+i{\varkappa}')$ for all $k'+i{\varkappa}'\in U^{(2)} _{\varepsilon}(k)$, $j=1, \dots, \widetilde{\mathcal N} (\lambda)$. Therefore, $\widetilde{\mathcal N} (\lambda;k'+i{\varkappa }')=\widetilde{\mathcal N} (\lambda)$. This proves the lemma. Lemma 5.2. If $k\in {\mathbb M}(\lambda)$, then $\mathcal N [\lambda;k]=\widetilde{\mathcal N} (\lambda)$. Proof. By Lemma 5.1 there exists a number $\varepsilon >0$ such that $\widetilde{\mathcal N} (\lambda;k'+i{\varkappa} ')=\widetilde{\mathcal N} (\lambda)$ for all $k'+i{\varkappa}'\in U^{(2)}_{ \varepsilon}(k)$. On the other hand, since $\mathbb C \setminus {\mathfrak M}_-(\lambda;k)$ is a discrete set, there exists $\zeta \in {\mathfrak M}_-(\lambda;k)$ such that $k+(\zeta /2)e_1+i(\zeta /2)e_2\in U^{(2)}_{\varepsilon}(k)$. Now from Lemma 4.1 we obtain $\mathcal N [\lambda;k]=\widetilde{\mathcal N} (\lambda;k+ (\zeta /2)e_1+i(\zeta /2)e_2)=\widetilde{\mathcal N} (\lambda)$. This proves the lemma. Now Theorem 1.1 is a direct consequence of Lemma 5.2 and Theorem 4.1. Lemma 5.3. If $\widetilde{\mathcal N} (\lambda)=P$, then $\mathcal N [\lambda;k]=P$ for all $k\in \mathbb R^2$. Proof. Since $v(\mathbb R^2\setminus {\mathbb M}(\lambda))=0$ and since the sets $\mathbb C \setminus {\mathfrak M}_-(\lambda;k')$ are discrete for all $k'\in \mathbb R^2$, it follows that for all $k\in \mathbb R^2$ and $\zeta \in {\mathfrak M}_-(\lambda;k)$ there exist vectors $k_j\in {\mathbb M}(\lambda)$ and numbers $\zeta_j\in {\mathfrak M}_-(\lambda; k_j)$, $j\in \mathbb N $, such that $k_j+(\zeta_j/2)e_1+i(\zeta_j/2)e_2\to k+(\zeta /2)e_1 +i(\zeta /2)e_2$ as $j\to +\infty $. By Lemma 5.2, $\widetilde{\mathcal N} (\lambda;k_j+(\zeta_j/2)e_1 +i(\zeta_j/2)e_2)=\mathcal N [\lambda;k_j]=\widetilde{\mathcal N} (\lambda)=P$, $j\in \mathbb N $. Hence we find from Lemma 2.2 that $\mathcal N [\lambda;k]=\widetilde{\mathcal N} (\lambda;k+(\zeta /2)e_1+i(\zeta /2)e_2)\geqslant P$. On the other hand, by Theorem 4.1, $\mathcal N [\lambda;k]\leqslant P$. Hence $\mathcal N [\lambda;k]=P$ for all $k\in \mathbb R^2$. This proves the lemma.
§ 6. Proof of Theorem 1.26.1. The definition and properties of the functions $\Phi (k,\psi;\,\cdot\,)$In what follows we assume that $\widetilde{\mathcal N} (\lambda) = P$. By Lemma 5.3 we also have $\mathcal N [\lambda;k] = P$ for all ${k \in \mathbb R^2}$. Let $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ be a nonconstant function, and let $k\in \mathbb R^2$. It follows from estimates (4.1) that $\mathcal H_B^{(0)}(k)\in {\mathbb L}(\lambda;k)$. Hence under the assumptions of Theorems 4.3 and 4.4 one can find a subspace ${\mathcal L}=\mathcal H_B^{(0)}(k)$ for which, by Theorem 4.3, for any function $\psi \in \mathcal H_B^{(0)}(k)\setminus \{0_B\} $ there exists a (unique) analytic function
$$
\begin{equation*}
{\mathfrak M}_-(\lambda;k)\setminus {\mathcal M}_{\mathcal H_B^{(0)}(k)}(\lambda;k)\ni \zeta \mapsto \Phi (k,\psi; \zeta)\in \mathcal H_-(\lambda,k;\zeta)\subset \mathcal H_B
\end{equation*}
\notag
$$
such that $\widehat P^{(0)}(k)\Phi (k,\psi;\zeta)=\psi $ for all $\zeta \in {\mathfrak M}_-(\lambda;k) \setminus {\mathcal M}_{\mathcal H_B^{(0)}(k)}(\lambda;k)$. The set ${(\mathbb C \setminus {\mathfrak M}_-(\lambda;k)) \cup {\mathcal M}_{\mathcal H_B^{(0)}(k)}(\lambda;k)}$ is discrete, and by Theorem 4.4 the functions ${\Phi (k,\psi;\,\cdot\,)}$ either have poles (of finite order) at points in this set, or extend analytically to some neighbourhoods of these points. For $|\zeta |\geqslant C_2'=C_2(\Lambda,B, B+ V- \lambda; 1/2)> 0$, from (4.1) we obtain
$$
\begin{equation}
\| (\widehat I_B-\widehat P^{(0)}(k)) \Phi (k,\psi;\zeta)\|_B\leqslant 2C_1\sqrt {\frac 2B}|\zeta |^{-1}\| B+V-\lambda \|_{L^2(K)}\| \psi \|_B
\end{equation}
\tag{6.1}
$$
(estimate (6.1) also holds for $\zeta \in (\mathbb C \setminus {\mathfrak M}_-(\lambda;k)) \cup {\mathcal M}_{\mathcal H_B^{(0)}(k)}(\lambda;k)$ if $|\zeta |\geqslant C_2'$, because the functions $\Phi (k,\psi;\,\cdot\,)$ are bounded, and therefore analytic in some neighbourhoods of the points $\zeta $). Hence the functions $\Phi (k,\psi;\,\cdot\,)$ can have only a finite number of poles, which we denote by $\zeta_1(k,\psi), \dots, \zeta_n(k,\psi)$, where $n=n(k,\psi)\in \mathbb N $ (the case $n=0$ can be excluded, since here, for all $\zeta \in \mathbb C$, the inclusion $\psi \in \mathcal H_-(\lambda,k;\zeta)$ would hold, and therefore $(B+V-\lambda)\psi=0_B$, which is impossible because $V$ is a nonconstant function and $\psi (k)\in C^{\omega}(\mathbb R^2;\mathbb C)$). Any analytic function $\chi \colon \mathbb C \to \mathcal H_B$ such that $\chi (\zeta) \to 0_B$ as $\zeta \to \infty $ is identical zero on $\mathcal H_B $, and so, employing estimate (6.1), we find that, for some functions ${\mathfrak A}_{j,\mu}(k,\psi)\in \mathcal H^2_B$, $j=1,\dots,n(k,\psi)$, $\mu=1,\dots, \mu_j(k,\psi)$ (where $\mu_j(k,\psi)\in \mathbb N $) for $\zeta \in \mathbb C \setminus \bigcup_{j=1}^{n(k,\psi)}\{\zeta_j(k,\psi)\} $, we have the equality
$$
\begin{equation}
\Phi (k,\psi;\zeta)=\psi +\sum_{j= 1}^{n(k,\psi)}\sum_{\mu= 1}^{\mu_j(k,\psi)}\frac {{\mathfrak A}_{j,\mu}(k,\psi)}{(\zeta -\zeta_j(k,\psi))^{\mu}}.
