Notation 2. Consider $\textbf {u},\textbf {v}$ two words in $\{0,1\}^{*}_{N}$ , and w some $(N,1)$ -cylindrical pattern of the subshift X. We say that the pattern w connects $\textbf {u}$ to $\textbf {v}$ , and we denote this by $\textbf {u} \mathcal {R}[w] \textbf {v}$ , when for all , $\textbf {u}_{k} = 1$ (respectively, $\textbf {v}_{k}=1$ ) if and only if w has an incoming (respectively, outgoing) curve on the bottom (respectively, top) of its kth symbol. This notation is illustrated in Figure 5.

Figure 5 Illustration for the definition of the notation $\boldsymbol {u} \mathcal {R}[w] \boldsymbol {v}$ .

Notation 3. Let $N \ge 1$ be an integer and $t>0$ . Let us denote by $\Omega _{N}$ the space $\mathbb {C}^{2} \bigotimes \cdots \bigotimes \mathbb {C}^{2}$ , the tensor product of N copies of $\mathbb {C}^{2}$ , whose canonical basis elements are denoted indifferently by $\boldsymbol {\epsilon }= \lvert {\boldsymbol {\epsilon }_{1} \cdots \boldsymbol {\epsilon }_{N}}\rangle $ or the words $\boldsymbol {\epsilon }_{1} \cdots \boldsymbol {\epsilon }_{N}$ , for $(\boldsymbol {\epsilon }_{1}, \ldots , \boldsymbol {\epsilon }_{N}) \in \{0,1\}^{N}$ , according to quantum mechanics notation, to distinguish them from the coordinate definition of vectors of $\Omega _{N}$ . For the definition of the matrices, we order the elements of this basis with the lexicographic order.

Notation 4. For all $\textbf {u} \in \{0,1\}_{N}^{*}$ , we denote by $|\textbf {u}|_{1}$ the number of such that $\textbf {u}_{k} = 1$ . If $|\textbf {u}|_{1} =n$ , we denote by $q_{1} [\textbf {u}]< \cdots < q_{n} [\textbf {u}]$ the integers such that $\textbf {u}_{k} = 1$ if and only if $k=q_{i}[\textbf {u}]$ for some .

Proof. $(\Rightarrow )$ : assume that $\boldsymbol {u} \mathcal {R}[w] \boldsymbol {v}$ for some w.

First, since w is an $(N,1)$ -cylindrical pattern, each of the curves that cross its bottom side also crosses its top side, which implies that $|\boldsymbol {u}|_{1} = |\boldsymbol {v}|_{1}$ .

We assume that $q_{1} [\boldsymbol {u}] \le q_{1} [\boldsymbol {v}]$ (the other case is processed similarly).

1. (1) The position ${q_{1}} \boldsymbol {[u]}$ is connected to ${q_{1}} \boldsymbol {[v]}$ or ${q_{1}} \boldsymbol {[u]} = {q_{1}} \boldsymbol {[v]}$ : Let us assume that $q_{1} [\boldsymbol {u}] \neq q_{1} [\boldsymbol {v}]$ and $q_{1} [\boldsymbol {u}]$ is not connected to $q_{1} [\boldsymbol {v}]$ . Then because $\boldsymbol {u} \mathcal {R}[w] \boldsymbol {v}$ , another curve would have to connect another position $q_{k} [\boldsymbol {u}]$ , $k \neq 1$ of $\boldsymbol {u}$ to $q_{1} [\boldsymbol {v}]$ . Since $q_{k} [\boldsymbol {u}]> q_{1} [\boldsymbol {u}]$ (by definition), this curve would cross the left border of w. It would imply that in the $q_{1} [\boldsymbol {u}]$ th symbol of w, two pieces of curves would appear: one horizontal, corresponding to the curve connecting the position $q_{k} [\boldsymbol {u}]$ to $q_{1} [\boldsymbol {v}]$ , and the one that connects $q_{1} [\boldsymbol {u}]$ to another position in $\boldsymbol {u}$ , which is not possible, by the definition of the alphabet of $X^{s}$ . This is illustrated in Figure 6.

   

   Figure 6 Illustration of (impossible) crossing situation, which would imply non-authorized symbols.

2. (2) $q_{1} \boldsymbol {[v]} \le q_{2} \boldsymbol {[u]} \le q_{2} \boldsymbol {[v]}$ : In both cases, this derives from similar arguments.

3. (3) Repetition: We can then repeat these arguments to obtain:

   $$ \begin{align*}q_{1} [\boldsymbol{u}] \le q_{1} [\boldsymbol{v}] \le q_{2} [\boldsymbol{u}] \le \cdots \le q_{n} [\boldsymbol{u}] \le q_{n} [\boldsymbol{v}],\end{align*} $$
   meaning that $\boldsymbol {u}$ and $\boldsymbol {v}$ are interlaced.

$(\Leftarrow )$ : if $|\boldsymbol {u}|_{1} = \boldsymbol {v}_{1}$ and $\boldsymbol {u},\boldsymbol {v}$ are interlaced, then we define w by connecting $q_{i} [\boldsymbol {u}]$ to $q_{i} [\boldsymbol {v}]$ for all . We thus have directly $\boldsymbol {u} \mathcal {R} [w] \boldsymbol {v}$ .

Proof. Consider words $\boldsymbol {u} \neq \boldsymbol {v}$ and w, such that $\boldsymbol {u} \mathcal {R}[w] \boldsymbol {v}$ . Because of Proposition 2, $\boldsymbol {u}$ and $\boldsymbol {v}$ are interlaced. Let us assume that we have (the other case is processed similarly)

$$ \begin{align*}q_{1} [\boldsymbol{u}] \le q_{1} [\boldsymbol{v}] \le q_{2} [\boldsymbol{u}] \le \cdots \le q_{n} [\boldsymbol{u}] \le q_{n} [\boldsymbol{v}].\end{align*} $$

Because the words are different, there is some j such that the position $q_{j} [\boldsymbol {u}]$ is connected to $q_{j} [\boldsymbol {v}]$ . This forces that the position $q_{j+1} [\boldsymbol {u}]$ is connected to $q_{j+1} [\boldsymbol {v}]$ if $j<n$ . If n, $q_{1} [\boldsymbol {u}]$ is connected to $q_{1} [\boldsymbol {v}]$ . By repeating this, we obtain that for all i, $q_{i} [\boldsymbol {u}]$ is connected to $q_{i} [\boldsymbol {v}]$ . This determines w, which implies that there is a unique w such that $\boldsymbol {u} \mathcal {R}[w] \boldsymbol {v}$ .

