THREE-TERM ASYMPTOTIC FORMULA FOR
LARGE EIGENVALUES OF THE TWO-PHOTON
QUANTUM RABI MODEL

Anne Boutet de Monvel, Lech Zielinski

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Anne Boutet de Monvel, Lech Zielinski. THREE-TERM ASYMPTOTIC FORMULA FOR LARGE EIGENVALUES OF THE TWO-PHOTON QUANTUM RABI MODEL. 2023. hal-04363130

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# THREE-TERM ASYMPTOTIC FORMULA FOR LARGE EIGENVALUES OF THE TWO-PHOTON QUANTUM RABI MODEL

ANNE BOUTET DE MONVEL¹ AND LECH ZIELINSKI²

**ABSTRACT.** We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences $(E_m^+)^\infty_{m=0}$, $(E_m^-)^\infty_{m=0}$ satisfying a three-term asymptotic formula with the remainder estimate $O(m^{-1} \ln m)$ when $m$ tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues of the two-photon quantum Rabi model.

**Keywords:** unbounded self-adjoint operators, discrete spectrum, asymptotic distribution of eigenvalues, quantum Rabi model

**Mathematics Subject Classification:** 47A75, 81Q10, 47B25, 47B36.

## 1. GENERAL PRESENTATION OF THE PAPER

1.1. **Introduction.** The simplest interaction between a two-level atom and a classical light field is described by the semi-classical Rabi model [22], [23]. The quantum Rabi model (QRM) couples a two-level system (TLS) with a quantized single-mode radiation and is considered as a particularly important model in quantum electrodynamics: we refer to [8] concerning the historical aspects of the QRM and to [21] for a list of recent research works and experimental realizations of the QRM.

The simplest QRM is defined by the one-photon Hamiltonian $\mathbf{H}_{\text{Rabi}}^{(1)}$ given in Definition 1.2(c). The operator $\mathbf{H}_{\text{Rabi}}^{(1)}$ is a self-adjoint operator depending on two real parameters: $g$ (the coupling constant) and $\Delta$ (the energy separation in the TLS). Its spectrum is discrete and the fundamental question is how to find a good approximation of the corresponding eigenvalues.

The first step in this direction, is the rotating-wave approximation (RWA) introduced in the famous paper of Jaynes and Cummings [20]. However, the RWA is a correct approximation only when $g$ is close to 0 and $\Delta$ close to 1 for $\mathbf{H}_{\text{Rabi}}^{(1)}$ given in Definition 1.2(c). The most popular idea of going beyond the limitations of RWA, bears the name of the generalized rotating-wave approximation (GRWA) after Irish [18]. The same idea was considered before by Feranchuk, Komarov, Ulyanenkov [13], under the name of the zeroth order approximation of the operator method (see [14]). According to the GRWA, the spectrum of $\mathbf{H}_{\text{Rabi}}^{(1)}$ is composed of two eigenvalue sequences: $(E_m^+)^\infty_{m=0}$ and $(E_m^-)^\infty_{m=0}$, satisfying $E_m^\pm \approx E_{m,\text{GRWA}}^\pm$ and the formula (25) in [13] claims that

$$E_{m,\text{GRWA}}^\pm \approx m - g^2 \pm \tau_m \text{ as } m \to \infty$$

<footnote>Date: December 23, 2023.</footnote>
holds with

$$ \mathbf{r}_m := \frac{\Delta \cos(4g\sqrt{m} - \frac{\pi}{4})}{2\sqrt{2\pi g\sqrt{m}}}. \qquad (1.1) $$

In [3]-[4] we proved that if $\mathbf{r}_m$ is given by (1.1) and $\varepsilon > 0$, then one has

$$ E_m^{\pm} = m - g^2 \pm \mathbf{r}_m + O(m^{-\frac{1}{2}+\varepsilon}) \quad \text{as } m \to \infty \qquad (1.2) $$

and

$$ E_{m, \text{GRWA}}^{\pm} = m - g^2 \pm \mathbf{r}_m + O(m^{-\frac{1}{2}}) \quad \text{as } m \to \infty \qquad (1.3) $$

Thus (1.2)-(1.3) allow one to estimate the quality of GRWA for large eigenvalues of the one-photon QRM. In this paper we will prove a similar result for the two-photon QRM defined by the Hamiltonian $\mathbf{H}_{\text{Rabi}}^{(2)}$ given in Definition 1.2(d).

We remark that the two-photon QRM was used in [15] to describe a two-level atom interacting with squeezed light and we refer to [9] for the overview of works concerning the two-photon quantum Rabi model (see also [21]). In what follows, we always assume that the coupling constant satisfies the condition $0 < g < 1/2$. This hypothesis implies that $\mathbf{H}_{\text{Rabi}}^{(2)}$ is self adjoint and has a discrete spectrum. In [6] we proved that the spectrum of $\mathbf{H}_{\text{Rabi}}^{(2)}$ is composed of two eigenvalue sequences: $(E_m^{+})_{m=0}^{\infty}$ and $(E_m^{-})_{m=0}^{\infty}$, satisfying

$$ E_m^{\pm} = \left(m + \frac{1}{2}\right) \sqrt{1 - 4g^2} - \frac{1}{2} + O(m^{-1/3}) \quad \text{as } m \to \infty \qquad (1.4) $$

In this paper we will prove the three-term asymptotic formula

$$ E_m^{\pm} = \left(m + \frac{1}{2}\right) \sqrt{1 - 4g^2} - \frac{1}{2} \pm \mathbf{r}_m + O(m^{-1 \ln m}) \quad \text{as } m \to \infty \qquad (1.5) $$

where $\mathbf{r}_m$ is given by (1.20). By analogy to (1.2)-(1.3), the expression

$$ \left(m + \frac{1}{2}\right) \sqrt{1 - 4g^2} - \frac{1}{2} \pm \mathbf{r}_m $$

can be viewed as a GRWA for large eigenvalues of the two-photon QRM.

## 1.2. Quantum Rabi Model.

**Notation 1.1.** (a) In what follows, $\mathbb{Z}$ is the set of integers and $\mathbb{N} := \{n \in \mathbb{Z} : n \ge 0\}$.

(b) We denote by $\ell^2(\mathbb{N})$ the complex Hilbert space of square-summable sequences $x : \mathbb{N} \to \mathbb{C}$ equipped with the scalar product

$$ \langle x, y \rangle_{\ell^2(\mathbb{N})} = \sum_{m=0}^{\infty} \overline{x(m)y(m)} \qquad (1.6) $$

and the norm $||x||_{\ell^2(\mathbb{N})} := \langle x, x \rangle_{\ell^2(\mathbb{N})}^{1/2}$. For $s > 0$ we denote

$$ \ell^{2,s}(\mathbb{N}) := \{x \in \ell^2(\mathbb{N}) : \sum_{m=0}^{\infty} (1+m^2)^s |x(m)|^2 < \infty\} \qquad (1.7) $$

(c) The canonical basis of $\ell^2(\mathbb{N})$ is denoted $\{e_n\}_{n \in \mathbb{N}}$ (i.e. $e_n(m) = \delta_{n,m}$ for $n, m \in \mathbb{N}$).

(d) The annihilation and creation operators, $\hat{a}$ and $\hat{a}^\dagger$, are the linear maps $\ell^{2,1/2}(\mathbb{N}) \to \ell^2(\mathbb{N})$ satisfying

$$ \hat{a}^\dagger e_n = \sqrt{n+1} e_{n+1} \quad \text{for } n \in \mathbb{N} \qquad (1.8) $$

$$ \hat{a} e_0 = 0 \quad \text{and} \quad \hat{a} e_n = \sqrt{n} e_{n-1} \quad \text{for } n \in \mathbb{N} \setminus \{0\}. \qquad (1.9) $$
(e) Using $(1,0) \in \mathbb{C}^2$ and $(0,1) \in \mathbb{C}^2$ as the canonical basis of the Euclidean space $\mathbb{C}^2$, we denote by $\sigma_x, \sigma_z, I_2$, the linear operators in $\mathbb{C}^2$ defined by the matrices

$$ \sigma_x := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_z := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad I_2 := \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \qquad (1.10) $$

**Definition 1.2.** (a) The two-level system (TLS) Hamiltonian is the linear map in $\mathbb{C}^2$ defined by the matrix

$$ H_{\text{TLS}} = \frac{1}{2} \begin{pmatrix} \Delta & 0 \\ 0 & -\Delta \end{pmatrix} = \frac{1}{2} \Delta \sigma_z \qquad (1.11) $$

where $\Delta$ is a real parameter.

(b) The Hamiltonian of the single-mode radiation is the linear map $H_{\text{rad}} : \ell^{2,1}(\mathbb{N}) \to \ell^2(\mathbb{N})$ defined by the formula

$$ H_{\text{rad}}e_n = \hat{a}^{\dagger}\hat{a}e_n = ne_n \text{ for } n \in \mathbb{N}. \qquad (1.12) $$

(c) The one-photon quantum Rabi Hamiltonian is defined as the linear map $\mathbf{H}_{\text{Rabi}}^{(1)}: \mathbb{C}^2 \otimes \ell^{2,1}(\mathbb{N}) \to \mathbb{C}^2 \otimes \ell^2(\mathbb{N})$ given by

$$ \mathbf{H}_{\text{Rabi}}^{(1)} = I_2 \otimes H_{\text{rad}} + H_{\text{TLS}} \otimes I_{\ell^2(\mathbb{N})} + g\sigma_x \otimes (\hat{a} + \hat{a}^{\dagger}), \qquad (1.13) $$

where $g > 0$ is the coupling constant.

(d) The two-photon quantum Rabi Hamiltonian is defined as the linear map $\mathbf{H}_{\text{Rabi}}^{(2)}: \mathbb{C}^2 \otimes \ell^{2,1}(\mathbb{N}) \to \mathbb{C}^2 \otimes \ell^2(\mathbb{N})$ given by

$$ \mathbf{H}_{\text{Rabi}}^{(2)} = I_2 \otimes H_{\text{rad}} + H_{\text{TLS}} \otimes I_{\ell^2(\mathbb{N})} + g\sigma_x \otimes (\hat{a}^2 + (\hat{a}^\dagger)^2), \qquad (1.14) $$

where $g > 0$ is the coupling constant.