\end{equation}
\tag{6.2}
$$
In addition, $\widehat P^{(0)}(k){\mathfrak A}_{j,\mu}(k,\psi)=0_B$ for all $j$ and $\mu $. For all $k\in \mathbb R^2$, $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $ and $Y\in 2\pi \Lambda^*$ we have
$$
\begin{equation*}
e^{-i(Y,x)}\Phi (k,\psi;\,\cdot\,)=\Phi (k+Y,e^{-i(Y,x)}\psi;\,\cdot\,),
\end{equation*}
\notag
$$
where $e^{-i(Y,x)}\psi \in \mathcal H^{(0)}_B(k+Y)\setminus \{0_B\} $. So we arrive at the following simple lemma. Lemma 6.1. If $k\in \mathbb R^2$, $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $ and $Y\in 2\pi \Lambda^*$, then $e^{-i(Y,x)}\psi \in \mathcal H^{(0)}_B(k+Y)\setminus \{0_B\} $, and also $n(k,\psi)=n(k+ Y,e^{-i(Y,x)}\psi)$, $\mu_j(k,\psi)=\mu_j({k+Y}, e^{-i(Y,x)}\psi)$ and $\zeta_j(k,\psi)=\zeta_j(k+Y,e^{-i(Y,x)}\psi)$, $j=1,\dots, n(k,\psi)$. The next lemma is a direct consequence of inequalities (6.1). Lemma 6.2. If $k\in \mathbb R^2$, $\psi \in \mathcal H_B^{(0)}(k)\setminus \{0_B\} $ and $j=1,\dots, n(k,\psi)$, then ${|\zeta_j(k,\psi)|<C_2'}$, where the constant $C_2'=C_2(\Lambda,B,B+V-\lambda;1/2) >0$ is independent of both $k$ and $\psi $. Let
$$
\begin{equation}
\mathbb C \ni \zeta \mapsto {\mathfrak P}(k,\psi;\zeta)=\prod_{j= 1}^{n(k,\psi)}(\zeta - \zeta_j(k,\psi))^{\mu_j(k,\psi)}=\sum_{j= 0}^{m(k,\psi)}{\mathfrak P}_j(k,\psi)\zeta^{m(k,\psi)-j}
\end{equation}
\tag{6.3}
$$
be a polynomial of degree $m(k,\psi)=\sum_{j= 1}^{n(k,\psi)}\mu_j(k,\psi)$, where ${\mathfrak P}_j(k,\psi)\in \mathbb C $, ${j=1,\dots,m(k,\psi)}$, and ${\mathfrak P}_0(k,\psi)=1$. From (6.2) we have
$$
\begin{equation}
{\mathfrak P}(k,\psi;\zeta)\Phi (k,\psi;\zeta)=\zeta^{m(k,\psi)}\psi +\sum_{j= 1}^{m(k,\psi)}\zeta^{m(k,\psi)-j}{\mathcal F}_j(k,\psi),
\end{equation}
\tag{6.4}
$$
where ${\mathcal F}_j(k,\psi)\in \mathcal H^2_B$. In addition, ${\mathfrak P}(k,\psi;\zeta)\Phi (k,\psi;\zeta)\in \mathcal H_-(\lambda,k;\zeta)$ for all $\zeta \in \mathbb C $. An appeal to (6.4) shows that
$$
\begin{equation}
\begin{gathered} \, (B+V-\lambda)\psi=-\widehat Z_-(k){\mathcal F}_1(k,\psi), \\ \notag (\widehat Z_+(k)\widehat Z_-(k)+B+V-\lambda){\mathcal F}_j(k,\psi)=-\widehat Z_-(k){\mathcal F}_{j+1}(k,\psi),\ \ j = 1,\dots, m(k,\psi)\,{-}\,1, \end{gathered}
\end{equation}
\tag{6.5}
$$
and
$$
\begin{equation*}
(\widehat Z_+(k)\widehat Z_-(k)+B+V-\lambda){\mathcal F}_{m(k,\psi)}(k,\psi)=0_B.
\end{equation*}
\notag
$$
6.2. Smoothness of the potential $V$Lemma 6.3. Let $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$, $k\in \mathbb R^2 $, $\psi \in \mathcal H_B^{(0)}(k)\setminus \{0_B\} $, and let the functions ${\mathcal F}_j(k,\psi)\!\in\! \mathcal H^2_B$, $j=1,\dots, m(k,\psi)$, satisfy (6.5). Then ${V\in C^{\infty}(\mathbb R^2\setminus \{x\colon \psi(x)=0\};\mathbb R)}$. In the proof of Lemma 6.3 we need the following two simple lemmas. Let $\mathcal O$ be a domain in $\mathbb R^2$. Lemma 6.4. Let $k\in \mathbb R^2$. If ${\mathcal F}\in H^2_{\mathrm{loc}}({\mathcal O};\mathbb C)$ and $\bigl(k_1-i\,{\partial}/{\partial x_1}\bigr)^2{\mathcal F}+\bigl(k_2-i\,{\partial}/{\partial x_2}-Bx_1 \bigr)^2{\mathcal F}\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb C)$ for $m\in \mathbb Z_+$, then ${\mathcal F}\in H^{m+2}_{\mathrm{loc}}({\mathcal O};\mathbb C)$. Lemma 6.5. If ${\mathcal W}\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb C)$ and ${\mathcal F}\in H^{m+1}_{\mathrm{loc}}({\mathcal O};\mathbb C)$ for some $m\in \mathbb N $, then ${\mathcal W}{\mathcal F}\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb C)$. Lemma 6.5 follows from the embeddings $H^{m+2}_{\mathrm{loc}}({\mathcal O};\mathbb C)\subset C^m({\mathcal O};\mathbb C)$, $m\in \mathbb Z_+$, and $H^1_{\mathrm{loc}}({\mathcal O};\mathbb C)\subset L^p_{\mathrm {loc}}({\mathcal O};\mathbb C)$, $p>2$ (the last embedding is used for $p=4$). Proof of Lemma 6.3. Let $\mathcal O$ be a domain in $\mathbb R^2$ not containing zeros of the function $\psi $ and such that $|\psi (x)| \geqslant \varepsilon >0$ for all $x\in {\mathcal O}$. Then ${\mathcal F}_j(k,\psi)(\cdot |_{\mathcal O})\in H^2_{\mathrm{loc}}({\mathcal O};\mathbb C)$, $j=1,\dots, n(k,\psi)$. Now from the first equality in (6.5) we have $(B+V-\lambda)\psi (\cdot |_{\mathcal O})\in H^1_{\mathrm{loc}}({\mathcal O};\mathbb C)$, and so $V(\cdot |_{\mathcal O})\in H^1_{\mathrm{loc}}({\mathcal O};\mathbb R)$. Let us now use induction on $m\in \mathbb N $. Suppose that ${\mathcal F}_j(k,\psi)(\cdot |_{\mathcal O}) \in H^{m+1}_{\mathrm{loc}}({\mathcal O};\mathbb C)$, $j=1,\dots, n(k,\psi)$, and $V(\cdot |_{\mathcal O})\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb R)$, $m\in \mathbb N $. We have $\widehat Z_-(k){\mathcal F}_j(k,\psi)(\cdot |_{\mathcal O})\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb C)$, $j=2,\dots, n(k,\psi)$, and (by Lemma 6.5) $(B+V-\lambda){\mathcal F}_j(k;\psi)(\cdot |_{\mathcal O})\in H^m_{\mathrm {loc}} ({\mathcal O};\mathbb C)$, $j=1,\dots, n(k,\psi)$, and so, using equalities (6.5), we find that $\widehat Z_+(k)\widehat Z_-(k) {\mathcal F}_j(k,\psi)(\,\cdot\,|_{\mathcal O})\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb C)$, $j=1,\dots, n(k,\psi)$. Therefore (see Lemma 6.4), ${\mathcal F}_j(k,\psi)(\cdot |_{\mathcal O})\in H^{m+2}_{\mathrm{loc}}({\mathcal O};\mathbb C)$. Now from the first equality in (6.5) we obtain $(B+V-\lambda)\psi (\cdot |_{\mathcal O})\in H^{m+1}_{\mathrm{loc}}({\mathcal O};\mathbb C)$, which implies that $V(\cdot |_{\mathcal O})\in H^{m+1}_{\mathrm{loc}}({\mathcal O};\mathbb R)$. Hence ${\mathcal F}_j(k,\psi) (\cdot |_{\mathcal O})\in H^{m+1}_{\mathrm{loc}}({\mathcal O};\mathbb C)$, $j=1,\dots, n(k,\psi)$, and $V(\cdot |_{\mathcal O})\in H^m_{\mathrm{loc}}({\mathcal O};\mathbb R)$ for all $m\in \mathbb N $, and therefore $V(\cdot |_{\mathcal O})\in \bigcap_{m\in \mathbb N}H^m_{\mathrm{loc}}({\mathcal O};\mathbb R)= C^{\infty}({\mathcal O};\mathbb R)$. This proves Lemma 6.3, because the union of all the above domains $\mathcal O$ coincides with $\mathbb R^2\setminus \{x\colon \psi (x)=0\} $. Proof of Theorem 1.2. We claim that in the case when $\widetilde{\mathcal N} (\lambda)=P=\eta $, where $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$, the potential $V\in L^2_{\Lambda}(\mathbb R^2;\mathbb R)$ belongs to $C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$. By Lemma 5.3, $\mathcal N [\lambda;k]=P$ for all $k\in \mathbb R^2$. Hence (for a vector $k\in \mathbb R^2$ and a function $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $) from (6.4) we can find functions ${\mathcal F}_j (k,\psi)\in \mathcal H^2_B$ satisfying equalities (6.5). Now it follows from Lemma 6.3 that ${V\in C^{\infty}(\mathbb R^2\setminus \{x\colon \psi (x)=0\};\mathbb R)}$. The set $\{x\colon \psi (x)=0\} $ is periodic4 with period lattice $\Lambda $, and $\psi \in C^{\omega}(\mathbb R^2;\mathbb C)$. Hence for any point $y\in\mathbb R^2$ there exist an open neighbourhood ${\mathcal O}(y)\subset \mathbb R^2$ and a vector $k'\in \mathbb R^2$ such that ${{\mathcal O}(y) \cap \{ x\colon \psi '(k';x)=0\}=\varnothing}$, where $\psi '(k')= \widehat U(k,k')\psi\in \mathcal H^{(0)}_B(k')\setminus \{0_B\} $ and
$$
\begin{equation*}
\{x\colon \psi '(k';x)=0\}=\{x\colon \psi (x)=0\} +\frac{k_2'-k_2}B\, e_1-\frac{k_1'-k_1}B\, e_2.