When $\boldsymbol {u} = \boldsymbol {v}$ , it is clear that there is a unique w connecting position $q_{i} [\boldsymbol {u}]$ to $q_{i} [\boldsymbol {v}]$ for all i. Any other w connecting $\boldsymbol {u}$ to $\boldsymbol {v}$ connects $q_{i} [\boldsymbol {u}]$ to $q_{j} [\boldsymbol {v}]$ for some $j \neq i$ . This j is forced to be $i+1$ (for similar arguments as in the first point of the proof of Proposition 2) and for similar arguments as above, this forces that $q_{i} [\boldsymbol {u}]$ to $q_{i+1} [\boldsymbol {v}]$ for all $i \le n$ and connects $q_{n} [\boldsymbol {u}]$ to $q_{1} [\boldsymbol {v}]$ .

Notation 5. For all N and $(N,1)$ -cylindrical pattern w, let us denote by $|w|$ the number of symbols

in this pattern. For instance, for the word w in Figure 5, $|w|=6$ .

Proof. This is a direct consequence of Proposition 2, since if $V_{N} (t) [\boldsymbol {\epsilon }, \boldsymbol {\eta }] \neq 0$ for $\boldsymbol {\epsilon },\boldsymbol {\eta }$ , two elements of the canonical basis of $\Omega _{N}$ , then $|\boldsymbol {\epsilon }|_{1} = |\boldsymbol {\eta }|_{1}$ .

Proof. Let us denote by $u \in \Omega _{N}$ the Perron–Frobenius eigenvector and $v \in \Omega _{N}$ another vector whose coordinates are all positive, associated to the eigenvalue $\mu $ . Then

$$ \begin{align*}\mu u^{t}.v = (Au)^{t}. v = u^{t} A v = \lambda u^{t}.v.\end{align*} $$

Thus, since $u^{t}.v>0$ , then $\mu = \lambda $ , and by (usual version of) Perron–Frobenius, there exists some $\alpha \in \mathbb {R}$ such that $v= \alpha .u$ . Since v has positive coordinates, $\alpha>0$ .

Proof. The non-negativity of the matrix is immediate when $t \ge 0$ is immediate. Let us prove the other properties.

Symmetry: since the interlacing relation is symmetric, for all $\boldsymbol {\epsilon },\boldsymbol {\eta } \in \{0,1\}^{*}_{N}$ , we have that $V_{N} (t) [\boldsymbol {\epsilon },\boldsymbol {\eta }]>0$ if and only if $V_{N} (t) [\boldsymbol {\eta },\boldsymbol {\epsilon }]>0$ . When this is the case, and $\boldsymbol {\epsilon } \neq \boldsymbol {\eta }$ (the case $\boldsymbol {\epsilon }=\boldsymbol {\eta }$ is trivial), there exists a unique (Proposition 3) w connecting $\boldsymbol {\epsilon }$ to $\boldsymbol {\eta }$ . The coefficient of this word is exactly $t^{2(n-|\{k: \boldsymbol {\epsilon }_{k} = \boldsymbol {\eta }_{k} = 1\}|)}$ , where $n= |\{k:\boldsymbol {\epsilon }_{k}=1\}|=|\{k:\boldsymbol {\eta }_{k}=1\}|$ , and this coefficient is indifferent to the exchange of $\boldsymbol {\epsilon }$ and $\boldsymbol {\eta }$ . This implies that $V_{N} (t) [\boldsymbol {\epsilon },\boldsymbol {\eta }] = V_{N} (t) [\boldsymbol {\eta },\boldsymbol {\epsilon }]$ . As a consequence, $V_{N}(t)$ is symmetric.

Irreducibility: Let $\boldsymbol {\epsilon }$ , $\boldsymbol {\eta }$ be two elements of the canonical basis of $\Omega _{N}$ such that $|\boldsymbol {\epsilon }|_{1} = |\boldsymbol {\eta }|_{1} = n$ . We shall prove that $V_{N}^{N^{n}} (t) [\boldsymbol {\epsilon },\boldsymbol {\eta }]>0$ .

1. (1) Interlacing case: If we have $\boldsymbol {\epsilon } \mathcal {R} \boldsymbol {\eta }$ , then $V_{N} (t) [\boldsymbol {\epsilon },\boldsymbol {\eta }]>0$ . Since $V_{N} (t) [\boldsymbol {\eta },\boldsymbol {\eta }]>0$ and $V_{N}(t)$ is non-negative, for all $k\ge 1$ ,

   $$ \begin{align*}V_{N} (t)^{k} [\boldsymbol{\epsilon},\boldsymbol{\eta}] \ge V_{N}(t) [\boldsymbol{\epsilon},\boldsymbol{\eta}] (V_{N} (t) [\boldsymbol{\eta},\boldsymbol{\eta}])^{k-1}>0.\end{align*} $$

   In particular, $V_{N} (t)^{N^{n}} [\boldsymbol {\epsilon },\boldsymbol {\eta }]>0$ .

2. (2) Non-interlacing case:

   • • Decreasing the interlacing degree: If we do not have the relation $\boldsymbol {\epsilon } \mathcal {R} \boldsymbol {\eta }$ , let us denote by $\omega (\boldsymbol {\epsilon },\boldsymbol {\eta })$ the following quantity (interlacing degree):

     
     Since the relation $\boldsymbol {\epsilon } \mathcal {R} \boldsymbol {\eta }$ does not hold, $\omega (\boldsymbol {\epsilon },\boldsymbol {\eta }) \ge 2$ . Let us also denote by $\lambda (\boldsymbol {\epsilon },\boldsymbol {\eta })$ the number of integers i such that i realizes the maximum in the definition of $\omega (\boldsymbol {\epsilon },\boldsymbol {\eta })$ .

     Let us see that there exists some $\boldsymbol {\epsilon }^{\prime }$ such that $\boldsymbol {\epsilon } \mathcal {R} \boldsymbol {\epsilon }^{\prime }$ and $\lambda (\boldsymbol {\epsilon }^{\prime },\boldsymbol {\eta }) < \lambda (\boldsymbol {\epsilon },\boldsymbol {\eta })$ if $\lambda (\boldsymbol {\epsilon },\boldsymbol {\eta }) \ge 2$ and otherwise $\omega (\boldsymbol {\epsilon }^{\prime },\boldsymbol {\eta }) < \omega (\boldsymbol {\epsilon },\boldsymbol {\eta })$ .