1.3. Main result. In what follows, we assume $0 < g < 1/2$ and introduce

$$ \beta := \sqrt{1 - 4g^2}, \qquad (1.15) $$

$$ \alpha := \arctan \left( \sqrt{\frac{1 - 2g}{1 + 2g}} \right). \qquad (1.16) $$

Let $\mathbf{H}_{0,\text{Rabi}}^{(2)}$ denote the operator given by (1.14) with $\Delta = 0$, i.e.

$$ \mathbf{H}_{0,\text{Rabi}}^{(2)} = I_2 \otimes H_{\text{rad}} + g\sigma_x \otimes (\hat{a}^2 + (\hat{a}^\dagger)^2). \qquad (1.17) $$

If $0 < g < 1/2$, then the spectrum of $\mathbf{H}_{0,\text{Rabi}}^{(2)}$ is explicitly known (see [11], [12]): it is composed of the sequence of eigenvalues

$$ E_m^0 = m\beta + (\beta - 1)/2, \quad m = 0, 1, 2, \dots \qquad (1.18) $$

and each eigenvalue $E_m^0$ is of multiplicity 2. Thus $0 < g < 1/2$ ensures the fact that $\mathbf{H}_{0,\text{Rabi}}^{(2)}$ is self-adjoint and has compact resolvent. Since $\mathbf{H}_{\text{Rabi}}^{(2)} - \mathbf{H}_{0,\text{Rabi}}^{(2)}$ is bounded, the operator $\mathbf{H}_{\text{Rabi}}^{(2)}$ is self-adjoint and has compact resolvent if $0 < g < 1/2$. The explicit values of eigenvalues of $\mathbf{H}_{\text{Rabi}}^{(2)}$ are not known when $\Delta \neq 0$, but we can describe their asymptotic behaviour. Our main result is the following
**Theorem 1.3.** Assume that $0 < g < 1/2$. Then one can find $\{v_m^+\}_{m \in \mathbb{N}} \cup \{v_m^-\}_{m \in \mathbb{N}}$, an orthonormal basis of $\mathbb{C}^2 \otimes \ell^2(\mathbb{N})$, such that

$$ \mathbf{H}_{\text{Rabi}}^{(2)} v_m^{\pm} = E_m^{\pm} v_m^{\pm}, \quad m = 0, 1, 2, \dots $$

and the eigenvalue sequences $(E_m^+)$$_{m \in \mathbb{N}}$, $(E_m^-)$$_{m \in \mathbb{N}}$, satisfy the large $m$ estimates

$$ E_m^{\pm} = m\beta + (\beta - 1)/2 \pm \mathbf{r}_m + O(m^{-1} \ln m) \quad (1.19) $$

with $\mathbf{r}_m$ given by the formula

$$ \mathbf{r}_m = \begin{cases} \frac{\Delta}{2} \sqrt{\frac{\beta}{\pi gm}} \cos((2m+1)\alpha) & \text{if } m \text{ is even} \\ \frac{\Delta}{2} \sqrt{\frac{\beta}{\pi gm}} \sin((2m+1)\alpha) & \text{if } m \text{ is odd,} \end{cases} \quad (1.20) $$

where $\beta$ is given by (1.15) and $\alpha$ is given by (1.16).

**Remarks.** (a) One has $E_m^{\pm} - E_m^0 = O(m^{-1/2})$ in spite of the fact that $\mathbf{H}_{\text{Rabi}}^{(2)} - \mathbf{H}_{0,\text{Rabi}}^{(2)}$ is not compact. A similar fact was established in [30] for the one-photon Rabi model.

(b) Following [25], one can prove that the spectrum of $\mathbf{H}_{\text{Rabi}}^{(2)}$ is not discrete if $g \ge \frac{1}{2}$.

1.4. **Overview of earlier results.** The distribution of large eigenvalues of the QRM has been a subject of many numerical investigations. Concerning the earliest physical considerations, we refer to Schmutz [26]. It is well known (see e.g [28]) that the one-photon quantum Rabi Hamiltonian can be expressed as a direct sum of two Jacobi operators, i.e. operators defined by infinite tridiagonal matrices. Similarly, the two-photon quantum Rabi Hamiltonian can be expressed as a direct sum of four Jacobi operators (see [6]). A mathematical study of large eigenvalues of Jacobi matrices was initiated by J. Janas and S. Naboko in the paper [19], which contains fundamental ideas of the method of approximate diagonalizations.

The question of the behaviour of large eigenvalues of $\mathbf{H}_{\text{Rabi}}^{(1)}$, was first posed by E. A. Tur [28]-[29] and it was considered by A. Boutet de Monvel, S. Naboko and L. O. Silva in [1]-[2]. Due to the difficulty of the problem, the authors give a solution for a simpler class of operators ("modified Jaynes-Cummings models"). The two-term asymptotic formula for large eigenvalues of the one-photon quantum Rabi model, was proved by E. A. (Tur) Yanovich in [30] (see also [29]). The three-term asymptotic formula (1.2) was proved in [3]-[4]. We remark that the formula (1.2) allows one to recover the values of parameters $g$ and $\Delta$ from the spectrum (see [5]). The two-term asymptotic formula (1.4) for large eigenvalues of the two-photon quantum Rabi model, was proved in [6]. The result of Theorem 1.3 and the main ideas of its proof were described in [7]. The same result was also described in [17].

1.5. **Organization of the paper.** Our approach is based on an analysis of operators acting in $\ell^2(\mathbb{Z})$. These operators are introduced in Section 2.1 and the corresponding three-term asymptotic formula is given in Theorem 2.3. In Theorem 2.3(a) we consider the operators $\tilde{J}_{\gamma}^0$, which are special case of operators considered by Edward [10]. In Section 2.2 we describe explicit expressions of their eigenvalues and eigenvectors by means of the discrete Fourier transform. In Theorem 2.3(b) we consider the operators
$\tilde{J}_{\gamma}^{\delta}$, which are viewed as perturbations of $\tilde{J}_{\gamma}^{0}$. In Section 3 we show that the assertion of Theorem 2.3(b) follows from a GRWA result given in Proposition 3.1. The proof of Proposition 3.1 begins in Section 4 and is completed in Section 5. In Section 6 we show how to deduce Theorem 1.3 from Theorem 2.3(b). More precisely, we follow the following scheme:

- we prove Proposition 3.1, using Lemma 4.3 and Kato-Temple estimate (Theorem 5.2)

- we prove Theorem 2.3, using Lemma 2.6, Proposition 3.1 and an asymptotic formula from [6]

- we prove Theorem 1.3, using Theorem 2.3, Lemma 6.2 and Lemma 6.3.

## 2. AN AUXILIARY PROBLEM IN $\ell^2(\mathbb{Z})$

### 2.1. Behaviour of large eigenvalues for auxiliary operators $\tilde{J}_{\gamma}^{\delta}$.

**Notation 2.1.** (a) We denote by $\mathcal{B}(\mathcal{V})$ the algebra of bounded operators on the Banach space $\mathcal{V}$ and $\|\cdot\|_{\mathcal{B}(\mathcal{V})}$ is the corresponding operator norm.

(b) We denote by $\ell^2(\mathbb{Z})$ the complex Hilbert space of square-summable sequences $x: \mathbb{Z} \to \mathbb{C}$ equipped with the scalar product

$$ \langle x, y \rangle_{\ell^2(\mathbb{Z})} = \sum_{k \in \mathbb{Z}} \overline{x(k)} y(k) \quad (2.1) $$

and the norm $\|x\|_{\ell^2(\mathbb{Z})} := \langle x, x \rangle_{\ell^2(\mathbb{Z})}^{1/2}$. For $s > 0$ we denote

$$ \ell^{2,s}(\mathbb{Z}) := \{x \in \ell^2(\mathbb{Z}) : \|x\|_{\ell^{2,s}(\mathbb{Z})} < \infty\} $$

where

$$ \|x\|_{\ell^{2,s}(\mathbb{Z})} := \left( \sum_{k \in \mathbb{Z}} (1+k^2)^s |x(k)|^2 \right)^{1/2} \quad (2.2) $$

(c) The canonical basis of $\ell^2(\mathbb{Z})$ is denoted $\{\tilde{e}_j\}_{j \in \mathbb{Z}}$ (i.e. $\tilde{e}_j(k) = \delta_{j,k}$ for $j, k \in \mathbb{Z}$) and $\ell_{\text{fin}}^2(\mathbb{Z})$ denotes the set of finite linear combinations of vectors belonging to $\{\tilde{e}_j\}_{j \in \mathbb{Z}}$.

(d) If $(\tilde{d}_j)_{j \in \mathbb{Z}}$ is real valued, then $\tilde{D} := \text{diag}(\tilde{d}_j)_{j \in \mathbb{Z}}$ is the self-adjoint operator in $\ell^2(\mathbb{Z})$ satisfying

$$ \tilde{D} \tilde{e}_j = \tilde{d}_j \tilde{e}_j \text{ for every } j \in \mathbb{Z}. $$

We denote

$$ \Lambda := \text{diag}(j)_{j \in \mathbb{Z}}. \quad (2.3) $$

(e) We denote by $S$ the shift defined in $\ell^2(\mathbb{Z})$ by the formula $(Sx)(j) = x(j-1)$.

**Definition 2.2.** In what follows, $\gamma$, $\delta$ and $g$ are fixed real numbers.

(a) We define $\tilde{J}_{\gamma}^{0}$ as the linear map $\ell^{2,1}(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ given by

$$ \tilde{J}_{\gamma}^{0} := \Lambda + g(S(\Lambda + \gamma) + (\Lambda + \gamma)S^{-1}) = \Lambda + g(S(\Lambda + \gamma) + \text{h.c.}) \quad (2.4) $$

(b) We define $\tilde{J}_{\gamma}^{\delta}$ as the linear map $\ell^{2,1}(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ given by the formula

$$ \tilde{J}_{\gamma}^{\delta} := \tilde{J}_{\gamma}^{0} + D_{\delta}, \quad (2.5) $$

where

$$ D_{\delta} := \text{diag}(\delta(-1)^j)_{j \in \mathbb{Z}}. \quad (2.6) $$
Using the above definition, we find that the action of $\tilde{J}_{\gamma}^{\delta}$ can be represented by the following tridiagonal, $\mathbb{Z} \times \mathbb{Z}$ matrix

$$ \left( \begin{array}{ccccccc} \cdots & & & & & & \\ -2 + \delta & g(-2 + \gamma) & 0 & 0 & 0 & & \\ g(-2 + \gamma) & -1 - \delta & g(-1 + \gamma) & 0 & 0 & & \\ 0 & g(-1 + \gamma) & \delta & g\gamma & 0 & & \\ 0 & 0 & g\gamma & 1 - \delta & g(1 + \gamma) & & \\ 0 & 0 & 0 & g(1 + \gamma) & 2 + \delta & & \\ & & & & & \ddots & \end{array} \right) $$

whose diagonal entries $\{\tilde{d}_{\delta}(j)\}_{j \in \mathbb{Z}}$ are given by

$$ \tilde{d}_{\delta}(j) := j + \delta(-1)^{j}, \qquad (2.7) $$

off-diagonal entries $\{\tilde{b}_{\gamma}(j)\}_{j \in \mathbb{Z}}$ are given by

$$ \tilde{b}_{\gamma}(j) := g(j + \gamma) \qquad (2.8) $$

and $\tilde{J}_{\gamma}^{\delta} \tilde{e}_{j} = \tilde{d}_{\delta}(j) \tilde{e}_{j} + \tilde{b}_{\gamma}(j) \tilde{e}_{j+1} + \tilde{b}_{\gamma}(j-1) \tilde{e}_{j-1}$ for every $j \in \mathbb{Z}$.