\end{equation*}
\notag
$$
In place of the vector $k$ and the function $\psi $ one can take the vector $k'$ and function $\psi '(k')$ (and define the corresponding functions $\Phi (k',\psi '(k');\,\cdot\,)$, and take the functions ${\mathcal F}_j(k',\psi '(k'))$, $j=1,\dots,m(k',\psi '(k'))$, satisfying equalities (6.5)), and so by Lemma 6.3 we have $V\in C^{\infty}(\mathbb R^2\setminus \{x\colon \psi '(k';x)=0\};\mathbb R)$. Hence $V(\,\cdot\,|_{{\mathcal O}(y)})\in C^{\infty}({\mathcal O}(y);\mathbb R)$ for all $y\in \mathbb R^2$, and therefore $V\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$. Theorem 1.2 is proved. § 7. Analytic functions with values in a Hilbert spaceLet $\mathcal H $ be a complex Hilbert space with inner product $(\cdot,\cdot)_{\mathcal H}$ (which is linear in the second argument). We denote the (closed) linear span of a set $A\subseteq \mathcal H $ by ${\mathcal L}(A)$. Lemma 7.1. Let ${\mathcal F}_j\colon \mathbb C \to \mathcal H $, $j=1, \dots, N$, $N\in \mathbb N $, be analytic functions. Then for each $z\in \mathbb C $ there exists a discrete set $M(z)\,{\subset}\,\mathbb C$ such that, for all $z'\in \mathbb C \setminus M(z)$, Proof. Let $z\in \mathbb C $ and $n\doteq \dim{\mathcal L}(\{{\mathcal F}_j(z)\colon j=1, \dots, N\}) \geqslant 1$. In the subspace ${\mathcal L}(\{{\mathcal F}_j(z)\colon j=1, \dots, N\})$ we choose some orthonormal basis $\Phi_1,\dots,\Phi_n$. Let ${\mathcal F}_{j_{\nu}}(z)$, $\nu=1,\dots,n$, be (different) functions from the above set of functions such that ${\mathcal L}(\{ {\mathcal F}_{j_{\nu}}(z)\colon \nu=1, \dots, n\})= {\mathcal L}(\{{\mathcal F}_j(z)\colon j=1, \dots, N\}) $. Then the function
$$
\begin{equation*}
\mathbb C \ni z'\mapsto \det\bigl((\Phi_{\mu},{\mathcal F}_{j_{\nu}}(z'))_{\mathcal H}\bigr) _{\mu,\nu= 1}^n\in \mathbb C
\end{equation*}
\notag
$$
is analytic and does not vanish for $z'=z$. If $M(z)$ is the zero set of this function, then $M(z)$ is discrete, and for all $z'\in \mathbb C \setminus M(z)$ the orthogonal projections of the functions ${\mathcal F}_{j_{\nu}}(z')$, $\nu=1,\dots,n$, onto the subspace ${\mathcal L}(\{ {\mathcal F}_j(z)\colon j=1, \dots, N\})$ are linearly independent. Therefore, inequality (7.1) holds. This proves the lemma. Lemma 7.2. Let ${\mathcal F}_j\colon \mathbb C \to \mathcal H $, $j\in \mathbb N $, be analytic functions, not all of which are equal to zero. Suppose that $\dim{\mathcal L}(\{{\mathcal F}_j(z)\colon j\in \mathbb N\})<+\infty $ for all $z\in \mathbb C $. Then there exist $N\in \mathbb N $ and a (possibly empty) discrete set $M\subset \mathbb C $ such that $\dim{\mathcal L}(\{{\mathcal F}_j(z)\colon {j\in \mathbb N}\})=N$ for all $z\in \mathbb C\setminus M$ and $\dim{\mathcal L}(\{{\mathcal F}_j(z)\colon j\in \mathbb N\})<N$ for $z\in M$. Proof. Suppose that there exist $z_n\,{\in}\, \mathbb C $, $n\,{\in}\, \mathbb N $, such that ${\dim {\mathcal L}(\{{\mathcal F}_j(z_n)\colon j\!\in\! \mathbb N\}) \!\geqslant\! n}$. By Lemma 7.1 there exist discrete sets $M_n\subset \mathbb C $, $n\in \mathbb N $, such that $z_n\not\in M_n$ and $\dim {\mathcal L}(\{{\mathcal F}_j(z)\colon j\in \mathbb N\}) \geqslant \dim{\mathcal L}(\{{\mathcal F}_j(z_n)\colon j\in \mathbb N\}) \geqslant n$ for all $z\in {\mathbb C \setminus M_n}$. Next, the set $\widetilde M\doteq \bigcup_{n= 1}^{+\infty} M_n$ is at most countable. Hence $\mathbb C \setminus \widetilde M\neq \varnothing $ and $\dim {\mathcal L}(\{{\mathcal F}_j(z)$: ${j\in \mathbb N}\})=+\infty $ for all $z\in \mathbb C \setminus \widetilde M$. But this contradicts the assumption of Lemma 7.2. Therefore, there exists a number $N\in \mathbb Z_+$ such that $\dim{\mathcal L}(\{{\mathcal F}_j(z)$: ${j\in \mathbb N}\})\leqslant N$ for all $z\in \mathbb C $. We assume that the above integer $N$ is smallest possible. Since not all functions ${\mathcal F}_j(\,\cdot\,)$ are zero, we have $N \in \mathbb N $. Hence by Lemma 7.1 there exists a discrete set $M\subset \mathbb C $ such that $\dim{\mathcal L} (\{{\mathcal F}_j(z)\colon j\in \mathbb N\})=N$ for all $z\in \mathbb C \setminus M$ and $\dim{\mathcal L} (\{{\mathcal F}_j(z)\colon j\in \mathbb N\})<N$ for $z\in M$. This proves the lemma. Lemma 7.3. Let ${\mathcal F}_j\colon \mathbb C \to \mathcal H $, $j=1, \dots, N$, $N\in \mathbb N $, be analytic functions and $M\subset \mathbb C $ be a discrete set. Suppose that $\dim {\mathcal L}(\{{\mathcal F}_j(z)\colon j=1, \dots, N\})=N$ for all $z\in \mathbb C \setminus M$ and ${\mathcal F}\colon \mathbb C \to \mathcal H $ is an analytic function such that ${\mathcal F}(z)\in {\mathcal L}(\{{\mathcal F}_j(z)\colon j=1, \dots, N\}) $ for all $z\in \mathbb C \setminus M$. Then there exist analytic or meromorphic functions ${\mathcal C}_j$, $j=1, \dots,N$, with values in $\mathbb C $ and with poles in the discrete set $M$ such that Proof. The functions ${\mathcal C}_j(z)$ are uniquely defined for all $z\in \mathbb C \setminus M$. Let $z_0\in \mathbb C \setminus M$. As in the proof of Lemma 7.1, we choose some orthonormal basis $\Phi_1, \dots, \Phi_N$ of the subspace ${\mathcal L}(\{ {\mathcal F}_j(z_0)\colon j=1,\dots, N\})$. Since is a nontrivial analytic function (this function is distinct from zero at $z=z_0$), its zeros form a discrete set $M(z_0)$. Let $\widehat P(z_0)$ be the orthogonal projection of the space $\mathcal H $ onto the subspace ${\mathcal L}(\{{\mathcal F}_j(z_0)\colon j=1, \dots, N\})$. Then for $z\in \mathbb C \setminus (M\cup M(z_0))$ the expansion ${\mathcal F}(z)=\sum_{j= 1}^N{\mathcal C}_j(z){\mathcal F}_j(z)$ is equivalent to the expansion $\widehat P(z_0){\mathcal F}(z)=\sum_{j= 1}^N{\mathcal C}_j(z)\widehat P(z_0){\mathcal F}_j(z)$, and the functions ${\mathcal C}_j(\,\cdot\,)$ are solutions of the system of linear equations where $z\mapsto (\Phi_{\mu},{\mathcal F}(z))_{\mathcal H}$ and $z\mapsto (\Phi_{\mu},{\mathcal F}_j(z))_{\mathcal H} $ are analytic functions. Hence by Cramer’s rule the ${\mathcal C}_j(\,\cdot\,)$, $j=1,\dots,N$, are analytic or meromorphic functions with poles in the discrete set $M\cup M(z_0)$. On the other hand $z_0\not\in M(z_0)$ for all $z_0\in \mathbb C \setminus M$, and so the poles of ${\mathcal C}_j(\,\cdot\,)$ can only lie in the set $M$. This proves the lemma.