     Since $|\boldsymbol {\epsilon }|_{1} = |\boldsymbol {\eta }|_{1}$ , there exist i and $i^{\prime }$ such that is maximal and is equal to $0$ . Let us assume that $i < i^{\prime }$ (the other case is processed similarly). We can also assume that there is no $i^{\prime \prime }$ such that $i < i^{\prime \prime }<i^{\prime }$ such that is equal to  $0$ .

     Let us denote $j_{0}$ and $j_{1}$ such that $q_{j_{0}} [\boldsymbol {\epsilon }]$ and $q_{j_{1}} [\boldsymbol {\epsilon }]$ are respectively the maxima of the sets and . There is a word w which connects $q_{j}[\boldsymbol {\epsilon }]$ to $q_{j+1}[\boldsymbol {\epsilon }]$ for all , and fixes $q_{j}[\boldsymbol {\epsilon }]$ for all other j. The words w thus connects $\epsilon $ to $\epsilon ^{\prime }$ which satisfies the above properties.

   • • A sequence with decreasing interlacing degree: As a consequence, since $\omega (\boldsymbol {\epsilon },\boldsymbol {\eta }) \le N$ , one can construct a finite sequence of words $\boldsymbol {\epsilon }^{(k)}$ , $k=1\ldots m$ such that $m \le N^{n}$ , $\boldsymbol {\epsilon }^{(1)} = \boldsymbol {\epsilon }$ , $\boldsymbol {\epsilon }^{(m)}$ and $\boldsymbol {\eta }$ are interlaced, and for all ${k <m}$ , $\boldsymbol {\epsilon }^{(k)} \mathcal {R} \boldsymbol {\epsilon }^{(k+1)}$ . This means that for all $k<m$ , $V_{N}[\boldsymbol {\epsilon }^{(k)}, \boldsymbol {\epsilon }^{(k+1)}]>0$ and $V_{N}[\boldsymbol {\epsilon }^{(m)},\boldsymbol {\eta }]>0$ . As a consequence, $V_{N} (t)^{N^{n}} [\boldsymbol {\epsilon },\boldsymbol {\eta }]>0$ .

This implies that $V_{N} (t)$ is irreducible on $\Omega _{N}^{(n)}$ for all $n \le N$ .

Notation 6. For all N and $n \le N$ , let us denote by $\overline {X}^{s}_{n,N}$ the subset (which is also a subshift) of $\overline {X}^{s}_{N}$ which consists in the set of configurations of $\overline {X}^{s}_{N}$ such that the number of curves that cross each of its rows is n, and $\overline {X}_{n,N}$ the subset of $\overline {X}_{N}$ such that the number of arrows pointing south in the south part of the symbols in any raw is n.

Notation 7. Let us denote, for all N and $n \le N$ , by $\lambda _{n,N} (t)$ the greatest eigenvalue of $V_{N} (t)$ on $\Omega _{N}^{(n)}$ .

Proof. Correspondence between elements of ${\overline {X}}^{{s}}_{{n,N}}$ and trajectories under action of ${V_{N}(1)}$ : Since for all N, $n \le N$ and $\boldsymbol {\epsilon },\boldsymbol {\eta }$ in the canonical basis of $\Omega _{N}^{(n)}$ , $V_{N} (1) [\boldsymbol {\epsilon },\boldsymbol {\eta }]$ is the number of ways to connect $\boldsymbol {\epsilon }$ to $\boldsymbol {\eta }$ by an $(N,1)$ -cylindrical pattern, and that there is a natural invertible map from the set of $(M,N)$ -cylindrical patterns to the sequences $(w_{i})_{i=1\ldots M}$ of $(N,1)$ -cylindrical patterns such that there exists some $(\boldsymbol {\epsilon }_{i})_{i=1\ldots M+1}$ such that for all i, $|\boldsymbol {\epsilon }_{i}|=n$ and for all $i \le M$ , $\boldsymbol {\epsilon }_{i} \mathcal {R}[w_{i}]\boldsymbol {\epsilon }_{i+1}$ ,

$$ \begin{align*}\|(V_{N}(1)_{\Omega_{N}^{(n)}}^{M})\|_{1}=\mathcal{N}_{M} (\overline{X}^{s}_{n,N}).\end{align*} $$

Gelfand’s formula: It is known (Gelfand’s formula) that $\|(V_{N} (1)_{\Omega _{N}^{(n)}}^{M})\|_{1}^{1/M} \rightarrow \lambda _{n,N} (1).$

As a consequence of the first point, $h (\overline {X}^{s}_{n,N}) = \log _{2} (\lambda _{n,N} (1)).$

Proof. We have the decomposition

$$ \begin{align*}\overline{X}^{s}_{N} = \displaystyle{\bigcup_{n=0}^{N}} \overline{X}^{s}_{n,N}.\end{align*} $$

Moreover, these subshifts are disjoint. As a consequence,

$$ \begin{align*}h(\overline{X}^{s}_{N}) = \max_{n \le N} h(\overline{X}^{s}_{n,N}).\end{align*} $$

From this, we deduce the statement.

Proof. For the purpose of notation, we also denote by $\pi _{s}$ the application from $\overline {X}_{n,N}$ to $\overline {X}^{s}_{n,N}$ that consists in an application of $\pi _{s}$ letter by letter. This map is invertible. Let us consider the application $\overline {\mathcal {T}}_{n,N}$ from $\overline {X}_{n,N}$ to $\overline {X}_{N-n,N}$ that inverts all the arrows. This map is an isomorphism, and thus the map $\pi _{s} \circ \overline {X}_{n,N} \circ \pi _{s}^{-1}$ is also an isomorphism from $\overline {X}^{s}_{n,N}$ to $\overline {X}^{s}_{N-n,N}$ . As a consequence, the two subshifts have the same entropy:

$$ \begin{align*}h(\overline{X}^{s}_{n,N}) = h(\overline{X}^{s}_{N-n,N}).\\[-40pt]\end{align*} $$

Proof. Let us fix some integer N and for all n between $1$ and $N/2+1$ , and consider the application that the set of patterns of $\overline {X}^{s}_{n,N}$ on $\mathbb {U}^{(1)}_{M}$ associates a pattern of $\overline {X}^{s}_{n-1,N}$ on $\mathbb {U}^{(1)}_{M}$ by suppressing the curve that crosses the leftmost symbol in the bottom row of the pattern crossed by a curve [see the schema in Figure 7].