**Theorem 2.3.** Let $\tilde{J}_{\gamma}^{0}$ be given by (2.4) and $\tilde{J}_{\gamma}^{\delta}$ by (2.5)-(2.6). If $0 < g < 1/2$ then (a) the spectrum of $\tilde{J}_{\gamma}^{0}$ is composed of a non-decreasing sequence of eigenvalues $\{d_{\gamma,j}\}_{j \in \mathbb{Z}}$ of the form

$$ d_{\gamma,j} := \beta j + \left( \gamma - \frac{1}{2} \right) (\beta - 1), \qquad (2.9) $$

where $\beta$ is given by (1.15).

(b) the spectrum of $\tilde{J}_{\gamma}^{\delta}$ is composed of a non-decreasing sequence of eigenvalues $\{\lambda_{j}(\tilde{J}_{\gamma}^{\delta})\}_{j \in \mathbb{Z}}$ which can be labeled so that

$$ \lambda_j(\tilde{J}_\gamma^\delta) = d_{\gamma,j} + \mathbf{r}_\gamma^\delta(j) + O(j^{-1} \ln j) \quad \text{as } j \to \infty \qquad (2.10) $$

where $d_{\gamma,j}$ is given by (2.9),

$$ \mathbf{r}_{\gamma}^{\delta}(j) = \delta \left( \frac{\beta}{2\pi gj} \right)^{1/2} \cos (4\alpha j + \hat{\theta}_{\gamma}), \qquad (2.11) $$

$\beta$ is given by (1.15), $\alpha$ by (1.16) and

$$ \hat{\theta}_{\gamma} = \left( \gamma - \frac{1}{2} \right) (4\alpha - \pi) + \frac{\pi}{4}. \qquad (2.12) $$

*Proof.* (a) This result was proved in [10]. We describe a simplified proof in Section 2.2. (b) See Section 3.2. $\square$

2.2. **Diagonalization of $\tilde{J}_{\gamma}^{0}$**. In what follows, $\mathbb{T} := \mathbb{R}/2\pi\mathbb{Z}$ is identified with $]-\pi, \pi]$ and $L^{2}(\mathbb{T})$ denotes the Hilbert space of Lebesgue square integrable functions $]-\pi, \pi] \to \mathbb{C}$ equipped with the scalar product

$$ \langle f, g \rangle_{L^2(\mathbb{T})} := \int_{-\pi}^{\pi} \overline{f(\theta)} g(\theta) \frac{d\theta}{2\pi} \qquad (2.13) $$

and the norm $||f||_{L^2(\mathbb{T})} = \langle f, f \rangle_{L^2(\mathbb{T})}^{1/2}$. We let $\mathcal{F}_{\mathbb{T}}$ denote the isometric isomorphism $L^2(\mathbb{T}) \to \ell^2(\mathbb{Z})$ given by

$$ (\mathcal{F}_{\mathbb{T}}f)(j) = \int_{-\pi}^{\pi} f(\theta) e^{-ij\theta} \frac{d\theta}{2\pi} \qquad (2.14) $$
and consider the operator

$$L_{\gamma}^{0} := \mathcal{F}_{\mathbb{T}}^{-1} \tilde{J}_{\gamma}^{0} \mathcal{F}_{\mathbb{T}} = -i \frac{d}{d\theta} + g \left( e^{i\theta} \left( -i \frac{d}{d\theta} + \gamma \right) + \text{h.c.} \right)$$

Similarly as in [10] and [6], we observe that the assumption $0 < g < 1/2$ ensures the fact that $L_{\gamma}^{0}$ is the first order linear elliptic differential operator,

$$L_{\gamma}^{0} = \frac{1}{2} \left( (1 + 2g \cos(\theta)) \left( -i \frac{d}{d\theta} \right) + \text{h.c.} \right) + (2\gamma - 1)g \cos(\theta)$$

and we introduce

$$\Phi(\theta) := \int_{0}^{\theta} \frac{\beta \, d\theta'}{1 + 2g \cos(\theta')}. \qquad (2.15)$$

An easy calculation allows one to express the right-hand side of (2.15),

$$\Phi(\theta) = 2 \arctan \left( \sqrt{\frac{1-2g}{1+2g}} \tan \left( \frac{\theta}{2} \right) \right) \quad \text{if} \quad -\pi < \theta < \pi. \qquad (2.16)$$

Moreover $\Phi(\pi) = \pi$ and $\Phi$ is odd, i.e. $\Phi(-\theta) = -\Phi(\theta)$. Using the change of variable $\eta = \Phi(\theta)$, we can define the unitary operator acting in $L^2(\mathbb{T})$ according to the formula

$$(U_{\Phi} f)(\theta) = \Phi'(\theta)^{1/2} f(\Phi(\theta)) \qquad (2.17)$$

and the direct computation (see [6]) gives

$$U_{\Phi}^{-1} L_{\gamma}^{0} U_{\Phi} = \beta \left( -i \frac{d}{d\eta} + q_{\gamma}(\eta) \right) \qquad (2.18)$$

with

$$q_{\gamma}(\eta) := \beta^{-1}(2\gamma - 1)g \cos(\Phi^{-1}(\eta)). \qquad (2.19)$$

In the following, $q_{\gamma}$ is given by (2.19) and

$$\tilde{q}_{\gamma}(\eta) := \int_{0}^{\eta} q_{\gamma}(\eta') \, d\eta'. \qquad (2.20)$$

We claim that

$$\tilde{q}_{\gamma}(\Phi(\theta)) = \left(\gamma - \frac{1}{2}\right)\left(\theta - \beta^{-1}\Phi(\theta)\right). \qquad (2.21)$$

Indeed,

$$\frac{d}{d\theta}(\tilde{q}_{\gamma}(\Phi(\theta))) = q_{\gamma}(\Phi(\theta))\Phi'(\theta) = \frac{(2\gamma - 1)g \cos(\theta)}{\beta} \frac{\beta}{1 + 2g \cos(\theta)} = \\ = \frac{2\gamma - 1}{2} \left(1 - \frac{1}{1 + 2g \cos(\theta)}\right) = \left(\gamma - \frac{1}{2}\right)\left(1 - \beta^{-1}\Phi'(\theta)\right)$$

implies $\tilde{q}_{\gamma}(\Phi(\theta)) = (\gamma - \frac{1}{2})(\theta - \beta^{-1}\Phi(\theta)) + C_0$ and $C_0 = 0$ holds due to $\Phi(0) = 0$ and $\tilde{q}_{\gamma}(\Phi(0)) = \tilde{q}_{\gamma}(0) = 0$.

Using $\Phi(\pm\pi) = \pm\pi$ in (2.21), we compute

$$\langle q_{\gamma} \rangle := \frac{\tilde{q}_{\gamma}(\pi) - \tilde{q}_{\gamma}(-\pi)}{2\pi} = \left(\gamma - \frac{1}{2}\right)(1 - \beta^{-1}) \qquad (2.22)$$

and remark that $\eta \to \langle q_{\gamma} \rangle \eta - \tilde{q}_{\gamma}(\eta)$ is a smooth $2\pi$-periodic function. Then we define $(f_{q_{\gamma},j})_{j \in \mathbb{Z}}$ to be the orthonormal basis in $L^2(\mathbb{T})$ given by

$$f_{q_{\gamma},j}(\eta) = e^{ij\eta} e^{i\langle q_{\gamma} \rangle \eta - \tilde{q}_{\gamma}(\eta)} \qquad (2.23)$$
and (see [7], [10]), for every $j \in \mathbb{Z}$, we get

$$ \beta \left( -i \frac{d}{d\eta} + q_{\gamma} \right) f_{q_{\gamma}, j} = d_{\gamma, j} f_{q_{\gamma}, j}, \qquad (2.24) $$

where $d_{\gamma,j} = \beta(j + \langle q_{\gamma} \rangle) = \beta j + (\gamma - 1/2)(\beta - 1)$ is given (2.9). Combining (2.24) with (2.18), we obtain

**Corollary 2.4.** Let $\{f_{\gamma,j}^0\}_{j \in \mathbb{Z}}$ be the orthonormal basis of $L^2(\mathbb{T})$ given by

$$ f_{\gamma,j}^0(\theta) := (U_{\Phi} f_{q_{\gamma},j})(\theta) = \Phi'(\theta)^{1/2} e^{ij\Phi(\theta)} e^{i(\langle q_{\gamma} \rangle \Phi(\theta) - \tilde{q}_{\gamma}(\Phi(\theta)))}. \qquad (2.25) $$

Then

$$ L_{\gamma}^{0} f_{\gamma, j}^{0} = d_{\gamma, j} f_{\gamma, j}^{0} \qquad (2.26) $$

holds with $d_{\gamma,j}$ given (2.9).

It is clear that the assertion of Theorem 2.3(a) follows from Corollary 2.4. Indeed, using (2.26) and $\tilde{J}_{\gamma}^0 = \mathcal{F}_{\mathbb{T}}L_{\gamma}^0 \mathcal{F}_{\mathbb{T}}^{-1}$, we get

$$ \tilde{J}_{\gamma}^{0} u_{\gamma, j}^{0} = d_{\gamma, j} u_{\gamma, j}^{0} \text{ with } u_{\gamma, j}^{0} := \mathcal{F}_{\mathbb{T}} f_{\gamma, j}^{0}. \qquad (2.27) $$

### 2.3. An auxiliary estimate.

**Notation 2.5.** (a) For $f \in L^2(\mathbb{T})$ we write $T_{\pi}f := f \circ \tau_{\pi}$ with $\tau_{\pi} : \mathbb{T} \to \mathbb{T}$ given by

$$ \tau_{\pi}(\theta + 2\pi\mathbb{Z}) := \theta + \pi + 2\pi\mathbb{Z} \qquad (2.28) $$

(b) For $j, n \in \mathbb{Z}$, we denote

$$ V_{\gamma}^{\delta}(j, n) := \delta \langle f_{\gamma, j}^{0}, T_{\pi} f_{\gamma, n}^{0} \rangle_{L^{2}(\mathbb{T})}, \qquad (2.29) $$

where $f_{\gamma,j}^0$ is given by (2.25).

**Lemma 2.6.** Let $\hat{c} > 0$ be small enough. Then there exists $\hat{C} > 0$ such that

$$ \sup_{\{k \in \mathbb{Z} : |k| \le \hat{c} |n|\}} |V_{\gamma}^{\delta}(n+k, n)| \le \hat{C} |n|^{-1/2} \qquad (2.30) $$

holds for every $n \in \mathbb{Z} \setminus \{0\}$.

