Tags: Fixed Point Method

Intro

We want to solve the Bethe Equations,

\(\xi_i^L = (-1)^{N-1} \prod_{j=1}^{N} \frac{1 + \xi_i \xi_j - \Delta \xi_j}{1 + \xi_i \xi_j - \Delta \xi_i}, \quad i= 1, \dots, N\),

via a fixed point method, i.e. a sequnce \(\xi(k), k \in \mathbb{Z}_{\geq 1}\) that converges to the solution. See this post (Post 1) and this post (Post 2).

Issue

In the previous post, we defined sequences that converges to the solution of the Bethe Equations. The only issue is that the sequences were defined via functions that were not continuous (and blow up to infinity) on certain subsets of the domain of the function. This means that we were not able to apply the Fixed Point Theorem (FPT) directly. We either need to show that the sequences stay in the domain where the funcition is continuous or we need to resolve the singularity.

Compactification Idea

Let us consider the second option of resolving the sigularity of the fucntion. In particular, we will compactify the domain and range.

We have a sequnce, \(\xi(k) \in \mathbb{C}^{N}\) defined as followns

\[\xi(k +1) = F(\xi(k))\]

for some function \(F: \mathbb{C}^{N} \rightarrow \mathbb{C}^{N}\). We aim to introduce an extension

\[\tilde{F}: (\mathbb{CP})^{N} \rightarrow (\mathbb{CP})^{N}\]

so that \(\tilde{F}(\xi) = F(\xi)\) for all \(\xi \in \mathbb{C}^{N} \subset (\mathbb{CP})^{N}\).

Coordinates

The coordinates of \(\mathbb{CP}\) are given by \([z_0 :z_1]\) with \(z_0, z_1 \in \mathbb{C}\) so that \(z_0 = z_1 = 0\) is not allowed and \([\lambda z_0 : \lambda z_1]\) is identified to be the same point for any \(\lambda \in \mathbb{C}\) with \(\lambda \neq 0\). In particualar, we have

\[\{ [z_0:1] \mid z_0 \in \mathbb{C} \} \cong \mathbb{C}\]

and the point \([z_0 : 0]\) is identified with \(\infty\). Moreover, we have the inclusion map \(\mathbb{C} \hookrightarrow \mathbb{CP}\) given by

\[z \mapsto [z:1]\]

and the projection map \(\{ [z_0:1] \mid z_0 \in \mathbb{C} \} \subset \mathbb{CP} \rightarrow \mathbb{C}\) given by

\[[z_0 : z_1] \mapsto z_0/z_1\]

when \(z_1 \neq 0\).

In our case, we will identify \(\xi_i = v_i/w_i \in \mathbb{C}\) with \([v_i:w_i] \in \mathbb{CP}\).

Let us extend the \(F\) function form this post (Post 2).

\[\begin{align} &\tilde{F}_i : ([v_i : w_i] \mid i =1, \dots, N) \mapsto \\ &[\prod_{j \neq i} \Delta - \tfrac{w_j}{v_j}: \Delta + (-1)^N \left((\tfrac{v_i}{w_i})^L \prod_{j=1}^{N}(\tfrac{w_j}{v_j} + \tfrac{v_i}{w_i} - \Delta \tfrac{v_i w_j}{w_i v_j}) + (-1)^N \prod_{j=1}^N(\tfrac{v_i}{w_i} - (\Delta - \tfrac{w_j}{v_i})) - \prod_{j =1}^N (\Delta - \tfrac{w_j}{v_j}) \right)] \end{align}\]

In shorthand, denote the function above as \(\tilde{F}_i : \xi \mapsto [\tilde{F}_{i, 0}(\xi): \tilde{F}_{i, 1}(\xi)]\).

Note that \(\tilde{F}_i\) is well-defined only if \(\tilde{F}_{i, 0}(\xi)\) and \(\tilde{F}_{i, 1}(\xi)\) are not simultaneously equal to zero (i.e. the zero sets of \(\tilde{F}_{i, 0}\) and \(\tilde{F}_{i, 1}\) are disjoint).

Question

  • Can we show that the zero sets of \(\tilde{F}_{i, 0}\) and \(\tilde{F}_{i, 1}\) are disjoint?

  • What if \(\Delta =0\)?