Figure 7 Illustration of the curve suppressing operation; the leftmost position of the bottom row crossed by a curve is colored gray on the left pattern.

For each pattern of $\overline {X}^{s}_{n-1,N}$ , the number of patterns in its preimage by this transformation is bounded from above by $N^{M}$ . As a consequence, for all M:

$$ \begin{align*}\mathcal{N}_{M} (\overline{X}^{s}_{n-1,N}). N^{M} \ge \mathcal{N}_{M} (\overline{X}^{s}_{n,N}),\end{align*} $$

and thus

$$ \begin{align*}h(\overline{X}^{s}_{n-1,N}) + \log_{2} (N) \ge h(\overline{X}^{s}_{n,N}).\end{align*} $$

As a consequence,

$$ \begin{align*} h(X^{s}) & = \lim_{N} \frac{1}{2N} \max ( \max_{2n+1 \le N} h(\overline{X}^{s}_{2n+1,2N}), \max_{2n \le N} h(\overline{X}^{s}_{2n,2N}) ) \nonumber \\& \le \lim_{N} \frac{1}{2N} \max ( \max_{2n+1 \le N} h(\overline{X}^{s}_{2n+1,2N}), \max_{2n \le N} h(\overline{X}^{s}_{2n-1,2N}) + \log_{2} (N) )\\ & \le \lim_{N} \frac{1}{2N} \max_{2n+1 \le N} h(\overline{X}^{s}_{2n+1,2N}). \end{align*} $$

Moreover, it is straightforward that

$$ \begin{align*}h(X^{s}) \ge \lim_{N} \frac{1}{2N} \max_{2n+1 \le N} h(\overline{X}^{s}_{2n+1,2N}),\end{align*} $$

thus we have the following equality:

$$ \begin{align*}h(X^{s}) = \lim_{N} \frac{1}{2N}\displaystyle{\max_{2n +1 \le N}} \log_{2} (\lambda_{2n+1,2N} (1)).\\[-3.6pc]\end{align*} $$

Notation 10. For all $(p_{1},\ldots ,p_{n}) \in I_{t}^{n}$ , let us denote by $\psi _{\mu _{t},n,N} (p_{1},\ldots ,p_{n})$ the vector in $\Omega _{N}$ such that for all $\boldsymbol {\epsilon } \in \{0,1\}^{*}_{N}$ ,

$$ \begin{align*}\psi_{\mu_{t},n,N}(p_{1},\ldots,p_{n})[\boldsymbol{\epsilon}] = \sum_{\sigma \in \Sigma_{n}} C_{\sigma} (t) [p_{1},\ldots,p_{n}] \prod_{k=1}^{n} e^{ip_{\sigma(k)} q_{k} [\boldsymbol{\epsilon}]},\end{align*} $$

where, denoting by $\epsilon (\sigma )$ the signature of $\sigma $ ,

$$ \begin{align*}C_{\sigma} (t) [p_{1},\ldots,p_{n}] = \epsilon(\sigma)\prod_{1 \le k <l\le n} ( 1 + e^{i(p_{\sigma(k)}+p_{\sigma(l)})} - 2\Delta_{t} e^{ip_{\sigma(k)}}).\end{align*} $$

Proof. The solutions are critical points of an auxiliary function ${\zeta _{t}}$ : Let us set, for all $t,p_{1},\ldots ,p_{n}$ (in the third sum, both k and j are arguments of the sum),

$$ \begin{align*} \zeta_{t} (p_{1},\ldots,p_{n}) & = N \sum_{j=1}^{n} \int_{0}^{\kappa^{-1}_{t} (p_{j})} \kappa_{t} (x) \,dx + \pi (n+1-2j)\sum_{j=1}^{n} \kappa_{t}^{-1} (p_{j})\\ &\quad + \sum_{k < j} \int_{0}^{\kappa_{t} {}^{-1} (p_{j})-\kappa_{t} {}^{-1} (p_{k})} \theta_{t} (x,0)\,dx. \end{align*} $$

The interest of this function lies in the fact that for all j (here the argument in each of the sums is k),

(10) $$ \begin{align} \frac{\partial \zeta_{t}}{\partial p_{j}} (p_{1}, \ldots, p_{n}) & = ( \kappa_{t}^{-1} )^{\prime} (p_{j}). \bigg( N p_{j} - 2\pi j + (n+1)\pi\nonumber\\ &\quad - \!\sum_{k < j}\! \theta_{t} (\kappa_{t} {}^{-1} (p_{j})-\kappa_{t} {}^{-1} (p_{k}),0) \bigg) + \!\sum_{k> j}\! \theta_{t} (\kappa_{t} {}^{-1} (p_{k})-\kappa_{t} {}^{-1} (p_{j}),0) .\nonumber\\ & = ( \kappa_{t}^{-1} )^{\prime} (p_{j}). \bigg( N p_{j} - 2\pi j + (n+1)\pi - \!\sum_{k < j } \theta_{t} (\kappa_{t} {}^{-1} (p_{k}),\kappa_{t} {}^{-1} (p_{j})) \bigg) \nonumber\\ &\quad + \sum_{k> j} \theta_{t} (\kappa_{t} {}^{-1} (p_{j}),\kappa_{t} {}^{-1} (p_{k})).\nonumber\\ \frac{\partial \zeta_{t}}{\partial p_{j}} (p_{1}, \ldots, p_{n}) & = ( \kappa_{t}^{-1} )^{\prime} (p_{j}). \bigg( N p_{j} - 2\pi j + (n+1)\pi + \sum_{k} \Theta_{t} (p_{j},p_{k})\bigg), \end{align} $$

since for all $x,y$ , $\Theta _{t} (x,y) = - \Theta _{t} (y,x)$ and for all x, $\Theta _{t}(x,x)=0$ (§5.1.1).

Hence, the system of Bethe equations is verified for the sequence $(p_{j})_{j}$ if and only for all j, $ {\partial \zeta _{t} }/{\partial p_{j}} (p_{1}, \ldots , p_{n} )=0$ .