**Proof.** Let us assume that $n \ne 0$. By definition (2.25), $f_{\gamma,n}^0 = e^{in\Phi}p_{\gamma}$ holds with

$$ p_{\gamma}(\theta) := e^{i\langle q_{\gamma} \rangle \Phi(\theta) - i\tilde{q}_{\gamma}(\Phi(\theta))} \beta^{1/2} (1 + 2g \cos \theta)^{-1/2} \qquad (2.31) $$

and we can express

$$ V_{\gamma}^{\delta}(n+k, n) = \delta \int_{-\pi}^{\pi} e^{in\Psi_{n,k}(\theta)} p_{\gamma}(\theta) \overline{(T_{\pi}p_{\gamma})(\theta)} \frac{d\theta}{2\pi} \qquad (2.32) $$

with

$$ \Psi_{n,k} = \Phi - T_{\pi}\Phi + \frac{k}{n}\Phi. \qquad (2.33) $$

Denote $\Psi := \Phi - T_{\pi}\Phi$. Then

$$ \Psi'(\theta) = \frac{\beta}{1 + 2g \cos \theta} - \frac{\beta}{1 - 2g \cos \theta} = 0 \Leftrightarrow \cos \theta = 0, \qquad (2.34) $$

i.e. $\Psi$ has two critical points: $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. Since

$$ \Psi''(\theta) = \left( \frac{2g\beta}{(1 + 2g \cos \theta)^2} + \frac{2g\beta}{(1 - 2g \cos \theta)^2} \right) \sin \theta, \qquad (2.35) $$
we obtain $\Psi''(\pm\frac{\pi}{2}) = \pm 4g\beta \neq 0$ and choosing $c_0 > 0$ small enough we can ensure

$$|\Psi''(\theta)| \geq c_0 \text{ for } \theta \in \mathcal{I}_{c_0} := [-\frac{\pi}{2} - c_0, -\frac{\pi}{2} + c_0] \cup [\frac{\pi}{2} - c_0, \frac{\pi}{2} + c_0].$$

However $\Psi_{n,k} = \Psi + \frac{k}{n}\Phi$ and denoting $C_0 = \sup |\Phi''|$ we can estimate

$$|\Psi''_{n,k}(\theta)| \geq |\Psi''(\theta)| - C_0 \left|\frac{k}{n}\right| \qquad (2.36)$$
$$\left|\frac{k}{n}\right| \leq \frac{c_0}{2C_0} \implies |\Psi''_{n,k}(\theta)| \geq \frac{c_0}{2} \text{ for } \theta \in \mathcal{I}_{c_0}.$$

Due to van der Corput Lemma (see Lemma 7.1(b)), there exists $C > 0$ such that

$$\left| \int_{\mathcal{I}_{c_0}} e^{in\Phi_{n,k}(\theta)} p_{\gamma}(\theta) \overline{(T_{\pi}p_{\gamma})(\theta)} \frac{d\theta}{2\pi} \right| \leq C|n|^{-1/2}. \qquad (2.37)$$

Moreover

$$c_1 := \inf\{|\Psi'(\theta)| : \theta \in [-\pi, \pi] \setminus \mathcal{I}_{c_0}\} > 0$$

and denoting $C_1 = \sup |\Phi'|$ we can estimate

$$|\Psi'_{n,k}(\theta)| \geq |\Psi'(\theta)| - C_1 \left|\frac{k}{n}\right|,$$

hence

$$\left|\frac{k}{n}\right| \leq \frac{c_1}{2C_1} \implies |\Psi'_{n,k}(\theta)| \geq \frac{c_1}{2} \text{ for } \theta \in [-\pi, \pi] \setminus \mathcal{I}_{c_0}. \qquad (2.38)$$

Due to the non-stationary phase estimate (see Lemma 7.1(a)), there exists a constant $C' > 0$ such that

$$\left| \int_{[-\pi, \pi] \setminus \mathcal{I}_{c_0}} e^{in\Phi_{n,k}(\theta)} p_{\gamma}(\theta) \overline{(T_{\pi}p_{\gamma})(\theta)} \frac{d\theta}{2\pi} \right| \leq C'|n|^{-1}. \qquad (2.39)$$

Combining (2.37), (2.39) and (2.32), we complete the proof. $\square$

### 3. THE GRWA APPROCH

**3.1. Statement of the result.** The idea of GRWA consists in using the diagonal entries of a perturbation as the first correction for eigenvalues of a perturbed diagonal matrix (see [18] and [14]). Our proof of Theorem 2.3 is based on the following variant of the GRWA :

**Proposition 3.1.** Let us fix $\beta_0 \in \mathbb{R}$, $\beta > 0$, $\hat{C} > 0$ and $\hat{c} > 0$. Let $J : \ell^{2,1}(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be given by the formula

$$J = \beta\Lambda + \beta_0 + V, \qquad (3.1)$$

where $\Lambda = \text{diag}(j)_{j \in \mathbb{Z}}$ and $V$ is a bounded self-adjoint operator in $\ell^2(\mathbb{Z})$ satisfying

$$\sup_{\{k \in \mathbb{Z} : |k| \leq \hat{c}|n|\}} |\langle \tilde{e}_{n+k}, V \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})}| \leq \hat{C}|n|^{-1/2} \qquad (3.2)$$

for every $n \in \mathbb{Z} \setminus \{0\}$. Assume that the spectrum of $J$ is composed of a non-decreasing sequence of eigenvalues $\{\lambda_j(J)\}_{j \in \mathbb{Z}}$ such that

$$\lambda_j(J) = j\beta + \beta_0 + o(1) \text{ as } j \to \infty. \qquad (3.3)$$

Then one has

$$\lambda_j(J) = j\beta + \beta_0 + \langle \tilde{e}_j, V\tilde{e}_j \rangle_{\ell^2(\mathbb{Z})} + O(j^{-1} \ln j) \text{ as } j \to \infty \qquad (3.4)$$
Proof. See Section 5.

□

3.2. **Proof of Theorem 2.3(b).** In what follows, we describe how to deduce the assertion of Theorem 2.3(b) from Proposition 3.1. We begin by introducing $\tilde{U}_{\gamma}$, the unitary operator in $\ell^2(\mathbb{Z})$ satisfying $\tilde{U}_{\gamma}\tilde{e}_j = u_{\gamma,j}^0$, where $\{\tilde{e}_j\}_{j \in \mathbb{Z}}$ is the canonical basis of $\ell^2(\mathbb{Z})$ and $\{u_{\gamma,j}^0\}_{j \in \mathbb{Z}}$ is the basis introduced in (2.27). We claim that the hypotheses of Proposition 3.1 are satisfied if

$$J = \tilde{U}_{\gamma}^{-1} \tilde{J}_{\gamma}^{\delta} \tilde{U}_{\gamma}, \qquad (3.5)$$

where $J_{\gamma}^{\delta}$ is as in Definition 2.2. Indeed, if $\beta$ is given by (1.15) and

$$\beta_0 = \left(\gamma - \frac{1}{2}\right)(\beta - 1), \qquad (3.6)$$

then $\tilde{U}_{\gamma}^{-1} \tilde{J}_{\gamma}^{\delta} \tilde{U}_{\gamma} = \beta\Lambda + \beta_0 + V_{\gamma}^{\delta}$ holds with

$$V_{\gamma}^{\delta} := \tilde{U}_{\gamma}^{-1} D_{\delta} \tilde{U}_{\gamma}, \qquad (3.7)$$

where $D_{\delta} = \text{diag}(\delta(-1)^j)_{j \in \mathbb{Z}}$ (see (2.6)). Therefore

$$\langle \tilde{e}_j, V_{\gamma}^{\delta} \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})} = \langle u_{j,\gamma}^0, D_{\delta} u_{j,\gamma}^0 \rangle_{\ell^2(\mathbb{Z})} = \langle f_{j,\gamma}^0, \mathcal{F}_{\mathbb{T}}^{-1} D_{\delta} \mathcal{F}_{\mathbb{T}} f_{j,\gamma}^0 \rangle_{\text{L}^2(\mathbb{T})}, \qquad (3.8)$$

where we used $u_{j,\gamma}^0 = \mathcal{F}_{\mathbb{T}} f_{j,\gamma}^0$ and the isometry $\mathcal{F}_{\mathbb{T}}$. Since $\mathcal{F}_{\mathbb{T}}^{-1} D_{\delta} \mathcal{F}_{\mathbb{T}} = \delta T_{\pi}$ holds with $T_{\pi}$ as in Notation 2.5, (3.8) gives

$$\langle \tilde{e}_j, V_{\gamma}^{\delta} \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})} = V_{\gamma}^{\delta}(j, k) \qquad (3.9)$$

with $V_{\gamma}^{\delta}(\cdot, \cdot)$ expressed by (2.29) and due to Lemma 2.6, the estimate (3.2) holds if $V = V_{\gamma}^{\delta}$ is given by (3.7). Moreover, the result of [6, Proposition 5.1] ensures the fact that the spectrum of $\tilde{J}_{\gamma}^{\delta}$ is composed of the non-decreasing sequence of eigenvalues $\{\lambda_j(\tilde{J}_{\gamma}^{\delta})\}_{j \in \mathbb{Z}}$, which can be labeled so that one has the estimate

$$\lambda_j(\tilde{J}_{\gamma}^{\delta}) = j\beta + \beta_0 + O(j^{-1/3}) \text{ as } j \to \infty, \qquad (3.10)$$

i.e. (3.3) holds if $J$ is given by (3.5). Thus all the hypotheses of Proposition 3.1 hold and choosing $\{\lambda_j(\tilde{J}_{\gamma}^{\delta})\}_{j \in \mathbb{Z}}$ to be the sequence of eigenvalues of $\tilde{J}_{\gamma}^{\delta}$ such that $\lambda_j(\tilde{J}_{\gamma}^{\delta}) = \lambda_j(J)$ for every $j \in \mathbb{Z}$, we obtain

$$\lambda_j(\tilde{J}_{\gamma}^{\delta}) = j\beta + \beta_0 + \langle \tilde{e}_j, V_{\gamma}^{\delta} \tilde{e}_j \rangle_{\ell^2(\mathbb{Z})} + O(j^{-1} \ln j) \text{ when } j \to \infty \qquad (3.11)$$

due to (3.4). The assertion of Theorem 2.3(b) follows from (3.11) and

$$\langle \tilde{e}_j, V_{\gamma}^{\delta} \tilde{e}_j \rangle_{\ell^2(\mathbb{Z})} = \mathbf{r}_{\gamma}^{\delta}(j) + O(j^{-1}) \text{ when } j \to \infty. \qquad (3.12)$$

where $\mathbf{r}_{\gamma}^{\delta}$ is given by (2.11)-(2.12). The proof of (3.12) is given in Section 3.3.