§ 8. Proof of Theorem 1.38.1. The definition of the functions ${\mathfrak B}_j(k,\psi)$ and ${\mathfrak F}_j(k,\psi;\,\cdot\,)$In what follows it is assumed that $V\in C^{\infty}_{\Lambda}({\mathbb R^2},\mathbb R)$ and $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$. We also choose a vector $k\in \mathbb R^2$. Let $\widehat Z_-^{-1}(k)$ be a right inverse of the operator $\widehat Z_-(k)$ such that, for all $\psi (k)\in \mathcal H^{(0)}_B(k)$ and $m\in \mathbb Z_+$,
$$
\begin{equation*}
\widehat Z_-^{-1}(k)\psi^{(m)}(k)=\bigl(2B(m+1)\bigr)^{-1/2}\psi^{(m+1)}(k);
\end{equation*}
\notag
$$
$\widehat Z_-(k)\widehat Z_-^{-1}(k)=\widehat I_B$. Right inverses of $\widehat Z_-(k)$ that are bounded linear operators are not uniquely defined. Any such operator can be expressed as $\widehat Z_-^{-1}(k)+\widehat Q(k)$, where $\widehat Q(k)$ is a bounded linear operator such that $\widehat Q(k)\Phi \in \mathcal H^{(0)}_B(k)$ for all $\Phi \in\mathcal H_B$. From (6.2) and estimate (6.1) (see also Lemma 6.2) it follows that for $|\zeta |\geqslant C_2'> \max_{j= 1,\dots,m(k,\psi)}|\zeta_j(k,\psi)|$ the function $\Phi (k,\psi;\,\cdot\,)$ (where $k\in \mathbb R^2$ and $\psi$ is a function in $\mathcal H^{(0)}_B(k)\setminus \{0_B\}$) expands in a series
$$
\begin{equation}
\Phi (k,\psi;\zeta)=\psi +\sum_{j= 1}^{+\infty}(-1)^j \zeta^{-j}\widehat Z_-^{-1}(k){\mathfrak B}_j(k,\psi).
\end{equation}
\tag{8.1}
$$
Since ${\mathfrak P}(k,\psi;\zeta)\Phi (k,\psi;\zeta)\in \mathcal H_-(\lambda,k;\zeta)$ for all $\zeta \in \mathbb C $, it follows that,
$$
\begin{equation}
{\mathfrak B}_j(k,\psi)=\bigl(\widehat Z_+(k)+(B+V-\lambda)\widehat Z_-^{-1}(k)\bigr)^{j-1}(B+V-\lambda)\psi, \qquad j\in \mathbb N
\end{equation}
\tag{8.2}
$$
(here we have used equality (6.5) and the inclusion $V\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb R)$). Lemma 8.1. The functions ${\mathfrak B}_1(k,\psi), \dots, {\mathfrak B}_{m(k,\psi)}(k,\psi)$ are linearly independent in $\mathcal H_B$. The remaining functions ${\mathfrak B}_j(k,\psi)$, $j>m(k,\psi)$, are linear combinations of them. In addition, Proof. It follows from (6.2) that for $j = 1,\dots,n(k,\psi)$ the functions $\widehat Z_-(k){\mathfrak A}_{j,\mu_j(k,\psi)}(k,\psi)$ are eigenfunctions of the operator ${\widehat Z_+(k)+(B\,{+}\,V\,{-}\,\lambda)\widehat Z_-^{-1}(k)}$ with eigenvalues $-\zeta_j(k,\psi)$, and therefore $(-1)^{\mu_j(k,\psi)-\mu}\widehat Z_-(k){\mathfrak A}_{j,\mu}(k,\psi)$, $\mu=1,\dots, \mu_j(k,\psi)-1$, are the associated functions of order $\mu_j(k,\psi)-\mu $, $\mu=1,\dots,\mu_j(k,\psi)-1$ (for $\mu_j(k,\psi)> 1$). All these functions are linearly independent. Let $L(k,\psi)$ be their linear span in $\mathcal H_B$, $\dim L(k,\psi)=m(k,\psi)$. It follows from (6.2) and (8.1) that the functions ${\mathfrak B}_j(k,\psi)$ are linear combinations of the functions $\widehat Z_-(k){\mathfrak A}_{j,\mu} (k,\psi)$. Therefore, ${\mathfrak B}_j(k,\psi)\in L(k,\psi)$, $j\in \mathbb N $. Using (6.2) and (8.1) we also find that each function
$$
\begin{equation*}
\begin{aligned} \, &\widehat Z_-(k){\mathfrak A}_{j,\mu}(k,\psi) =\frac1{2\pi i}\biggl(\prod_{j'\colon j'\neq j} \bigl(\zeta_j(k,\psi)-\zeta_{j'}(k,\psi)\bigr) \biggr)^{-1} \\ &\qquad\qquad \times\oint_{\gamma_R}\biggl(\prod_{j'\colon j'\neq j}\bigl(\zeta -\zeta_{j'}(k,\psi)\bigr)^{\mu_{j'(k,\psi)}}\biggr) \frac{\widehat Z_-(k)\Phi (k,\psi;\zeta)}{(\zeta -\zeta_j(k,\psi))^{\mu -1}}\,d\zeta \end{aligned}
\end{equation*}
\notag
$$
(where $\gamma_R$ is the circle in $\mathbb C $ of radius $R>C_2'$ with centre at the origin which is oriented counterclockwise) is a finite linear combination of the functions ${\mathfrak B}_j(k,\psi)$. If $s\in \mathbb N $ is the smallest number such that the functions ${\mathfrak B}_1(k,\psi), \dots,$ ${\mathfrak B}_s(k,\psi)$ are linearly independent in $\mathcal H_B$ and the function ${\mathfrak B}_{s+1}(k,\psi)$ is a linear combination of them, then the equalities
$$
\begin{equation*}
{\mathfrak B}_{j+1}(k,\psi)=\bigl(\widehat Z_+(k)+(B+V-\lambda)\widehat Z_-^{-1}(k)\bigr) {\mathfrak B}_j(k,\psi), \qquad j\in \mathbb N,
\end{equation*}
\notag
$$
imply that for $j\geqslant s+1$ all functions ${\mathfrak B}_j(k,\psi)$ are also linear combinations of the functions ${\mathfrak B}_1(k,\psi), \dots, {\mathfrak B}_s(k,\psi)$. Hence $s=m(k,\psi)$. The operator $\widehat Z_+(k)+(B+V-\lambda)\widehat Z_-^{-1}(k)$, which acts on the invariant subspace $L(k,\psi)$, has eigenvalues $-\zeta_j(k,\psi)$ of algebraic multiplicity $\mu_j(k,\psi)$, $j=1,\dots, n(k,\psi)$. Hence by the Cayley-Hamilton theorem the restriction of the operator
$$
\begin{equation*}
\prod_{j= 1}^{n(k,\psi)}\bigl(\widehat Z_+(k)+\zeta_j(k,\psi)+(B+V-\lambda)\widehat Z_-^{-1}(k)\bigr)^{\mu_j(k,\psi)}
\end{equation*}
\notag
$$
to the subspace $L(k,\psi)$ is the zero operator. Now (8.3) follows, which proves the lemma. Proof. From (6.5), for $j=1,\dots,m(k,\psi)$, we obtain in succession that there exist numbers $q_0(k,\psi)=1$ and $q_j(k,\psi)\in \mathbb C $, $j=1,\dots,m(k,\psi)$, such that
$$
\begin{equation*}
{\mathcal F}_j(k,\psi)\,{=}\sum_{\nu= 0}^{j-1}(-1)^{j-\nu}q_{\nu}(k,\psi)\widehat Z_-^{-1}(k) {\mathfrak B}_{j-\nu}(k,\psi)+q_j(k,\psi)\psi,\quad\ \ j = 1,\dots,m(k,\psi).