Uniqueness of the local minimum of ${\zeta _{t}}$ using convexity: Let us set $\tilde {\zeta }_{t} : \mathbb {R}^{n} \rightarrow \mathbb {R}$ such that for all $\alpha _{1}, \ldots , \alpha _{n}$ :

$$ \begin{align*}\tilde{\zeta}_{t} (\alpha_{1} , \ldots , \alpha_{n}) = \zeta_{t} (\kappa_{t} (\alpha_{1}), \ldots , \kappa_{t} (\alpha_{n})).\end{align*} $$

From equation (10), we have that for all sequence $(\alpha _{k})_{k}$ and all j,

$$ \begin{align*}\frac{\partial \tilde{\zeta}_{t}}{\partial p_{j}} (\alpha_{1}, \ldots , \alpha_{n}) = N \kappa_{t} (\alpha_{j}) - 2\pi j + (n+1)\pi + \sum_{k} \theta_{t} (\alpha_{j},\alpha_{k}).\end{align*} $$

As a consequence, using Computation 3, for all $k \neq j$ ,

(11) $$ \begin{align}\frac{\partial {}^{2} \tilde{\zeta}_{t}}{\partial p_{k} \partial p_{j}} (\alpha_{1}, \ldots , \alpha_{n}) = \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{j},\alpha_{k}) = \frac{\sin(2\mu_{t})}{\cosh(\alpha_{j}-\alpha_{k}) - \cos(2\mu_{t})}. \end{align} $$

Moreover, for all j,

(12) $$ \begin{align} \frac{\partial^{2} \tilde{\zeta}_{t}}{\partial^{2} p_{j}} (\alpha_{1}, \ldots , \alpha_{n} , t) = N \kappa^{\prime}_{t} (\alpha_{j}) + \sum_{k \neq j} \frac{\partial \theta_{t}}{\partial \alpha} (\alpha_{j}, \alpha_{k}) = N \kappa^{\prime}_{t} (\alpha_{j}) - \sum_{k \neq j} \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{j}, \alpha_{k}). \end{align} $$

Let us denote by $\tilde {H}_{t} (\alpha _{1} , \ldots , \alpha _{n})$ the Hessian matrix of $\tilde {\zeta }_{t}$ . For any $(x_{1}, \ldots , x_{n}) \in \mathbb {R}^{n}$ , we have, from equations (11) and (12),

$$ \begin{align*} |&|(x_{1},\ldots,x_{n}) \cdot \tilde{H}_{t} (\alpha_{1} , \ldots , \alpha_{n} ) \cdot (x_{1},\ldots,x_{n})^{t} \\ &\quad= N \sum_{j} \kappa^{\prime}_{t} (\alpha_{j}) x_{j} {}^{2} - \sum_{j \neq k} \bigg( \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{j}, \alpha_{k}) x_{j} (x_{j}-x_{k}).\bigg)\\ &\quad = N \sum_{j} \kappa^{\prime}_{t} (\alpha_{j}) x_{j} {}^{2} - \sum_{j < k} \bigg( \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{j}, \alpha_{k}) x_{j} (x_{j}-x_{k}) \bigg)\\ & \qquad - \sum_{j < k} \bigg( \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{k}, \alpha_{j}) x_{k} (x_{k}-x_{j}) \bigg)\\ &\quad = N \sum_{j} \kappa^{\prime}_{t} (\alpha_{j}) x_{j} {}^{2} - \sum_{j < k} \bigg( \frac{\partial \theta_{t}}{\partial \beta} (\alpha_{j}, \alpha_{k}) (x_{j}-x_{k})^{2} \bigg)> 0. \end{align*} $$

As a consequence, $\tilde {\zeta }_{t}$ is a convex function. Thus, if it has a local minimum, it is unique. Since $\kappa _{t}$ is increasing, this property is also true for $\zeta _{t}$ . The function ${\zeta _{t}}$ has a minimum in ${I_{t}^{n}}$ : Let us consider $(C_{l})_{l}$ as an increasing sequence of compact intervals such that $\bigcup _{l} C_{l} = I_{t}$ .

Let us assume that $\zeta _{t}$ has no local minimum in $I_{t}^{n}$ . As a consequence, for all j, the minimum $\textbf {p}^{(l)}$ of $\zeta _{t}$ on $(C_{l})^{n}$ is on its border. Without loss of generality, we can assume that there exists some $\textbf {p}^{(\infty )} \in \overline {I_{t}}^{n}$ such that $\textbf {p}^{(l)} \rightarrow \textbf {p}^{(\infty )} $ .

We can assume without loss of generality that there exists some such that $j \le j_{0}$ if and only if $\textbf {p}^{(\infty )}_{j} = \pi -\mu _{t}$ . The number $j_{0}$ is the number of j such that $\textbf {p}^{(\infty )}_{j} = \pi -\mu _{t}$ . We can assume furthermore that $j_{0} \le n/2$ : if it is not the case, then we use a reasoning similar to the one that follows, replacing $\pi -\mu _{t}$ by $-(\pi -\mu _{t})$ .

Thus, there exists $l_{0}$ such that for all l and $j \le j_{0}$ , $\textbf {p}_{j}^{(l)} \ge 0$ . Since $\tilde {\zeta }_{t}$ is convex and that $\textbf {p}^{(l)}$ is a minimum for this function on the compact set $(C_{l})^{n}$ , then for all $j \le j_{0}$ ,

(13)j $$\begin{align} \frac{\partial \zeta_{t}}{\partial p_{j}} ({\kern1pt}\textbf{p}^{(l)}_{1}, \ldots , \textbf{p}^{(l)}_{n}) \le 0. \end{align}$$

This is a particular case of the fact that for a convex and continuously differentiable function $f : I \mapsto \mathbb {R}$ , where I is a compact interval of $\mathbb {R}$ , if its minimum on I occurs at maximal element of I, then $f^{\prime }$ is non-positive on this point, as illustrated in Figure 8.

Figure 8 Illustration of the fact that the minimum of a convex continuously differentiable function on a real compact interval has non-positive derivative.