3.3. **Proof of (3.12).** Due to (2.32),

$$V_{\gamma}^{\delta}(n, n) = \delta \int_{-\pi}^{\pi} e^{in\Psi(\theta)} h_{\gamma}(\theta) \frac{d\theta}{2\pi}, \qquad (3.13)$$

where $\Psi := \Phi - T_{\pi}\Phi$ and $h_{\gamma}(\theta) = p_{\gamma}(\theta) \overline{(T_{\pi}p_{\gamma})(\theta)}$ with $p_{\gamma}$ given by (2.31). Since $\pm \frac{\pi}{2}$ are non-degenerated critical points of $\Psi$ (see (2.34), (2.35))), the stationary phase formula gives

$$V_{\gamma}^{\delta}(n, n) = \delta \sum_{\nu=\pm 1} \frac{h_{\gamma}(\nu\pi/2) e^{in\Psi(\nu\pi/2) + i\nu\pi/4}}{\sqrt{2\pi n |\Psi''(\nu\pi/2)|}} + O(n^{-1}). \qquad (3.14)$$
Since $\Phi$ is odd, we get

$$ \Psi(\pm\pi/2) = \Phi(\pm\pi/2) - \Phi(\mp\pi/2) = 2\Phi(\pm\pi/2) = \pm 2\Phi(\pi/2) = \pm 4\alpha \quad (3.15) $$

where (2.16) was used to write $\Phi(\pi/2) = 2\alpha$ with $\alpha$ given by (1.16). By definition (2.31), one has $p_{\gamma}(\pi/2) = \sqrt{\beta} e^{i\theta_{\gamma}}$ with

$$ \tilde{\theta}_{\gamma} := \langle q_{\gamma} \rangle \Phi(\pi/2) - \tilde{q}_{\gamma}(\Phi(\pi/2)) \quad (3.16) $$

and $\tilde{q}_{\gamma} \circ \Phi$ is odd, hence $p_{\gamma}(-\pi/2) = \sqrt{\beta} e^{-i\tilde{\theta}_{\gamma}}$ and $h_{\gamma}(\pi/2) = \beta e^{2i\tilde{\theta}_{\gamma}}$. Moreover, one has $h_{\gamma}(-\pi/2) = \tilde{h}_{\gamma}(\pi/2)$, hence

$$ h_{\gamma}(\pm\pi/2) = \beta e^{\pm 2i\tilde{\theta}_{\gamma}} \quad (3.17) $$

and we get two conjugated terms corresponding to $\nu = \pm 1$ in (3.14). Therefore, using (3.17), (3.15) and $|\Psi''(\pm\pi/2)| = 4g\beta$ in (3.14), we get

$$ \begin{aligned} V_{\gamma}^{\delta}(n, n) &= \delta \frac{2\operatorname{Re}(\beta e^{2i\tilde{\theta}_{\gamma}} e^{i4n\alpha + i\pi/4})}{\sqrt{2\pi n \cdot 4g\beta}} + O(n^{-1}) \\ &= \delta \frac{\sqrt{\beta} \cos(4n\alpha + 2\tilde{\theta}_{\gamma} + \pi/4)}{\sqrt{2\pi n g}} + O(n^{-1}). \end{aligned} $$

To complete the proof of (3.12), it remains to check that $\hat{\theta}_{\gamma} := 2\tilde{\theta}_{\gamma} + \pi/4$ is given by (2.12). However, (2.22) gives

$$ \langle q_{\gamma} \rangle \Phi(\pi/2) = \left(\gamma - \frac{1}{2}\right) \left(\Phi(\pi/2) - \beta^{-1}\Phi(\pi/2)\right), \quad (3.18) $$

(2.21) gives

$$ -\tilde{q}_{\gamma}(\Phi(\pi/2)) = \left(\gamma - \frac{1}{2}\right) \left(\beta^{-1}\Phi(\pi/2) - \pi/2\right), \quad (3.19) $$

and summing up (3.18), (3.19), we find that (3.16) gives

$$ \tilde{\theta}_{\gamma} = \left(\gamma - \frac{1}{2}\right) \left(\Phi(\pi/2) - \pi/2\right) = \left(\gamma - \frac{1}{2}\right) \left(2\alpha - \pi/2\right) \quad (3.20) $$

and $\hat{\theta}_{\gamma} = 2\tilde{\theta}_{\gamma} + \frac{\pi}{4} = (\gamma - \frac{1}{2})(4\alpha - \pi) + \frac{\pi}{4}$ as claimed in (2.12).

### 4. AUXILIARY OPERATORS $Q_n$

4.1. **Definition of $Q_n$.** Our aim is to introduce a sequence of auxiliary self-adjoint operators $\{Q_n\}_{n \in \mathbb{N}^*}$ which will be used in Section 5 to estimate $\lambda_n(J)$ thanks to an analysis of the operator

$$ J'_n := e^{-iQ_n} J e^{iQ_n}. \quad (4.1) $$

**Definition 4.1.** Let $V$ be a self-adjoint bounded operator in $\ell^2(\mathbb{Z})$ and denote

$$ V(j, k) := (V \tilde{e}_k)(j) = \langle \tilde{e}_j, V \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})} \quad \text{for } (j, k) \in \mathbb{Z}^2. $$

For $n \in \mathbb{N}^*$ define $(Q_n(j, k))_{(j,k) \in \mathbb{Z}^2}$ by

$$ Q_n(j, k) = \begin{cases} i \frac{V(j, n)}{\beta(j - n)} & \text{when } j \neq n \text{ and } k = n \\ i \frac{V(n, k)}{\beta(n - k)} & \text{when } j = n \text{ and } k \neq n \\ 0 & \text{otherwise} \end{cases} \quad (4.2) $$
Then

$$Q_n(j, k) \neq 0 \Rightarrow \text{ (either } (j \neq n \text{ and } k = n) \text{ or } (j = n \text{ and } k \neq n)\text{)} \quad (4.3)$$

and the relation

$$(Q_n \tilde{e}_k)(j) = \langle \tilde{e}_j, Q_n \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})} = Q_n(j, k) \text{ for } (j, k) \in \mathbb{Z}^2$$

defines in $\ell^2(\mathbb{Z})$ a self-adjoint Hilbert-Schmidt operator $Q_n$ due to

$$\sum_{(j,k) \in \mathbb{Z}^2} |Q_n(j,k)|^2 \le \frac{2}{\beta^2} \sum_{j \in \mathbb{Z}} |V(j,n)|^2 = \frac{2}{\beta^2} \|V \tilde{e}_n\|_{\ell^2(\mathbb{Z})}^2 \le \frac{2}{\beta^2} \|V\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}^2$$

**Notation 4.2.** If $n \in \mathbb{Z}$ then $\hat{\Pi}_n \in \mathcal{B}(\ell^2(\mathbb{Z}))$ (respectively $\tilde{\Pi}_n \in \mathcal{B}(\ell^2(\mathbb{Z}))$) denotes the orthogonal projection on $\mathcal{C}\tilde{e}_n$ (respectively on $(\mathcal{C}\tilde{e}_n)^\perp$), i.e.

$$\hat{\Pi}_n x := \langle \tilde{e}_n, x \rangle_{\ell^2(\mathbb{Z})} \tilde{e}_n = x(n) \tilde{e}_n \quad \text{and} \quad \tilde{\Pi}_n := I - \hat{\Pi}_n.$$

**4.2. Properties of $Q_n$.** We observe that (4.3) implies

$$\hat{\Pi}_n Q_n \hat{\Pi}_n = 0 = \tilde{\Pi}_n Q_n \tilde{\Pi}_n \quad (4.4)$$

and

$$Q_n = \tilde{\Pi}_n Q_n \hat{\Pi}_n + \hat{\Pi}_n Q_n \tilde{\Pi}_n. \quad (4.5)$$

More properties of $Q_n$ are given in

**Lemma 4.3.** Assume that $V$ satisfies the assumption of Proposition 3.1, i.e. $V$ is a bounded self-adjoint operator in $\ell^2(\mathbb{Z})$ satisfying

$$|j - n| \le \hat{c}|n| \Rightarrow |V(j, n)| \le \hat{C}|n|^{-1/2}, \quad (4.6)$$

where $\hat{C}$, $\hat{c} > 0$ are independent of $n$. Let $Q_n \in \mathcal{B}(\ell^2(\mathbb{Z}))$ be as in Definition 4.1. Then (a) one has large $n$ estimate

$$\|Q_n\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} = O(n^{-1/2}) \quad (4.7)$$

(b) $\ell^{2,1}(\mathbb{Z})$ is an invariant subspace of $Q_n$ and for every $x \in \ell^{2,1}(\mathbb{Z})$,

$$i\beta[Q_n, \Lambda]x = i\beta(Q_n\Lambda - \Lambda Q_n)x = V_n x \quad (4.8)$$

holds with

$$V_n := \tilde{\Pi}_n V \hat{\Pi}_n + \hat{\Pi}_n V \tilde{\Pi}_n. \quad (4.9)$$

(c) $\ell^{2,1}(\mathbb{Z})$ is an invariant subspace of $e^{itQ_n}$ and $t \to e^{itQ_n}x$ is of class $C^\infty(\mathbb{R}; \ell^{2,1}(\mathbb{Z}))$ if $x \in \ell^{2,1}(\mathbb{Z})$.

**Proof.** (a) By definition of $Q_n$,

$$\|Q_n \tilde{e}_n\|_{\ell^2(\mathbb{Z})}^2 = \sum_{j \in \mathbb{Z} \setminus \{n\}} |Q_n(j, n)|^2 = \sum_{j \in \mathbb{Z} \setminus \{n\}} \frac{|V(j, n)|^2}{\beta^2 (j - n)^2} \le \beta^{-2} (M_n + M'_n)$$

holds with

$$M_n := \sum_{\{j \in \mathbb{Z}: 0 < |j-n| \le \hat{c}n\}} \frac{|V(j, n)|^2}{(j-n)^2} \le \sum_{m \in \mathbb{Z} \setminus \{0\}} \frac{\hat{C}^2 n^{-1}}{m^2} = C_1 n^{-1}$$

due to (4.6) and

$$M'_n := \sum_{\{j \in \mathbb{Z}: |j-n| > \hat{c}n\}} \frac{|V(j, n)|^2}{(j-n)^2} \le \sum_{j \in \mathbb{Z}} \frac{|V(j, n)|^2}{\hat{c}^2 n^2} = \frac{\|V \tilde{e}_n\|_{\ell^2(\mathbb{Z})}^2}{\hat{c}^2 n^2} \le C_2 n^{-2}.$$
This shows that $||Q_n \hat{\Pi}_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} = O(n^{-1/2})$. Moreover, due to (4.5),

$$ ||Q_n \tilde{\Pi}_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} = ||\hat{\Pi}_n Q_n \tilde{\Pi}_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le ||\hat{\Pi}_n Q_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} = ||Q_n \hat{\Pi}_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} $$

is $O(n^{-1/2})$, which completes the proof of (4.7).