\end{equation*}
\notag
$$
From the last equality in (6.5) we see that
$$
\begin{equation*}
\sum_{\nu= 0}^{m(k,\psi)}(-1)^{\nu}q_{\nu}(k,\psi){\mathfrak B}_{m(k,\psi)+1-\nu}(k,\psi)=0_B.
\end{equation*}
\notag
$$
On the other hand, in view of (8.3) (and the functions ${\mathfrak B}_s(k,\psi)$, $s=1,\dots,m(k,\psi)$, are linearly independent in $\mathcal H_B$). Hence $q_{\nu}(k,\psi)={\mathfrak P}_{\nu}(k,\psi)$, $\nu=0,\dots,m(k,\psi)$. This proves the lemma. Corollary 8.1. For all $\zeta \in \mathbb C \setminus \bigcup_{j= 1}^{n(k,\psi)}\{\zeta_j(k,\psi)\} $, Given a vector $k\in \mathbb R^2$ and a function $\psi \in \mathcal H^{(0)}_B(k)\setminus \{0_B\} $, with a vector $k'\in \mathbb R^2$ one can uniquely associate the number $z=k_1'-k_1+i(k_2'-k_2)\in \mathbb C $. Next, using Lemma 3.3, for each vector $k'\in \mathbb R^2$ we can choose a function $\widetilde \psi (k')=\exp(-({z}/{(2B)})\widehat Z_+(k))\psi \in \mathcal H^{(0)}_B(k')\setminus \{ 0_B\} $. By Lemma 8.1, for all $k'\in \mathbb R^2$ the functions
$$
\begin{equation}
{\mathfrak B}_j(k',\widetilde \psi (k'))=\bigl(\widehat Z_+(k')+(B+V-\lambda)\widehat Z_-^{-1} (k')\bigr)^{j-1}(B+V-\lambda)\widetilde \psi (k')
\end{equation}
\tag{8.4}
$$
are linearly independent for $j=1,\dots, m(k',\widetilde \psi (k'))$, and the functions ${\mathfrak B}_j(k',\widetilde \psi (k'))$, $j> m(k',\widetilde \psi (k'))$, are linear combinations of them. Consider the operators
$$
\begin{equation}
(\widehat Z_-(k)+z)^{-1}_*\doteq \widehat Z_-^{-1}(k)+\sum_{s= 1}^{+\infty}(-z)^s\widehat Z_- ^{-1-s}(k), \qquad z\in \mathbb C.
\end{equation}
\tag{8.5}
$$
The norm of the compact operator $\widehat Z_-^{-s}(k)\doteq (\widehat Z_-^{-1}(k))^s$, $s\in \mathbb N $, is $(2B)^{-s/2}(s!)^{-1/2}$, and so for each $R>0$ the series (8.5) converges absolutely and uniformly in the operator norm for all $z\in U^{(1)}_R(0)$. Since $\widehat Z_-(k')(\widehat Z_-(k)+z)^{-1}_*=\widehat I_B$, we have
$$
\begin{equation*}
\widehat Z_-^{-1}(k')=(\widehat I_B-\widehat P^{(0)}(k'))(\widehat Z_-(k)+z)^{-1}_*.
\end{equation*}
\notag
$$
Also, a direct calculation shows that
$$
\begin{equation}
(\widehat Z_-(k)+z)^{-1}_* \widehat Z_-(k')(\widehat I_B-\widehat P^{(0)}(k))=(\widehat I_B-\widehat P^{(0)}(k))
\end{equation}
\tag{8.6}
$$
and
$$
\begin{equation}
\begin{gathered} \, (\widehat Z_-(k)+z)^{-1}_* \psi=-z^{-1}\biggl(\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)-\widehat I_B\biggr) \psi, \\ \psi \in \mathcal H^{(0)}_B(k), \qquad z\in \mathbb C \setminus \{0\}. \end{gathered}
\end{equation}
\tag{8.7}
$$
Proof. Using (8.6) and (8.7) we have
$$
\begin{equation*}
\begin{aligned} \, &(\widehat Z_-(k)+z)^{-1}_*= (\widehat Z_-(k)+z)^{-1}_* \widehat Z_-(k')(\widehat I_B-\widehat P^{(0)}(k)) \widehat Z_-^{-1}(k') \\ &\qquad\qquad +(\widehat Z_-(k)+z)^{-1}_* \widehat Z_-(k')\widehat P^{(0)}(k) \widehat Z_-^{-1}(k') \\ &\qquad = (\widehat I_B-\widehat P^{(0)}(k)) \widehat Z_-^{-1}(k')+z(\widehat Z_-(k)+z)^{-1}_* \widehat P^{(0)}(k)\widehat Z_-^{-1}(k') \\ &\qquad =(\widehat I_B-\widehat P^{(0)}(k)) \widehat Z_-^{-1}(k')-\biggl(\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)-\widehat I_B\biggr) \widehat P^{(0)}(k) \widehat Z_-^{-1}(k') \\ &\qquad = \widehat Z_-^{-1}(k')-\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\widehat P^{(0)}(k)\widehat Z_-^{-1}(k'), \end{aligned}
\end{equation*}
\notag
$$
which proves the lemma. In what follows we assume that $\eta=P=1$. We also choose a vector ${\psi \in \mathcal H^{(0)}_B(k)}$ such that $\|\psi\|_B\!=\!1$. In this case the numbers $m(k',\widetilde \psi(k'))$, $n(k',\widetilde \psi(k'))$, ${\zeta}_j(k',\widetilde \psi (k'))$ and ${\mu}_j(k',\widetilde \psi (k'))$ are independent of the functions $\widetilde \psi (k')\in \mathcal H^{(0)}_B(k')\setminus \{0_B\} $. So in what follows we denote these numbers by $m(k')$, $n(k')$, ${\zeta}_j(k')$ and ${\mu}_j(k')$, $j=1,\dots, n(k')$, $k'\in \mathbb R^2$. For $\eta=P=1$ (and all $k'\in \mathbb R^2$) we have $\dim\mathcal H^{(0)}_B(k')=1$, and therefore
$$
\begin{equation}
\begin{aligned} \, \notag &\widehat Z_-^{-1}(k')\Phi= (\widehat Z_-(k)+z)^{-1}_*\Phi \\ &\qquad\qquad -\exp\biggl(-\frac{|z|^2}{2B}\biggr) \bigl(\widetilde \psi (k'),(\widehat Z_-(k)+z)^{-1}_*\Phi \bigr)_B\widetilde \psi (k'), \qquad \Phi \in \mathcal H_B. \end{aligned}
\end{equation}
\tag{8.8}
$$
For each $j\in \mathbb N $ consider the function
$$
\begin{equation}
\begin{aligned} \, \notag &\mathbb C \ni z \mapsto {\mathfrak F}_j(k,\psi;z) \\ &\qquad \doteq \bigl(\widehat Z_+(k)+(B+V-\lambda)(\widehat Z_-(k)+z)^{-1}_*\bigr)^{j-1} (B+V-\lambda)\widetilde \psi (k'). \end{aligned}
\end{equation}
\tag{8.9}
$$
An appeal to (8.4) and (8.8) shows that
$$
\begin{equation}
{\mathfrak B}_j(k',\widetilde \psi (k'))=\sum_{s= 1}^ja_{js}{\mathfrak F}_s(k,\psi;z), \qquad j\in \mathbb N,
\end{equation}
\tag{8.10}
$$
where $a_{jj}=1$ and $a_{js}=a_{js}(k;z)\in \mathbb C $ for $s<j$. Now we have the following result. Lemma 8.4. For all $z\in \mathbb C $ the functions ${\mathfrak F}_j(k,\psi;z)\in \mathcal H^{\infty}_B$, $j=1, \dots, m(k')$, are linearly independent, and the functions ${\mathfrak F}_j(k,\psi;z)\in \mathcal H^{\infty}_B$, $j>m (k')$, are linear combinations of them. 8.2. The analyticity of the functions ${\mathfrak F}_j(k,\psi;\,\cdot\,)$Let ${\mathcal S}^{\infty}_B$ be the set of functions $\mathbb C \ni z\mapsto F(z)\in \mathcal H^{\infty}_B$ such that, for (a vector $k\in \mathbb R^2$ and) all $R>0$ and $s\in \mathbb Z_+$, there exist positive numbers $c_s(k,R;F)=c_s(B,\Lambda;k,R;F)$ such that, for all $z\in U^{(1)}_{R}(0)$, The set ${\mathcal S}^{\infty}_B$ is independent of the choice of the vector $k\in \mathbb R^2$. In addition, the function $z\mapsto F(z)\in \mathcal H^{\infty}_B$ lies in the space ${\mathcal S}^{\infty}_B$ if and only if, for some (and, therefore, each) $k'\in \mathbb R^2$ and all $R>0$ and $s\in \mathbb Z_+$, there exists $\widetilde c_s(k',R;F)=\widetilde c_s(B,\Lambda;k',R;F)>0$ such that, for all $z\in U^{(1)}_{R}(0)$ and $m\in \mathbb Z_+$,
$$
\begin{equation}
\| \widehat P^{(m)}(k')F(z)\|_B \leqslant \widetilde c_s(k',R;F)(m+1)^{-s}.