Since $\Theta _{t}$ cannot be defined on $\{(x,x): x \in \partial I_{t}\}$ , to have an inequality that can be transformed by continuity into an inequality on $\textbf {p}$ , we sum the inequalities in equation (13) _j :

$$ \begin{align*}\sum_{j=1}^{j_{0}} \frac{\partial \zeta_{t}}{\partial p_{j}} ({\kern1pt}\textbf{p}^{(l)}_{1}, \ldots , \textbf{p}^{(l)}_{n}) \le 0.\end{align*} $$

According to the first point of the proof (equation (10)), this inequality can be re-written:

$$ \begin{align*}N \sum_{j=1}^{j_{0}} \textbf{p}_{j} - 2\pi \sum_{j=1}^{j_{0}} j + j_{0} (n+1) \pi + \sum_{j=1}^{j_{0}} \sum_{k} \Theta_{t} ({\kern1pt}\textbf{p}_{j}^{(l)}, \textbf{p}_{k}^{(l)}) \le 0.\end{align*} $$

For all $j,j^{\prime } \le j_{0}$ , the terms $\Theta _{t} ({\kern1pt}\textbf{p}_{j}^{(l)},\textbf {p}_{j^{\prime }}^{(l)})$ and $\Theta _{t} ({\kern1pt}\textbf{p}_{j^{\prime }}^{(l)},\textbf {p}_{j}^{(l)})$ cancel out in this sum. As a consequence,

$$ \begin{align*}N \sum_{j=1}^{j_{0}} \textbf{p}_{j} - 2\pi \sum_{j=1}^{j_{0}} j + j_{0} (n+1) \pi + \sum_{j=1}^{j_{0}} \sum_{k>j_{0}} \Theta_{t} ({\kern1pt}\textbf{p}_{j}^{(l)}, \textbf{p}_{k}^{(l)}) \le 0.\end{align*} $$

This time, the inequality can be extended by continuity and we obtain

$$ \begin{align*}N \sum_{j=1}^{j_{0}} \textbf{p}_{j} - 2\pi \sum_{j=1}^{j_{0}} j + j_{0} (n+1) \pi + \sum_{j=1}^{j_{0}} \sum_{k>j_{0}} \Theta_{t} ({\kern1pt}\textbf{p}_{j}^{(\infty)}, \textbf{p}_{k}^{(\infty)}) \le 0.\end{align*} $$

From Computation 1, we have

$$ \begin{align*}N j_{0} (\pi-\mu_{t}) - 2\pi \sum_{j=1}^{j_{0}} j + j_{0} (n+1) \pi + j_{0} (n-j_{0})\cdot (2\mu_{t} - \pi) \le 0.\end{align*} $$

Thus,

$$ \begin{align*}j_{0} (n+1) \pi + j_{0} (n-j_{0})\pi + (N j_{0}- 2j_{0} (n-j_{0})) (\pi-\mu_{t} ) \le 2\pi \sum_{j=1}^{j_{0}} j.\end{align*} $$

Since $\mu _{t} \le \pi $ and that $2j_{0} (n-j_{0}) - Nj_{0} = -2 j_{0}^{2} < 0$ , this last inequality implies

(14) $$ \begin{align} j_{0} (n+1) \pi + j_{0} (n-j_{0}) \pi \le 2\pi \sum_{j=1}^{j_{0}} j. \end{align} $$

However, we have

(15) $$ \begin{align} \sum_{j=1}^{j_{0}} j \le n j_{0} - \sum_{j=1}^{j_{0}} j = nj_{0} - \frac{j_{0} (j_{0}+1)}{2}.\end{align} $$

As a consequence of equations (14) and (15),

$$ \begin{align*} j_{0} (n+1) \pi + j_{0} (n-j_{0})\cdot \pi & \le 2\pi n j_{0} - j_{0} (j_{0} +1) \pi\\ (2n+1) j_{0} \pi - j_{0} {}^{2} \pi & \le 2\pi n j_{0} - j_{0} {}^{2} \pi - j_{0} \pi \\ j_{0} \pi & \le -j_{0} \pi. \end{align*} $$

Since this last inequality is impossible, this means that $\zeta _{t}$ has a minimum in $I_{t}^{n}$ .

Characterization of the solutions with an analytic differential equation: Let us denote by $\textbf {p} (t) = ({\kern1pt}\textbf{p}_{1} (t) , \ldots , \textbf {p}_{n} (t))$ , for all $t \in (0, \sqrt {2})$ , the unique minimum of the function $\zeta _{t}$ in $I_{t}^{n}$ . Let us denote by $t \mapsto \textbf {s} (t) = (\textbf {s}_{1} (t), \ldots , \textbf {s}_{n} (t))$ the unique solution of the differential equation:

(16) $$ \begin{align} {\textbf{s}}^{\prime}(t) = -( H_{t} (\textbf{s}_{1} (t) , \ldots , \textbf{s}_{n} (t))) {}^{-1} \cdot \bigg( \frac{\partial^{2} \zeta_{t}} {\partial t \partial p_{j} } (\textbf{s}_{1} (t), \ldots , \textbf{s}_{n} (t)) \bigg)_{j}, \end{align} $$

such that $\textbf {s}(t)$ is the minimum of the function $\zeta _{t}$ when $t=\sqrt {2}/2$ , where $H_{t}$ is the Hessian matrix of $\zeta _{t}$ (existence and uniqueness are provided by classical theorems on first-order nonlinear differential equations). Since this is an analytic differential equation, its solution $\textbf {s}$ is analytic.

Let us rewrite equation (16):

$$ \begin{align*}H_{t} (\textbf{s}_{1} (t) , \ldots , \textbf{s}_{n} (t)) \cdot \textbf{s}^{\prime}(t) = -\bigg(\frac{\partial^{2} \zeta_{t}} {\partial t \partial p_{j} } (\textbf{s}_{1} (t), \ldots , \textbf{s}_{n} (t)) \bigg)_{j} \end{align*} $$

$$ \begin{align*}\frac{\partial^{2} \zeta_{t}} {\partial t \partial p_{j} } (\textbf{s}_{1} (t), \ldots , \textbf{s}_{n} (t)) + \sum_{k} \textbf{s}^{\prime}_{k} (t) \cdot \frac{\partial^{2} \zeta_{t}} {\partial p_{k} \partial p_{j} } (\textbf{s}_{1} (t), \ldots , \textbf{s}_{n} (t)) = 0. \end{align*} $$

This means that for all j,

$$ \begin{align*}\frac{\partial \zeta_{t}} {\partial p_{j} } (\textbf{s}_{1} (t), \ldots , \textbf{s}_{n} (t) )\end{align*} $$

is a constant. Since $\textbf {s}(t)$ is the minimum of $\zeta _{t}$ when $t=\sqrt {2}/2$ , this constant is zero. As a consequence, by uniqueness of the minimum of $\zeta _{t}$ for all t, $\textbf {s}(t) = \textbf {p}(t)$ . This means that $t \mapsto \textbf {p} (t)$ is analytic.