(b) Let $V_n$ be given by (4.9). Then one has

$$ V_n(j, k) := (V_n \tilde{e}_k)(j) = \langle \tilde{e}_j, V_n \tilde{e}_k \rangle_{\ell^2(\mathbb{Z})} = \begin{cases} V(j, n) & \text{when } j \ne n \text{ and } k = n \\ V(n, k) & \text{when } j = n \text{ and } k \ne n \\ 0 & \text{otherwise} \end{cases} \quad (4.10) $$

and combining (4.10) with (4.2) we obtain

$$ i\beta(j-k)Q_n(j,k) = -V_n(j,k) \text{ for every } j,k \in \mathbb{Z}. \quad (4.11) $$

However, (4.11) implies

$$ i\beta\left(j + \frac{1}{2}\right)(Q_n \tilde{e}_k)(j) = i\beta\left(Q_n\left(k + \frac{1}{2}\right)\tilde{e}_k\right)(j) - (V_n \tilde{e}_k)(j) \quad (4.12) $$

and ensures $Q_n \tilde{e}_k \in \ell^{2,1}(\mathbb{Z})$. Moreover, (4.12) ensures the equality

$$ i\beta\left(\Lambda + \frac{1}{2}\right)Q_n x = i\beta Q_n\left(\Lambda + \frac{1}{2}\right)x - V_n x \quad (4.13) $$

for $x \in \ell_{\text{fin}}^2(\mathbb{Z})$. If $y \in \ell_{\text{fin}}^2(\mathbb{Z})$, then using (4.13) with $x = (\Lambda + \frac{1}{2})^{-1}y$, we obtain

$$ i\beta\left(\Lambda + \frac{1}{2}\right)Q_n\left(\Lambda + \frac{1}{2}\right)^{-1}y = i\beta Q'_n y \quad (4.14) $$

with $Q'_n \in \mathcal{B}(\ell^2(\mathbb{Z}))$ given by

$$ Q'_n := Q_n - (i\beta)^{-1}V_n\left(\Lambda + \frac{1}{2}\right)^{-1}. \quad (4.15) $$

Due to (4.14), for every $y \in \ell_{\text{fin}}^2(\mathbb{Z})$ one has

$$ i\beta Q_n \left( \Lambda + \frac{1}{2} \right)^{-1} y = i\beta \left( \Lambda + \frac{1}{2} \right)^{-1} Q'_n y \quad (4.16) $$

and by continuity, (4.16) holds for every $y \in \ell^2(\mathbb{Z})$. Consider now $x \in \ell^{2,1}(\mathbb{Z})$ and take $y = (\Lambda + \frac{1}{2})x$ in (4.16). This gives

$$ i\beta Q_n x = i\beta \left(\Lambda + \frac{1}{2}\right)^{-1} Q'_n \left(\Lambda + \frac{1}{2}\right) x \in \ell^{2,1}(\mathbb{Z}) \quad (4.17) $$

and

$$ i\beta\left(\Lambda + \frac{1}{2}\right)Q_n x = i\beta Q'_n\left(\Lambda + \frac{1}{2}\right)x = i\beta Q_n\left(\Lambda + \frac{1}{2}\right)x - V_n x, \quad (4.18) $$

implying (4.8) for every $x \in \ell^{2,1}(\mathbb{Z})$.

(c) We first observe that (4.16) holds for every $y \in \ell^2(\mathbb{Z})$ and implies (4.14) for every $y \in \ell^2(\mathbb{Z})$. Since (4.14) implies $(Q'_n)^m y = (\Lambda + \frac{1}{2})(Q_n)^m (\Lambda + \frac{1}{2})^{-1}y$ for every $m \in \mathbb{N}$ and $y \in \ell^2(\mathbb{Z})$, we have

$$ \begin{aligned} e^{itQ_n} \left(\Lambda + \frac{1}{2}\right)^{-1} y &= \lim_{N \to \infty} \sum_{m=0}^{N} \frac{(it)^m}{m!} (Q_n)^m \left(\Lambda + \frac{1}{2}\right)^{-1} y = \\ &= \lim_{N \to \infty} \sum_{m=0}^{N} \frac{(it)^m}{m!} \left(\Lambda + \frac{1}{2}\right)^{-1} (Q'_n)^m y = \left(\Lambda + \frac{1}{2}\right)^{-1} e^{itQ'_n} y \in \ell^{2,1}(\mathbb{Z}) \end{aligned} $$

if $y \in \ell^2(\mathbb{Z})$ and setting $x = (\Lambda + \frac{1}{2})^{-1}y$, we find that $t \to (\Lambda + \frac{1}{2})e^{itQ_n}x = e^{itQ'_n}y$ is $C^\infty(\mathbb{R}; \ell^2(\mathbb{Z}))$ for every $x \in \ell^{2,1}(\mathbb{Z})$. $\square$
5. PROOF OF PROPOSITION 3.1

5.1. **Taylor's expansion formula.** We will investigate $J'_n = e^{-iQ_n}Je^{iQ_n}$, using a special version of the Taylor's expansion formula. Assume that $B$ and $Q \in \mathcal{B}(\ell^2(\mathbb{Z}))$ and denote

$$F_{tQ}(B) := e^{-itQ} B e^{itQ} \quad \text{for } t \in \mathbb{R}, \qquad (5.1)$$

$$\mathrm{ad}_{iQ}^{1}(B) := [B, iQ] = i(BQ - QB)$$

and

$$\mathrm{ad}_{iQ}^{m+1}(B) := [\mathrm{ad}_{iQ}^{m}(B), iQ] \quad \text{for } m \in \mathbb{N}^*.$$

Then

$$\frac{\mathrm{d}^m}{\mathrm{d}t^m} F_{tQ}(B) = e^{-itQ} \mathrm{ad}_{iQ}^m(B) e^{itQ} = F_{tQ}(\mathrm{ad}_{iQ}^m(B)) \qquad (5.2)$$

and the Taylor formula gives

$$F_{tQ}(B) = B + \sum_{m=1}^{N-1} \frac{t^m}{m!} \mathrm{ad}_{iQ}^{m}(B) + \mathcal{R}_{Q}^{t,N}(B) \qquad (5.3)$$

with

$$\mathcal{R}_{Q}^{t,N}(B) := \frac{t^N}{(N-1)!} \int_{0}^{1} F_{stQ}(\mathrm{ad}_{iQ}^{N}(B)) (1-s)^{N-1} \mathrm{d}s. \qquad (5.4)$$

We can also consider the case when $B$ is an unbounded symmetric operator in $\ell^2(\mathbb{Z})$, defined on a dense domain $\mathcal{D}(B)$. Suppose that $\mathcal{D}(B)$ is an invariant subspace for $Q$ and $e^{itQ}$ for every $t \in \mathbb{R}$. If $t \to B e^{itQ}x$ is $C^1(\mathbb{R}, \ell^2(\mathbb{Z}))$ for every $x \in \mathcal{D}(B)$, then

$$\frac{\mathrm{d}}{\mathrm{d}t} \langle e^{itQ}x, Be^{itQ}y \rangle_{\ell^2(\mathbb{Z})} = \langle Be^{itQ}x, iQe^{itQ}y \rangle_{\ell^2(\mathbb{Z})} + \langle iQe^{itQ}x, Be^{itQ}y \rangle_{\ell^2(\mathbb{Z})}$$

holds for every $x, y \in \mathcal{D}(B)$. If the form $(x, y) \to \langle Bx, iQy \rangle_{\ell^2(\mathbb{Z})} + \langle iQx, By \rangle_{\ell^2(\mathbb{Z})}$ can be extended from $\mathcal{D}(B) \times \mathcal{D}(B)$ to a bounded form on $\ell^2(\mathbb{Z}) \times \ell^2(\mathbb{Z})$, then we can introduce $[B, iQ] \in \mathcal{B}(\ell^2(\mathbb{Z}))$ defined by this form and we can write

$$\frac{\mathrm{d}}{\mathrm{d}t} F_{tQ}(B) = e^{-itQ} [B, iQ] e^{itQ} = F_{tQ}([B, iQ]),$$

hence (5.3)-(5.4) still hold for every $N \in \mathbb{N}^*$.

5.2. **Step 1 of the proof of Proposition 3.1.**

**Notation 5.1.** If $L_n, L'_n$ are linear maps defined on a dense subspace of the Banach space $\mathcal{V}$ and $L_n - L'_n$ can be extended to a bounded operator on $\mathcal{V}$, then $||L_n - L'_n||_{\mathcal{B}(\mathcal{V})}$ denotes the norm of this extension and the notation $L_n = L'_n + O(\eta_n)$ means that choosing $n_0 \in \mathbb{N}$ and $C > 0$ large enough, we ensure $||L_n - L'_n||_{\mathcal{B}(\mathcal{V})} \le C\eta_n$ for all $n \ge n_0$. If $x_n, x'_n \in \mathcal{V}$, then we write $x_n = y_n + O(\eta_n)$ if and only if we can find $C > 0$, $n_0 \in \mathbb{N}$, such that $||x_n - x'_n||_\mathcal{V} \le C\eta_n$ holds for all $n \ge n_0$.

In what follows, $\mathcal{V}$ satisfies the hypotheses of Proposition 3.1 and without any loss of generality we assume $\beta_0 = 0$, i.e. $J = \beta\Lambda + V$. We consider

$$J'_{n} = e^{-iQ_{n}} (\beta\Lambda + V) e^{iQ_{n}} \qquad (5.5)$$

and claim the large $n$ estimate

$$J'_{n} = \beta\Lambda + V - V_{n} + [V - \frac{1}{2}V_{n}, iQ_{n}] + O(n^{-1}), \qquad (5.6)$$
where $V_n = \tilde{\Pi}_n V \hat{\Pi}_n + \hat{\Pi}_n V \tilde{\Pi}_n$ is given by (4.9).