\end{equation}
\tag{8.12}
$$
It follows from estimates (8.11) that the set ${\mathcal S}^{\infty}_B$ contains, together with the function $F(\,\cdot\,)\in {\mathcal S}^{\infty}_B$, all functions $\widehat Z_+^s(k)F(\,\cdot\,)$, $s\in \mathbb Z_+$. Lemma 8.5. The function $z\mapsto \widetilde \psi (k')=\exp(-(z/(2B))\widehat Z_+(k))\psi $ belongs to ${\mathcal S}^{\infty}_B$. Proof. From (3.1), for all $m\in \mathbb N $ and $z\in \mathbb C $ we obtain
$$
\begin{equation*}
\biggl\| \widehat P^{(m)}(k)\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr) \psi \biggr\|_B=(2B)^{-m/2}(m!)^{-1/2}|z|^m,
\end{equation*}
\notag
$$
which implies (8.12). This proves the lemma. Lemma 8.6. If $F(\,\cdot\,)\in {\mathcal S}^{\infty}_B$ and ${\mathcal W}\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb C)$, then ${\mathcal W}F(\,\cdot\,)\in {\mathcal S}^{\infty}_B$. Proof. If $F(\,\cdot\,)$ satisfies (8.11), then for all $z\in U^{(1)}_R(0)$, $R>0$, and $s\in \mathbb Z_+$,
$$
\begin{equation*}
\begin{aligned} \, &\| \widehat Z_+^s(k){\mathcal W}F(z)\|_B \leqslant \sum_{\mu= 0}^s \binom{s}{\mu} \biggl\| \biggl(\biggl(\frac{\partial}{\partial x_1}-i\,\frac{\partial}{\partial x_2}\biggr)^{\mu} {\mathcal W}\biggr) \widehat Z_+^{s-\mu}(k)F(z)\biggr\|_B \\ &\qquad \leqslant \sum_{\mu= 0}^s\binom{s}{\mu} \biggl\| \biggl(\frac{\partial}{\partial x_1}-i\,\frac{\partial}{\partial x_2}\biggr)^{\mu} {\mathcal W}\biggr\|_{L^{\infty}(K)} c_{s-\mu}(k,R;F), \end{aligned}
\end{equation*}
\notag
$$
where the symbols $\binom{s}{\mu}$ are binomial coefficients. This proves the lemma. Lemma 8.7. If $F(\,\cdot\,)\in {\mathcal S}^{\infty}_B$, then the function $z\mapsto (\widehat Z_-(k)+z)^{-1}_*F(z)$ also belongs to the space ${\mathcal S}^{\infty}_B$. Proof. Let $s\in \mathbb Z_+$. If $F(\,\cdot\,)$ satisfies (8.11), then for all $\nu \in \mathbb N $ and $z\in U^{(1)}_R(0)$,
$$
\begin{equation*}
\| \widehat Z_+^s(k)\widehat Z_-^{-\nu}(k)F(z)\|_B\leqslant (2B)^{-\nu}\| \widehat Z_+^{s+\nu}(k)F(z)\|_B\leqslant (2B)^{-\nu}c_{s+\nu}(k,R;F).
\end{equation*}
\notag
$$
On the other hand, if $\mathbb N \ni \nu >2s$, then the norm of the compact operator $\widehat Z_+^s(k)\widehat Z_-^{-\nu}(k)$ is $(2B)^{(s-\nu)/2}\sqrt {(\nu +s)!}\, (\nu !)^{-1}<(2B)^{(s-\nu)/2}3^s((\nu -s-1)!)^{-1/2}$. Hence for any $R>0$ the series converges absolutely and uniformly in the operator norm for $z\in U^{(1)}_R(0)$. Therefore, there exists a positive number $c_s'(k,R;F)=c_s'(B,\Lambda;k,R;F)$ such that for all $z\in U^{(1)}_R(0)$. This proves the lemma. Lemma 8.8. For all $j\in \mathbb N $ the functions ${\mathfrak F}_j(k,\psi;\,\cdot\,)$ lie in the space ${\mathcal S}^{\infty}_B$. This result is a direct consequence of Lemmas 8.5, 8.6 and 8.7. Lemma 8.9. For all $j\in \mathbb N $ the functions $\mathbb C \ni z\mapsto {\mathfrak F}_j(k,\psi;z)\in \mathcal H^{\infty}_B\subset \mathcal H_B$ are analytic. Proof. By Lemma 8.8, for each $R>0$ the sequence $\sum_{m= 0}^M\widehat P^{(m)}{\mathfrak F}_j(k,\psi;\,\cdot\,)$, $j\in \mathbb N $, $M\in \mathbb Z_+$, converges uniformly as $M\to +\infty $ to the function ${\mathfrak F}_j(k,\psi;\,\cdot\,)$ on $U^{(1)}_R(0)$. So, to prove Lemma 8.9 it suffices to show that for each $m\in \mathbb Z_+$ the functions $\widehat P^{(m)}(k){\mathfrak F}_j(k,\psi;\,\cdot\,)$ are analytic. Since
$$
\begin{equation*}
\widehat P^{(m)}(k)\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi =(2B)^{-m/2}(m!)^{-1/2}(-z)^m\psi^{(m)}(k)
\end{equation*}
\notag
$$
and since the function $z\mapsto \exp(-(z/(2B))\widehat Z_+(k))\psi $ belongs to ${\mathcal S}^{\infty }_B$ by Lemma 8.5, it follows that for any $\Phi \in \mathcal H^{(m)}_B(k)$ the function
$$
\begin{equation*}
\mathbb C \ni z\mapsto (\Phi,{\mathfrak F}_1(k,\psi;z))_B =\biggl((B+V-\lambda)\Phi,\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi \biggr)_B
\end{equation*}
\notag
$$
is analytic, and so the functions $\widehat P^{(m)}(k){\mathfrak F}_1(k,\psi;\,\cdot\,)$, $m\in \mathbb Z_+$, and therefore also ${\mathfrak F}_1(k,\psi;\,\cdot\,)$, are analytic too. If we assume that the function ${\mathfrak F}_j(k,\psi;\,\cdot\,)$ is analytic for some $j\in \mathbb N $, then for any $\Phi \in \mathcal H^{(m)}_B(k)$ the function
$$
\begin{equation*}
\begin{aligned} \, & \mathbb C \ni z\mapsto (\Phi,{\mathfrak F}_{j+1}(k,\psi;z))_B= (\widehat Z_-(k)\Phi,{\mathfrak F}_j(k,\psi;z))_B \\ &\qquad\qquad +\bigl((B+V-\lambda)\Phi,(\widehat Z_-(k)+z)^{-1}_*{\mathfrak F}_j(k,\psi;z)\bigr)_B \in \mathbb C \end{aligned}
\end{equation*}
\notag
$$
is analytic (the analyticity of the function $z\,{\mapsto}\, (\widehat Z_-(k)+z)^{-1}_*{\mathfrak F}_j(k,\psi;z)$ can be derived from the expansion (8.5) and the inclusion ${\mathfrak F}_j(k,\psi;\,\cdot\,) \in \mathcal S^{\infty}_B$; in addition, the function $z\,{\mapsto}\, (\widehat Z_-(k)+z)^{-1}_*{\mathfrak F}_j(k,\psi;z)$ also belongs to ${\mathcal S}^{\infty}_B$). Hence the functions $\widehat P^{(m)}(k){\mathfrak F}_{j+1}(k,\psi;\,\cdot\,)$, $m\in \mathbb Z_+$, are analytic, and therefore so is the function ${\mathfrak F}_{j+1}(k,\psi;\,\cdot\,)$. Consequently, all functions ${\mathfrak F}_j(k,\psi;\,\cdot\,)$, $j\in \mathbb N $ are analytic. This proves the lemma. From Lemmas 7.2, 8.4 and 8.9 it follows that there exist $\widetilde m(k)\in \mathbb N $ and a discrete set $\widetilde M(k)\subset \mathbb C $ such that $m(k',\widetilde \psi (k'))=\widetilde m(k)$ for all $z\in \mathbb C \setminus \widetilde M(k)$ and $m(k',\widetilde \psi (k'))<\widetilde m(k)$ for $z\in \widetilde M(k)$. Thus, we have the following result. Lemma 8.10. The numbers $\widetilde m(k)$ are independent of $k\in \mathbb R^2$, $\widetilde m(k)\equiv \widetilde m \in \mathbb N $. In addition, for each $k''\in \mathbb R^2$, the discrete set $\widetilde M(k'')=\widetilde M(k)-(k_1''-k_1)-i(k_2''-k_2)$ can be chosen. For all $z\in \mathbb C \setminus \widetilde M(k)$ the functions ${\mathfrak F}_j(k,\psi;z)$, $j=1,\dots, \widetilde m$, are linearly independent, and the functions ${\mathfrak F}_j(k,\psi;z)$, $j>\widetilde m$, are linear combinations of them. By Lemma 7.3, for each $z\in \mathbb C \setminus \widetilde M(k)$ we have the expansion