Antisymmetry of the solutions: For all $t,j$ , since $\textbf {p} (t)$ is the minimum of $\zeta _{t}$ , using equation (10) and the fact that $\kappa _{t}$ is increasing, we can write successively the following equations:

$$ \begin{align*} N\textbf{p}_{n-j+1} - 2\pi (n-j+1) + (n+1)\pi +\sum_{k} \Theta_{k} ({\kern1pt}\textbf{p}_{n-j+1}, \textbf{p}_{n-k+1}) & = 0;\\ N\textbf{p}_{n-j+1} + 2\pi j - (n+1)\pi +\sum_{k} \Theta_{k} ({\kern1pt}\textbf{p}_{n-j+1}, \textbf{p}_{n-k+1}) & = 0;\\ -N\textbf{p}_{n-j+1} - 2\pi j + (n+1)\pi -\sum_{k} \Theta_{k} ({\kern1pt}\textbf{p}_{n-j+1}, \textbf{p}_{n-k+1}) & = 0;\\ -N\textbf{p}_{n-j+1} - 2\pi j + (n+1)\pi +\sum_{k} \Theta_{k} (-\textbf{p}_{n-j+1}, -\textbf{p}_{n-k+1}) & = 0. \end{align*} $$

This means that the sequence $(-\textbf {p}_{n-j+1} (t))_{j}$ is a minimum for $\zeta _{t}$ , and as a consequence, for all j, $\textbf {p}_{n-j+1} = - \textbf {p}_{j}$ .

The numbers $\textbf {p}_{{j(t)}}$ , ${j \le n}$ , are all distinct: Let us consider the function

$$ \begin{align*}\chi_{t} : \alpha \mapsto N \kappa_{t} (\alpha) + \sum_{k=1}^{n} \theta_{t} (\alpha,\boldsymbol{\alpha}_{k} (t)),\end{align*} $$

where for all k, $\boldsymbol {\alpha }_{k} (t)$ is equal to $\kappa _{t}^{-1} ({\kern1pt}\textbf{p}_{k} (t))$ . For all j, this function has value ${\pi (2j-(n+1))}$ in $\boldsymbol {\alpha }_{j} (t)$ (by the Bethe equations). The finite sequence ${(\pi (2j-(n+1)))_{j}}$ is increasing, thus it is sufficient to prove that the function $\chi _{t}$ is increasing. Its derivative is

$$ \begin{align*}\chi^{\prime}_{t} : \alpha \mapsto N \kappa^{\prime}_{t} (\alpha) + \sum_{k=1}^{n} \frac{\partial \theta_{t}}{\partial \alpha} (\alpha, \boldsymbol{\alpha}_{k} (t)).\end{align*} $$

Since $t \in (0,\sqrt {2})$ , $\sin (\mu _{t})<0$ , and thus this function is positive. As a consequence, $\chi _{t}$ is increasing.

Proof. (1) The Bethe vector is ${\neq 0}$ for ${t}$ in a neighborhood of ${\sqrt {2}}$ :

• • Limit of the Bethe vector in $\boldsymbol {\sqrt {2}}$ : Let us denote by $({\kern1pt}\textbf{p}_{j} (t))_{j}$ the solution of the system of equations $(E_{j})[t,2n+1,N]$ .

  Let us recall [Theorem 3] that for all t, and $\boldsymbol {\epsilon }$ in the canonical basis of $\Omega _{N}$ ,

  $$ \begin{align*}\psi_{t,2n+1,N} ({\kern1pt}\textbf{p}_{1} (t) , \ldots , \textbf{p}_{2n+1} (t)) [\boldsymbol{\epsilon}] = \sum_{\sigma \in \Sigma_{2n+1}} C_{\sigma}(t)[{\kern1pt}\textbf{p} (t)] \prod_{k=1}^{2n+1} e^{i\textbf{p}_{\sigma(k) (t)} \cdot q_{k} [\boldsymbol{\epsilon}]}.\end{align*} $$
  This expression admits a limit when $t \rightarrow \sqrt {2}$ , given by
  $$ \begin{align*}\sum_{\sigma \in \Sigma_{2n+1}} C_{\sigma}(\sqrt{2})[{\kern1pt}\textbf{p}(\sqrt{2})] \prod_{k=1}^{2n+1} e^{i\textbf{p}_{\sigma(k) (\sqrt{2})} \cdot q_{k} [\boldsymbol{\epsilon}]},\end{align*} $$
  where $({\kern1pt}\textbf{p}_{k} (\sqrt {2}))_{k}$ is solution of the system of equations $(E_{k})[\sqrt {2},2n+1,N]$ .

• • The expression ${\epsilon (\sigma ) C_{\sigma } (\sqrt {2})} {[}\textbf {p}{(\sqrt {2})]}$ is independent from ${\sigma }$ : Indeed, from the definition of $C_{\sigma }(t)$ we have

  $$ \begin{align*} &\prod_{1\le k < l \le 2n+1} (1 + e^{i ({\kern1pt}\textbf{p}_{\sigma(k)} (\sqrt{2})+\textbf{p}_{\sigma(l)} (\sqrt{2}))})\\[6pt] &\ \qquad = \prod_{1\le \sigma^{-1}(k) < \sigma^{-1}(l) \le 2n+1} (1 + e^{i ({\kern1pt}\textbf{p}_{k} (\sqrt{2})+\textbf{p}_{l} (\sqrt{2}))}) \\[6pt] &\ \qquad = \prod_{1\le k < l \le 2n+1} (1 + e^{i ({\kern1pt}\textbf{p}_{k} (\sqrt{2})+\textbf{p}_{l} (\sqrt{2}))}). \end{align*} $$

  Indeed, for all $l \neq k$ , one of the conditions $\sigma ^{-1} (k) < \sigma ^{-1} (l)$ or $\sigma ^{-1} (l) < \sigma ^{-1} (k)$ is verified, exclusively. This means that $(1 + e^{i ({\kern1pt}\textbf{p}_{k} (\sqrt {2})+\textbf {p}_{l} (\sqrt {2}))})$ appears exactly once in the product for each $l,k$ such that $l \neq k$ .