We first use (5.3)-(5.4) with $N=2$ and write

$$e^{-iQ_n} V e^{iQ_n} = V + [V, iQ_n] + \mathcal{R}_{Q_n}^{1,2}(V), \qquad (5.7)$$

$$\|\mathcal{R}_{Q_n}^{1,2}(V)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le \|\mathrm{ad}_{iQ_n}^2(V)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le C \|Q_n\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}^2. \qquad (5.8)$$

We also use (5.3)-(5.4) with $N=3$ and write

$$e^{-iQ_n} \beta \Lambda e^{iQ_n} = \beta \Lambda + [\beta \Lambda, iQ_n] + \frac{1}{2} [[\beta \Lambda, iQ_n], iQ_n] + \mathcal{R}_{Q_n}^{1,3}(\beta \Lambda), \qquad (5.9)$$

$$\|\mathcal{R}_{Q_n}^{1,3}(\beta\Lambda)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le \|\mathrm{ad}_{iQ_n}^3(\beta\Lambda)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}. \qquad (5.10)$$

Using $[\beta\Lambda, iQ_n] = -V_n$ (see Lemma 4.3b) in (5.9)-(5.10), we obtain

$$e^{-iQ_n} \beta \Lambda e^{iQ_n} = \beta \Lambda - V_n - \left[\frac{1}{2} V_n, iQ_n\right] + \mathcal{R}_{Q_n}^{1,3}(\beta \Lambda), \qquad (5.11)$$

$$\|\mathcal{R}_{Q_n}^{1,3}(\beta\Lambda)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le \|\mathrm{ad}_{iQ_n}^2(V_n)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le C \|Q_n\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}^2. \qquad (5.12)$$

and summing up (5.11), (5.7), we obtain

$$e^{-iQ_n} J e^{iQ_n} = \beta \Lambda + V - V_n + \left[V - \frac{1}{2} V_n, iQ_n\right] + \mathcal{R}_{Q_n}^{1,2}(V) + \mathcal{R}_{Q_n}^{1,3}(\beta \Lambda), \qquad (5.13)$$

$$\|\mathcal{R}_{Q_n}^{1,2}(V) + \mathcal{R}_{Q_n}^{1,3}(\beta\Lambda)\|_{\mathcal{B}(\ell^2(\mathbb{Z}))} \le C \|Q_n\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}^2. \qquad (5.14)$$

To complete the proof, we observe that $\|Q_n\|_{\mathcal{B}(\ell^2(\mathbb{Z}))}^2 = O(n^{-1})$ follows from (4.7).

### 5.3. Step 2 of the proof of Proposition 3.1. We denote

$$W_n := [V - \frac{1}{2}V_n, iQ_n] \qquad (5.15)$$

and we claim that the following estimate

$$W_n(n, n) := (W_n \tilde{e}_n)(n) = \langle \tilde{e}_n, W_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} = O(n^{-1} \ln n) \qquad (5.16)$$

holds when $n \to \infty$.

Indeed, $W_n = [V'_n, iQ_n]$ holds with $V'_n := V - \frac{1}{2}V_n$ and

$$|V'_n(j, k)| \le 2|V(j, k)| \quad \text{for all } j, k \in \mathbb{Z}, \qquad (5.17)$$

hence we can estimate

$$|(V'_n Q_n)(n, n)| \le \sum_{j \in \mathbb{Z}} 2|V(n, j) Q_n(j, n)| \le$$

$$\le \sum_{j \in \mathbb{Z} \setminus \{n\}} \frac{2|V(j, n)|^2}{\beta|j - n|} \le 2\beta^{-1}(M_n + M'_n)$$

with

$$M_n := \sum_{\{j \in \mathbb{Z}: 0 < |j-n| \le \hat{c}n\}} \frac{|V(j, n)|^2}{|j-n|} \le \sum_{\{m \in \mathbb{Z}: 0 < |m| \le \hat{c}n\}} \frac{\hat{C}^2 n^{-1}}{|m|} \le C_1 n^{-1} \ln n$$

due to (4.6) and

$$M'_n := \sum_{\{j \in \mathbb{Z}: |j-n| > \hat{c}n\}} \frac{|V(j, n)|^2}{|j-n|} \le \sum_{j \in \mathbb{Z}} \frac{|V(j, n)|^2}{\hat{c}n} = \frac{\|V \tilde{e}_n\|_{\ell^2(\mathbb{Z})}^2}{\hat{c}n} \le C_2 n^{-1}.$$
5.4. **Step 3 of the proof of Proposition 3.1.** Let $J'_n$ be as above and denote

$$d'_n := \beta n + V(n, n). \qquad (5.18)$$

We claim that the following estimates

$$\langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} = d'_n + O(n^{-1} \ln n) \qquad (5.19)$$

$$(J'_n - \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})}) \tilde{e}_n = O(n^{-1/2}) \qquad (5.20)$$

hold when $n \to \infty$.

Indeed, writing

$$\begin{aligned} \beta \Lambda + V - V_n &= \beta \Lambda + V - \tilde{\Pi}_n V \tilde{\Pi}_n - \tilde{\Pi}_n V \tilde{\Pi}_n \\ &= \beta \Lambda + \tilde{\Pi}_n V \tilde{\Pi}_n + \tilde{\Pi}_n V \tilde{\Pi}_n = \beta \Lambda + V(n, n) \tilde{\Pi}_n + \tilde{\Pi}_n V \tilde{\Pi}_n \end{aligned}$$

in (5.6), we find

$$J'_n \tilde{e}_n = (\beta \Lambda + V(n, n) \tilde{\Pi}_n + \tilde{\Pi}_n V \tilde{\Pi}_n + W_n) \tilde{e}_n + O(n^{-1}), \qquad (5.21)$$

where $W_n = [V - \frac{1}{2}V_n, iQ_n]$ is given by (5.15). However (5.21) implies

$$J'_n \tilde{e}_n = d'_n \tilde{e}_n + W_n \tilde{e}_n + O(n^{-1}), \qquad (5.22)$$

hence

$$\langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} = d'_n + \langle \tilde{e}_n, W_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} + O(n^{-1}) \qquad (5.23)$$

and (5.19) follows from (5.16).

In order to prove (5.20), we observe that (5.19) implies

$$(J'_n - \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})}) \tilde{e}_n = (J'_n - d'_n) \tilde{e}_n + O(n^{-1} \ln n) \qquad (5.24)$$

and (5.22) ensures

$$(J'_n - d'_n) \tilde{e}_n = W_n \tilde{e}_n + O(n^{-1}). \qquad (5.25)$$

Thus (5.20) follows from (5.24), (5.25) and

$$||W_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} \leq C ||Q_n||_{\mathcal{B}(\ell^2(\mathbb{Z}))} = O(n^{-1/2}).$$

5.5. **End of the proof of Proposition 3.1.** In what follows, $\sigma(L)$ denotes the spectrum of the operator $L$. We assume that the hypotheses of Proposition 3.1 hold and $\beta_0 = 0$. Then (3.3) ensures $\lambda_n(J) = \beta n + o(1)$ as $n \to \infty$ and

$$\sigma(J) \cap [\beta(n - \frac{1}{2}), \beta(n + \frac{1}{2})] = \{\lambda_n(J)\} \text{ for } n \geq n_0 \qquad (5.26)$$

if $n_0 \in \mathbb{N}$ is chosen large enough. Let $\{\lambda_j(J'_n)\}_{j \in \mathbb{Z}}$ be the sequence of eigenvalues of $J'_n$ labeled so that $\lambda_j(J'_n) = \lambda_j(J)$ for every $j \in \mathbb{Z}$. Then (5.26) gives

$$\sigma(J'_n) \cap [\beta(n - \frac{1}{2}), \beta(n + \frac{1}{2})] = \{\lambda_n(J'_n)\} \text{ for } n \geq n_0 \qquad (5.27)$$

and we will use $L = \tilde{J}'_n$, $x = e_n$, $d' = \beta(n - \frac{1}{2})$ and $d'' = \beta(n + \frac{1}{2})$ with $n \geq n_0$ in

**Theorem 5.2. (Kato-Temple)** Assume that the operator $L$ is self-adjoint in the Hilbert space $\mathcal{H}$ and has exactly one eigenvalue $\lambda$ in the interval $[d', d'']$. If $x$ is an element of the domain of $L$ such that $\eta := \langle x, Lx \rangle_{\mathcal{H}}$ belongs to $[d', d'']$, then

$$\eta - \frac{\|(L - \eta)x\|^2}{\eta - d'} \leq \lambda \leq \eta + \frac{\|(L - \eta)x\|^2}{d'' - \eta}. \qquad (5.28)$$

*Proof.* See [16]. □
We claim that the assumptions of Theorem 5.2 are satisfied for $n \ge n_0$ if $n_0$ is chosen large enough. Indeed, (5.27) holds and

$$ \eta := \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} \in [\beta(n - \frac{1}{4}), \beta(n + \frac{1}{4})] \text{ for } n \ge n_0 \quad (5.29) $$

holds due to (5.19) and $V(n, n) = O(n^{-1/2})$ as $n \to \infty$. Since (5.29) implies the inequality $\min\{d'' - \eta, \eta - d'\} \ge \frac{1}{4}\beta$, the estimate (5.28) gives

$$ |\lambda_n(J'_n) - \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})}| \le 4\beta^{-1} \|(J'_n \tilde{e}_n - \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})}) \tilde{e}_n\|^2 = O(n^{-1}), \quad (5.30) $$

where the last estimate is due to (5.20). The estimate (5.30) implies

$$ \lambda_n(J'_n) = \langle \tilde{e}_n, J'_n \tilde{e}_n \rangle_{\ell^2(\mathbb{Z})} + O(n^{-1}) \quad (5.31) $$

and using (5.19) in the right hand side of (5.31), we obtain

$$ \lambda_n(J) = \lambda_n(J'_n) = d'_n + O(n^{-1} \ln n), $$

completing the proof of Proposition 3.1.

## 6. PROOF OF THEOREM 1.3

**6.1. Step 1 of the proof of Theorem 1.3.** As before $0 < g < \frac{1}{2}$ and $\{e_m\}_{m \in \mathbb{N}}$ is the canonical basis of $\ell^2(\mathbb{N})$.