$$
\begin{equation}
{\mathfrak F}_{\widetilde m+1}(k,\psi;z)=\sum_{j= 1}^{\widetilde m}{\mathcal C}_j(k;z) {\mathfrak F}_j(k,\psi;z),
\end{equation}
\tag{8.13}
$$
where ${\mathcal C}_j(k;z)\in \mathbb C $, and the functions ${\mathcal C}_j(k;\,\cdot\,)$ are either analytic or meromorphic with poles in the discrete set $\widetilde M(k)$. 8.3. An explicit expression for the function ${\mathcal C}_{\widetilde m}(k;\,\cdot\,)$Let ${\mathcal W} \in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb C)$. Consider the functions
$$
\begin{equation*}
\begin{aligned} \, & \mathbb C \ni z\mapsto {\mathcal G}({\mathcal W},k;z) =\exp\biggl(-\frac{|z|^2}{2B}\biggr) \\ &\qquad\qquad\times \biggl(\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi, (\widehat Z_-(k)+z)^{-1}_*{\mathcal W} \exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi \biggr)_B. \end{aligned}
\end{equation*}
\notag
$$
Both the functions
$$
\begin{equation*}
z\mapsto \exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi\quad\text{and}\quad z\mapsto (\widehat Z_-(k)+z)^{-1}_*{\mathcal W}\exp\biggl(-\frac{z}{2B}\widehat Z_+(k)\biggr)\psi
\end{equation*}
\notag
$$
lie in the space ${\mathcal S}^{\infty}_B$ (see Lemmas 8.5, 8.6 and 8.7) and are analytic (see the proof of Lemma 8.9). Hence ${\mathcal G}({\mathcal W},k;\,\cdot\,)\in C^{\omega}(\mathbb R^2;\mathbb C)$. For $z\in \mathbb C \setminus \widetilde M(k)$ we set $\Xi (k;z)=\sum_{j= 1}^{n(k')}\mu_j(k')\zeta_j(k')$, $\widetilde m=\widetilde m(k)=m(k')=\sum_{j= 1}^{n(k')}\mu_j(k')$, and $\widetilde V(x)=V(x)-V_0$, $x\in \mathbb R^2$, where $\displaystyle V_0=(v(K))^{-1}\int_KV(x)\,dx $. If in (8.3), in place of the vector $k$ and the function $\psi $ we take any vector $k'\in \mathbb R^2$ such that $z\in \mathbb C \setminus \widetilde M(k)$ and consider the function $\widetilde \psi (k')$, then using (8.8) we arrive at the expansion (8.13) for the function $\mathbb C \setminus \widetilde M(k)\ni z\mapsto{\mathfrak F}_{\widetilde m+1}(k,\psi;z)$, where
$$
\begin{equation}
{\mathcal C}_{\widetilde m}(k;z)=-\widetilde m\overline z -\Xi (k;z)+ {\mathcal G}(B+V-\lambda,k;z).
\end{equation}
\tag{8.14}
$$
Since ${\mathcal G}(B+V-\lambda,k;\,\cdot\,)$ is a locally bounded function in $\mathbb C $, it follows that (see Lemma 6.2) for each $R>0$ the function $z\mapsto {\mathcal C}_{\widetilde m}(k;z)$ is bounded on the set $(\mathbb C \setminus \widetilde M(k))\cap U^{(1)}_{R}(0)$. Hence ${\mathcal C}_{\widetilde m}(k;\,\cdot\,)$ extends analytically to neighbourhoods of points of the discrete set $\widetilde M(k)$ and is analytic in $\mathbb C $. From (8.14) it also follows that the function $\Xi (k;\,\cdot\,)$ extends to the set $\mathbb C \setminus \widetilde M(k)$ by continuity and is continuous in $\mathbb C $. For any vector $\widetilde k\in \mathbb R^2$,
$$
\begin{equation*}
\Xi (\widetilde k;z)=\Xi (k;z+{\widetilde k}_1-k_1+i({\widetilde k}_2-k_2)).
\end{equation*}
\notag
$$
By Lemma 6.2, $|\Xi (k;z)|\leqslant C_2'\widetilde m$, $z\in \mathbb C $. The numbers $n(k')$, $\mu_j(k')$ and $\zeta_j(k')$ (for $\eta=P=1$) are independent of the functions $\psi (k')\in \mathcal H^{(0)}_B(k')\setminus \{0_B\} $. Now the next result follows from Lemma 6.1. We assume that $P=1$, and so for all $Y\in 2\pi \Lambda^*$ and $k\in \mathbb R^2$, under the hypotheses of Lemma 3.4 we have $\widehat U^{(Y)}(k)\psi = \theta^{(Y)}(k)\psi $, $\psi \in \mathcal H^{(0)}_B$, where $\theta^{(Y)}(k) \in \mathbb C $ and $|\theta^{(Y)}(k)|=1$. Proof. By Lemma 3.4 (for $\eta=P=1$)
$$
\begin{equation*}
\begin{aligned} \, &\theta^{(Y)}(k') =\exp\biggl(\frac{|Y|^2}{4B}\biggr) \bigl(\psi (k'), \exp(i(Y,x))\psi (k')\bigr)_B \\ &\quad=\exp\biggl(\frac{|Y|^2}{4B}\biggr) \bigl(\widehat U(k,k')\psi, \exp(i(Y,x))\widehat U(k,k')\psi \bigr)_B \\ &\quad=\exp\biggl(\frac{|Y|^2}{4B}\biggr) \int_Ke^{i(Y,x)}\biggl| \psi \biggl(x-\frac{\operatorname{Im} z}B \, e_1+\frac{\operatorname{Re} z}B\, e_2\biggr) \biggr|^2\,dx \\ &\quad=\exp\biggl(\frac{|Y|^2}{4B}\biggr) \exp(iB^{-1}Y_1\operatorname{Im} z) \exp(-iB^{-1}Y_2\operatorname{Re} z) \int_Ke^{i(Y,x)}|\psi (x)|^2\,dx \\ &\quad= \exp\biggl(\frac{|Y|^2}{4B}\biggr) \exp\biggl(\frac{z}{2B}(Y_1-iY_2)\biggr) \exp\biggl(-\frac{\overline z}{2B}(Y_1+iY_2)\biggr)(\psi,e^{ i(Y,x)}\psi)_B \\ &\quad=\exp\biggl(\frac{z}{2B}(Y_1-iY_2)\biggr) \exp\biggl(-\frac{\overline z}{2B}(Y_1+iY_2)\biggr)\theta^{(Y)}(k), \end{aligned}
\end{equation*}
\notag
$$
where $\psi (k')=\exp(-|z|^2/(4B))\widetilde \psi (k')$. This proves the lemma. The next result is a direct consequence of Lemma 8.12. Lemma 8.14. For each function ${\mathcal W}\in C^{\infty}_{\Lambda}(\mathbb R^2;\mathbb C)$ (and all $k\in \mathbb R^2$ and $z\in \mathbb C $) Proof. By Lemma 8.3,
$$
\begin{equation*}
{\mathcal G}({\mathcal W},k;z) =-\exp\biggl(\frac{|z|^2}{4B}\biggr)(\psi,\widehat Z_-^{-1}(k'){\mathcal W}\psi (k'))_B.