• • The terms ${(}{1 + e}^{{i} {(}\textbf {p}_{{k}} {(\sqrt {2})+}\textbf {p}_{{l}} {(\sqrt {2}))}}{)}$ , ${k \neq l}$ , are all non-zero: Indeed, none of the $\textbf {p}_{k} (\sqrt {2}) + \textbf {p}_{l} (\sqrt {2})$ can be equal to $\pm \pi $ . This comes from the fact that the system of Bethe equations $(E_{k})[\sqrt {2},2n+1,N]$ has a unique simple solution given by

  $$ \begin{align*}\textbf{p}_{k} (\sqrt{2}) = \frac{\pi}{N} \bigg( 2k - \frac{(2n+1+1)}{2}\bigg) = \frac{2\pi}{N} ( k - (n+1) ).\end{align*} $$
  These numbers are bounded respectively above and below by
  $$ \begin{align*}\textbf{p}_{1} (\sqrt{2})= - 2\pi \frac{n-1}{N}, \quad \textbf{p}_{n} (\sqrt{2}) = 2\pi \frac{n-1}{N}.\end{align*} $$
  Since $2n+1 \le N/2$ , these numbers are in $[-\pi /2,\pi /2]$ , and the possible sums of two different values of these numbers is in $]{-}\pi ,\pi [$ .

• • The limit of Bethe vectors is non-zero: As a consequence of the last points, we have that the $\boldsymbol {\epsilon }$ coordinate of the limit of Bethe vectors when $t \rightarrow \sqrt {2}$ is, up to a non-zero constant,

  $$ \begin{align*}\sum_{\sigma \in \Sigma_{2n+1}} \epsilon(\sigma)\cdot \prod_{k=1}^{2n+1} e^{i\textbf{p}_{\sigma(k) (\sqrt{2})} \cdot q_{k} [\boldsymbol{\epsilon}]},\end{align*} $$
  which is the determinant of the matrix $(e^{i\textbf {p}_{\sigma (k) (\sqrt {2})}. Here, q_{l} [\boldsymbol {\epsilon }]})_{k,l}$ , which is a submatrix of the matrix , where $(s_{k})$ is a sequence of distinct numbers in $]{-}\pi /2,\pi /2[$ such that for all $k \le 2n+1$ ,
  $$ \begin{align*}s_{k} = \textbf{p}_{\sigma(k) (\sqrt{2})},\end{align*} $$
  and $(s^{\prime }_{l})_{l}$ is a sequence of distinct integers such that for all $l \le n$ ,
  $$ \begin{align*}s^{\prime}_{l} = q_{l} [\boldsymbol{\epsilon}].\end{align*} $$
  If the determinant is non-zero, then the sum above is non-zero. This is the case since this last matrix is obtained from the Vandermonde matrix , whose determinant is
  $$ \begin{align*}\prod_{k<l} ( e^{is_{l}} - e^{is_{k}}) \neq 0,\end{align*} $$
  by a permutation of the columns.

(2) From the Hamiltonian to the transfer matrix:

• • Eigenvector of ${V_{N} (\sqrt {2})}$ and ${H_{N}}$ : Since the limit of Bethe vectors is not equal to zero, and it satisfies an equation which is the ‘limit’ of equations which make the Bethe vectors candidate eigenvectors, it is an eigenvector of the matrix $V_{N} (\sqrt {2})$ . It is also an eigenvector of the Hamiltonian $H_{N}$ , for the eigenvalue

  (17) $$ \begin{align} 2\bigg(\displaystyle{\sum_{k=1}^{n-1}} \cos(\frac{2\pi k}{N}) + \displaystyle{\sum_{k=N-n+1}^{N}} \cos\bigg(\frac{2\pi k}{N}\bigg)\bigg). \end{align} $$
  This is a consequence of Lemma 11, since for all j, $N\textbf {p}_{j} (\sqrt {2}) = 2\pi (j-(n+1)),$ the eigenvalue is
  $$ \begin{align*} 2\sum_{k=1}^{2n+1} \cos{( \textbf{p}_{k} (\sqrt{2}))} & = 2 \sum_{k=1}^{n} \cos{( \textbf{p}_{k} (\sqrt{2}))}+ 2 \sum_{k=n+1}^{2n+1} \cos{( N - \textbf{p}_{k} (\sqrt{2}))}\\[6pt] & = 2\bigg(\displaystyle{\sum_{k=1}^{n-1}} \cos\bigg(\frac{2\pi k}{N}\bigg) + \displaystyle{\sum_{k=N-n+1}^{N}} \cos\bigg(\frac{2\pi k}{N}\bigg)\bigg). \end{align*} $$

• • Comparison with the other eigenvalues of ${H}$ : From Theorem 5, we know that the number given by the expression in equation (17) is the largest eigenvalue of $H_{N}$ on $\Omega _{N}^{(2n+1)}$ . Indeed, it is straightforward that $\psi _{\alpha }$ is in $\Omega _{N}^{(2n+1)}$ if and only if the number of k such that $\alpha _{k} = 1$ is $2n+1$ . The sum in the statement of Theorem 5 is maximal among these sequences when

  $$ \begin{align*}\alpha_{1} = \cdots = \alpha_{n-1}=1 = \alpha_{N-n+1} = \cdots = \alpha_{N}\end{align*} $$
  and the other $\alpha _{k}$ are equal to $0$ .

• • Identification: As a consequence, from Perron–Frobenius theorem, the limit of Bethe vectors in $\sqrt {2}$ is positive, thus this is also true for t sufficiently close to $\sqrt {2}$ . From the same theorem, the Bethe vector is associated to the maximal eigenvalue of $V_{N} (t)$ . As a consequence, the Bethe value $\Lambda _{2n+1,N} (t) [{\kern1pt}\textbf{p}_{1} (t), \cdots \textbf {p}_{2n+1} (t)]$ is equal to the largest eigenvalue $\lambda _{2n+1,N} (t)$ of $V_{N} (t)$ on $\Omega _{N}^{(2n+1)}$ for these values of t. Since these two functions are analytic in t (by the implicit functions theorem on the characteristic polynomial, using the fact that the largest eigenvalue is simple), one can identify these two functions on the interval $(0,\sqrt {2})$ .