**Definition 6.1.** For $\nu = \pm 1$, $\mu = 0, 1$, we define $\hat{J}_{\nu,\mu}$ as the linear map $\hat{J}_{\nu,\mu} : \ell^{2,1}(\mathbb{N}) \to \ell^2(\mathbb{N})$ given by

$$ \left\{ \begin{array}{l} \hat{J}_{\nu,\mu} e_m = \hat{d}_{\nu,\mu}(m) e_m + \hat{b}_{\mu}(m) e_{m+1} + \hat{b}_{\mu}(m-1) e_{m-1} \text{ for } m \in \mathbb{N}^*, \\ \hat{J}_{\nu,\mu} e_0 = \hat{d}_{\nu,\mu}(0) e_0 + \hat{b}_{\mu}(0) e_1 \end{array} \right. \quad (6.1) $$

where

$$ \left\{ \begin{array}{l} \hat{d}_{\nu,\mu}(m) := 2m + \mu + (-1)^m \nu \Delta/2 \\ \hat{b}_{\mu}(m) := g \sqrt{(2m + 1 + \mu)(2m + 2 + \mu)}. \end{array} \right. \quad (6.2) $$

**Lemma 6.2.** The operator $\mathbf{H}_{\text{Rabi}}^{(2)}$ is similar to the direct sum

$$ \hat{J}_{1,0} \oplus \hat{J}_{-1,0} \oplus \hat{J}_{1,1} \oplus \hat{J}_{-1,1} \quad (6.3) $$

*Proof.* See Section 2.2 in [6]. □

<figure><img src="image_3.png" /></figure>

**6.2. Step 2 of the proof of Theorem 1.3.** Let $\tilde{J}_{\gamma}^{\delta}$, $\tilde{d}_{\delta}$, $\tilde{b}_{\gamma}$ be as in Definition 2.2. If $\hat{d}_{\nu,\mu}$ and $\hat{b}_{\mu}$ are given by (6.2), then it is easy to check the following relations

$$ \left\{ \begin{array}{l} \hat{d}_{\pm 1,0}(m) = 2\tilde{d}_{\pm \Delta/4}(m) \\ \hat{b}_0(m) = 2\tilde{b}_{3/4}(m) + O(m^{-1}) \text{ when } m \to \infty \end{array} \right. \quad (6.4) $$

$$ \left\{ \begin{array}{l} \hat{d}_{\pm 1,1}(m) = 2\tilde{d}_{\pm \Delta/4}(m) + 1 \\ \hat{b}_1(m) = 2\tilde{b}_{5/4}(m) + O(m^{-1}) \text{ when } m \to \infty \end{array} \right. \quad (6.5) $$

The above properties allow us to prove
**Lemma 6.3.** Let $0 < g < \frac{1}{2}$ and let $\{\lambda_n(\hat{J}_{\nu,\mu})\}_{n \in \mathbb{N}}$ denote the non-decreasing sequence of eigenvalues of $\hat{J}_{\nu,\mu}$. If $\{\lambda_j(\tilde{J}_{\gamma}^{\delta})\}_{j \in \mathbb{Z}}$ is the sequence of eigenvalues of $\tilde{J}_{\gamma}^{\delta}$ satisfying (2.10), then one has

$$ \lambda_n(\hat{J}_{\pm 1,0}) = 2\lambda_n(\tilde{J}_{3/4}^{\pm \Delta/4}) + O(n^{-1}) \text{ as } n \to \infty \quad (6.6) $$

and

$$ \lambda_n(\hat{J}_{\pm 1,1}) = 2\lambda_n(\tilde{J}_{5/4}^{\pm \Delta/4}) + 1 + O(n^{-1}) \text{ as } n \to \infty. \quad (6.7) $$

*Proof.* We identify $\ell^2(\mathbb{N})$ with $\{x \in \ell^2(\mathbb{Z}) : x(k) = 0 \text{ for } k \in \mathbb{Z} \setminus \mathbb{N}\}$. Let $\Pi_{\ge 0}$ be the orthogonal projection $\ell^2(\mathbb{Z}) \to \ell^2(\mathbb{N})$ and let $\hat{J}_{\gamma}^{\delta} := \Pi_{\ge 0} \tilde{J}_{\gamma}^{\delta}|_{\ell^2(\mathbb{N})}$. Identifying $\tilde{e}_m$ with $e_m$ when $m \in \mathbb{N}$, we get

$$ \begin{cases} \hat{J}_{\gamma}^{\delta} e_m = \hat{d}_{\delta}(m)e_m + \hat{b}_{\gamma}(m)e_{m+1} + \hat{b}_{\gamma}(m-1)e_{m-1} \text{ for } m \in \mathbb{N}^{*}, \\ \hat{J}_{\gamma}^{\delta} e_0 = \hat{d}_{\delta}(0)e_0 + \hat{b}_{\gamma}(0)e_1 \end{cases} \quad (6.8) $$

We observe that (6.4)-(6.5) imply

$$ 2\hat{J}_{3/4}^{\pm \Delta/4} - \hat{J}_{\pm 1,0} \in \mathcal{B}(\ell^2(\mathbb{N}), \ell^{2,1}(\mathbb{N})), \quad (6.9) $$

$$ 2\hat{J}_{5/4}^{\pm \Delta/4} + 1 - \hat{J}_{\pm 1,1} \in \mathcal{B}(\ell^2(\mathbb{N}), \ell^{2,1}(\mathbb{N})) \quad (6.10) $$

and using the result of Rozenblum [24, Theorem 1.1] as indicated in the proof of (3.6) in [6, Section 3.2], we find that (6.9)-(6.10) ensure

$$ 2\lambda_n(\hat{J}_{3/4}^{\pm \Delta/4}) = \lambda_n(2\hat{J}_{3/4}^{\pm \Delta/4}) = \lambda_n(\hat{J}_{\pm 1,0}) + O(n^{-1}) \text{ as } n \to \infty, \quad (6.11) $$

$$ 2\lambda_n(\hat{J}_{5/4}^{\pm \Delta/4}) + 1 = \lambda_n(2\hat{J}_{5/4}^{\pm \Delta/4} + 1) = \lambda_n(\hat{J}_{\pm 1,1}) + O(n^{-1}) \text{ as } n \to \infty, \quad (6.12) $$

where $\{\lambda_n(\hat{J}_{\gamma}^{\delta})\}_{n \in \mathbb{N}}$ is the non-decreasing sequence of eigenvalues of $\hat{J}_{\gamma}^{\delta}$.

Let $\{\lambda_j(\tilde{J}_{\gamma}^{\delta})\}_{j \in \mathbb{Z}}$ be the sequence of eigenvalues of $\tilde{J}_{\gamma}^{\delta}$ satisfying (2.10). We complete the proof, combining (6.11)-(6.12) and the estimate

$$ \lambda_n(\hat{J}_{\gamma}^{\delta}) - \lambda_n(\tilde{J}_{\gamma}^{\delta}) = O(n^{-N}) \text{ as } n \to \infty, \quad (6.13) $$

which holds for every $N \ge 1$. The estimate (6.13) is proved in Section 6.1 of [6]. $\square$

**6.3.** **End of the proof of Theorem 1.3.** Using the asymptotic estimates (2.10) in the right hand side of (6.6) and (6.7), we obtain

$$ \lambda_n(\hat{J}_{\pm 1,0}) = 2 \left( n\beta + \frac{1}{4}(\beta - 1) + \mathbf{r}_{3/4}^{\pm \Delta/4}(n) \right) + O(n^{-1} \ln n), \quad (6.14) $$

$$ \lambda_n(\hat{J}_{\pm 1,1}) = 2 \left( n\beta + \frac{3}{4}(\beta - 1) + \mathbf{r}_{5/4}^{\pm \Delta/4}(n) \right) + 1 + O(n^{-1} \ln n). \quad (6.15) $$

Due to (2.12),

$$ \hat{\theta}_{3/4} = \frac{1}{4}(4\alpha - \pi) + \frac{\pi}{4} = \alpha, $$

$$ \hat{\theta}_{5/4} = \frac{3}{4}(4\alpha - \pi) + \frac{\pi}{4} = 3\alpha - \frac{\pi}{2} $$

and (6.15)-(6.15) give

$$ \lambda_n(\hat{J}_{\pm 1,0}) = 2n\beta + \frac{1}{2}(\beta - 1) \pm \frac{\Delta}{2} \left( \frac{\beta}{2\pi gn} \right)^{1/2} \cos((4n + 1)\alpha) + O(n^{-1} \ln n), $$

$$ \lambda_n(\hat{J}_{\pm 1,1}) = (2n + 1)\beta + \frac{1}{2}(\beta - 1) \pm \frac{\Delta}{2} \left( \frac{\beta}{2\pi gn} \right)^{1/2} \sin((4n + 3)\alpha) + O(n^{-1} \ln n) $$
Due to Lemma 6.2, $\sigma(H_{\text{Rabi}}^{(2)}) = \sigma(\hat{J}_{1,0}) \cup \sigma(\hat{J}_{-1,0}) \cup \sigma(\hat{J}_{1,1}) \cup \sigma(\hat{J}_{-1,1})$ and we complete the proof, taking

$$ \begin{cases} E_{2n}^{\pm} = \lambda_n(\hat{J}_{\pm 1,0}) \\ E_{2n+1}^{\pm} = \lambda_n(\hat{J}_{\pm 1,1}) \end{cases} \qquad (6.16) $$

for $n \in \mathbb{N}$.

## 7. APPENDIX: ESTIMATES OF OSCILLATORY INTEGRALS

**Lemma 7.1.** Let us fix real numbers $\theta_0, \theta_1$ and $c_0 > 0$. Assume that $\{\Psi_{\tau}\}_{\tau \in T}$ is a family of functions $[\theta_0, \theta_1] \to \mathbb{R}$, bounded in $C^2([\theta_0, \theta_1])$, i.e. $\sup_{\theta_0 \le \theta \le \theta_1} |\Psi_{\tau}^{(k)}(\theta)| < \infty$ for $k = 0, 1, 2$. For $\lambda \in \mathbb{R}$ and $h \in C^2([\theta_0, \theta_1])$ we denote

$$ M_{\tau}(\lambda) := \int_{\theta_0}^{\theta_1} e^{i\lambda\Psi_{\tau}(\theta)} h(\theta) \, d\theta. \qquad (7.1) $$

(a) If $|\Psi'_{\tau}(\theta)| \ge c_0 > 0$ holds for every $\theta \in [\theta_0, \theta_1]$ and $\tau \in T$, then there is a constant $C$ such that the estimate

$$ |M_{\tau}(\lambda)| \le C|\lambda|^{-1} \qquad (7.2) $$

holds for all $\tau \in T$ and $\lambda \in \mathbb{R} \setminus \{0\}$.

(b) If $|\Psi''_{\tau}(\theta)| \ge c_0 > 0$ holds for every $\theta \in [\theta_0, \theta_1]$ and $\tau \in T$, then there is a constant $C$ such that the estimate

$$ |M_{\tau}(\lambda)| \le C|\lambda|^{-1/2} \qquad (7.3) $$

holds for all $\tau \in T$ and $\lambda \in \mathbb{R} \setminus \{0\}$.

*Proof.* (a) The non-stationary phase estimate (7.2) follows by using the standard integration by parts.

(b) The estimate (7.3) follows from van der Corput Lemma (see [27, Section VIII.1.2, p. 354]). □

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<footnote n="1">¹UNIVERSITÉ PARIS CITÉ, BÂTIMENT SOPHIE GERMAIN, BOÎTE COURRIER 7012, 75205 PARIS CEDEX 13, FRANCE, E-MAIL: ANNE.BOUTET-DE-MONVEL@IMJ-PRG.FR</footnote>

<footnote>2LABORATOIRE DE MATHÉMATIQUES PURES ET APPLIQUÉES JOSEPH LIOUVILLE UR 2597, UNIVERSITÉ DU LITTORAL CÔTE D'OPALE, F-62228 CALAIS, FRANCE, E-MAIL: LECH.ZIELINSKI@UNIVERSITTORAL.FR</footnote>