Consider the matrices $A(n)$, which are $2n \times 2n$ square matrices with entries
$$A(n) = \begin{pmatrix} 0 & v & 0 & 0 & 0 & 0 & \dots & 0 \\ v & 0 & w & 0 & 0 & 0 & \dots & 0 \\ 0 & w & 0 & v & 0 & 0 & \dots & 0 \\ 0 & 0 & v & 0 & w & 0 & \dots & 0 \\ 0 & 0 & 0 & w & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \end{pmatrix},$$
i.e. those two band diagonals alternate between $v$ and $w$.
I am interested in finding the eigenvalues of these matrices, particularly as we take the limit $n\rightarrow\infty$. Even if there are no exact solutions, maybe there are some other ways of expressing the eigenvalue in an equation form or something. Any signposting to related identities or papers is appreciated. Thanks!
It is actually easier to deal with the infinite-dimensional matrices, $n \to \infty$. The eigenvalues are basically arbitrary and continuous, just as the ones of $\hat x$ in quantum mechanics which you may represent by $a + a^\dagger$ as is well known there.
If $w=0$, then $A= v( \sigma_1\oplus \sigma_1\oplus\sigma_1\oplus\sigma_1\oplus ...)$ with the standard obvious eigenvalues of the Pauli matrix, $\pm 1$, so $\pm v$. Likewise for $v=0$, when $A= w( 0\oplus \sigma_1\oplus\sigma_1\oplus\sigma_1\oplus,...)$ so $\pm w$.
If $w\neq 0\neq v$, then you may take $w$ out of the matrix, and redefine $v/w\mapsto v$, so this becomes a one-parameter problem, $$A/w = \begin{pmatrix} 0 & v & 0 & 0 & 0 & 0 & \dots & 0 \\ v & 0 & 1 & 0 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & v & 0 & 0 & \dots & 0 \\ 0 & 0 & v & 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \vdots \end{pmatrix},$$
To find your bearings, first consider the null eigenvector $(1,0,-v,0,v^2,0, ...)^T$.
your then appreciate the recursive structure of the eigenvector $\vec E_\lambda=(1,e_2,e_3,e_4, ...)^T $ yielding $$ \lambda= v e_2 \\ \lambda e_2= v +e_3 \\ \lambda e_3= e_2+ v e_4 \\ ... $$ which allows finding the eigenvectors for arbitrary eigenvalue $\lambda$, $\vec E_\lambda= (1,~\lambda/v, ~\lambda^2/v-v, ~\lambda (\lambda^2-v^2-1)/v, ~v^2+ \lambda^2(\lambda^2-v^2 -v-1)/v , ...)^T$.
The components $e_m$ should not increase with m without bound, which might provide restrictions on the range of allowable eigenvalues $\lambda$.