\end{equation*}
\notag
$$
For all $j\in \mathbb Z_+$ we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl(\psi,\widehat Z_-^{-1}(k')\psi^{(j)}(k')\bigr)_B= (2B)^{-1-j/2}\bigl((j+1)(j+1)!\bigr)^{-1/2}\bigl(\psi,\widehat Z_+^{j+1}(k')\psi (k')\bigr)_B \\ &\qquad = (2B)^{-1-j/2}\bigl((j+1)(j+1)!\bigr)^{-1/2}\bigl((\widehat Z_-(k)+z)^{j+1}\psi,\psi (k')\bigr)_B \\ &\qquad = (2B)^{-1-j/2}\bigl((j+1)(j+1)!\bigr)^{-1/2}\overline z^{j+1}\exp\biggl(-\frac{|z|^2}{4B}\biggr) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, &\bigl(\psi^{(j)}(k'),\exp(i(Y,x))\psi (k')\bigr)_B \\ &\qquad = (2B)^{-j/2}(j!)^{-1/2}\bigl(\psi (k'),\exp(i(Y,x))(\widehat Z_-(k')+(Y_1+iY_2))^j\psi (k')\bigr)_B \\ &\qquad = (2B)^{-j/2}(j!)^{-1/2}(Y_1+iY_2)^j\exp\biggl(-\frac{|Y|^2}{4B}\biggr)\theta^{(Y)}(k'). \end{aligned}
\end{equation*}
\notag
$$
Therefore,5
$$
\begin{equation*}
\begin{aligned} \, &-{\mathcal G}({\mathcal W},k;z) =\exp\biggl(\frac{|z|^2}{4B}\biggr)\sum_{j= 0}^{+\infty}(\psi,\widehat Z_-^{-1}(k')\psi^{(j)}(k'))_B(\psi^{(j)}(k'),{\mathcal W}\psi (k'))_B \\ &\qquad = \overline z(2B)^{-1}{\mathcal W}_0+\sum_{Y\in 2\pi \Lambda^*\setminus \{0\}} {\mathcal W}_Y\biggl(\sum_{j= 0}^{+\infty}\frac{{\overline z}^{j+1}(Y_1+iY_2)^j}{(2B)^{j+1}(j+1)!}\biggr) \exp\biggl(-\frac{|Y|^2}{4B}\biggr)\theta^{(Y)}(k') \\ &\qquad = \overline z(2B)^{-1}{\mathcal W}_0+\sum_{Y\in 2\pi \Lambda^*\setminus \{0\}} \exp\biggl(-\frac{|Y|^2}{4B}\biggr) (Y_1+iY_2)^{-1}{\mathcal W}_Y\theta^{(Y)}(k') \\ &\qquad\qquad\times \biggl(\exp\biggl(\frac{\overline z}{2B}(Y_1+iY_2)\biggr)-1\biggr). \end{aligned}
\end{equation*}
\notag
$$
Lemma 8.14 follows from the last equality and Lemma 8.12. Proof. From (8.14) and Lemma 8.14, for (a vector $k\in \mathbb R^2$ and) all $z\in \mathbb C$ we have
$$
\begin{equation}
{\mathcal C}_{\widetilde m}(k;z)=-(2B)^{-1}\bigl(V_0+(2\widetilde m +1)B-\lambda \bigr) \overline z -\Xi (k;z)-{\mathcal G}_0(V,k;z) +{\mathcal G}_1(V,k;z),
\end{equation}
\tag{8.17}
$$
where ${\mathcal G}_0(V,k;\,\cdot\,)$ is an analytic function, and the function ${\mathcal G}_1(V,k;\,\cdot\,)$ is continuous and bounded. For $z=re^{i\phi}$ ($r>0$, $\phi \in [0,2\pi)$) we have
$$
\begin{equation}
\begin{gathered} \, \frac1{2\pi}\int_0^{2\pi}{\mathcal C}_{\widetilde m}(k;re^{i\phi})e^{i\phi}\,d\phi= \frac1{2\pi}\int_0^{2\pi} {\mathcal G}_0(V,k;re^{i\phi})e^{i\phi}\,d\phi=0, \\ \frac1{2\pi}\int_0^{2\pi}\overline ze^{i\phi}d\phi=r, \qquad \biggl|\frac1{2\pi}\int_0^{2\pi}\Xi (k;re^{i\phi})e^{i\phi}\,d\phi \biggr| \leqslant C_2'\widetilde m, \\ \biggl|\frac1{2\pi}\int_0^{2\pi} {\mathcal G}_1(V,k;re^{i\phi})e^{i\phi}\,d\phi \biggr| \leqslant \sum_{Y\in 2\pi \Lambda^*\setminus \{0\}} \exp\biggl(-\frac{|Y|^2}{4B}\biggr)|Y|^{-1}|{\mathcal W}_Y|. \end{gathered}
\end{equation}
\tag{8.18}
$$
Now letting $r\to +\infty $ and using (8.17) and (8.18) we obtain equality (8.16). Theorem 8.1 is proved. Theorem 1.3 (for an ‘enlarged’ lattice $\Lambda $) is a direct consequence of Theorem 8.1. Theorem 8.2. The equality ${\mathcal C}_{\widetilde m}(k;z)=\widetilde C -{\mathcal G}_0({V},k;z)$, $z\in \mathbb C$, holds, where the number $\widetilde C\in \mathbb C $ is independent of both $k\in \mathbb R^2$ and $z\in \mathbb C $. Proof. By (8.16) and (8.17)
$$
\begin{equation}
{\mathcal C}_{\widetilde m}(k;z)=-\Xi (k;z)-{\mathcal G}_0({V,k;z)}+{\mathcal G}_1({V},k;z),
\end{equation}
\tag{8.19}
$$
where ${\mathcal G}_0(V,k;\,\cdot\,)$ is an analytic function in $\mathbb C $, and $\Xi (k;\,\cdot\,)$ and ${\mathcal G}_1(V,k;\,\cdot\,)$ are continuous bounded functions. Now from (8.19), for all $n\in \mathbb N $ we obtain
$$
\begin{equation*}
\frac1{2\pi i}\oint_{{\gamma}_r}\bigl({\mathcal C}_{\widetilde m}(k;z)+{\mathcal G}_0(V,k;z)\bigr) \,\frac{dz}{z^{n+1}} \to 0
\end{equation*}
\notag
$$
as $r\to +\infty $. Hence ${\mathcal C}_{\widetilde m}(k;z)=\widetilde C(k)-{\mathcal G}_0(V,k;z)$, $z\in \mathbb C $. In addition, $\widetilde C(k)=-\Xi (k;z)+{\mathcal G}_1(V,k;z)$ (for all $z\in \mathbb C $). The continuous function ${\mathbb R^2\ni \xi \!\mapsto\! \Xi (k;\xi_1\!+\!i\xi_2)}$ is periodic with period lattice $2\pi \Lambda^*$ (see Lemma 8.11). It follows from the definition of the function ${\mathcal G}_1(V,k;\,\cdot\,)$ and Lemma 8.13 that the continuous function $\mathbb R^2 \ni \xi \mapsto {\mathcal G}_1(V,k;\xi_1+i\xi_2)$ is also periodic with period lattice $2\pi \Lambda^*$, and
$$
\begin{equation*}
\iint_{2\pi K^*}{\mathcal G}_1(V,k;\xi_1+i\xi_2)\,d\xi_1\,d\xi_2=0.
\end{equation*}
\notag
$$
Hence the quantity
$$
\begin{equation*}
\widetilde C=\widetilde C(k)=-(2\pi)^{-2}v(K)\iint_{2\pi K^*}\Xi (k;\xi_1+i\xi_2) \,d\xi_1\,d\xi_2
\end{equation*}
\notag
$$
is independent of $k$. Theorem 8.2 is proved. 
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\Bibitem{Dan23} |
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