Ferguson in his paper in 1980 studied properties of periodic Jacobi matrices which is the following matrix
$$
A = \begin{bmatrix}
a_1&b_1&0&0&0&0& \cdots & b_n\\
b_1&a_2&b_2&0&0&0&\cdots&0\\
0&b_2&a_3&b_3&0&0&\cdots&0\\
0&0&b_3&a_4&b_4&0&\cdots&0\\
0&0&0&b_4&a_5&b_5&\cdots&0\\
0&0&0&\ddots&\ddots&\ddots&\ddots&\vdots\\
0&0&0&0&0&b_{n-2}&a_{n-1}&b_{n-1}\\
b_n&0&0&0&0&0&b_{n-1}&a_n
\end{bmatrix}
$$
For a given eigenpair $(\lambda, X)$ of $A$ where $X = [x_1, \dots, x_n]$ we have
$$
b_{i-1}x_{i-1} + a_ix_i + b_ix_{i+1} = \lambda x_i, \text{ for } i=1, \cdots, n
$$
such that $b_0 = b_n, x_0=x_n, x_{n+1} = x_1$. He solved this system of linear equations to build such matrix. In fact it is an inverse eigenvalue problem, constructing a matrix of a specific form from spectral data.
There is something I can't understand is this statement
By analogy with Floquet theory, which analyzes the analogous problem for ordinary differential equations, let us consider the nontrivial solutions of this recurrence relation which satisfy the boundary conditions
$$
z_n = \rho z_0, z_{n+1} = \rho z_1
$$
Here the parameter $\rho$ is called the Floquet multiplier of the solution $x$. For $\rho \neq 0$ a nontrivial solution exists if and only if $\lambda$ is an eigenvalue of the matrix $$
L = \begin{bmatrix}
a_1&b_1&0&0&0&0& \cdots & \dfrac{1}{\rho} b_n\\
b_1&a_2&b_2&0&0&0&\cdots&0\\
0&b_2&a_3&b_3&0&0&\cdots&0\\
0&0&b_3&a_4&b_4&0&\cdots&0\\
0&0&0&b_4&a_5&b_5&\cdots&0\\
0&0&0&\ddots&\ddots&\ddots&\ddots&\vdots\\
0&0&0&0&0&b_{n-2}&a_{n-1}&b_{n-1}\\
\rho b_n&0&0&0&0&0&b_{n-1}&a_n
\end{bmatrix}
$$
How did he apply the Floquet theory to this problem? There is no function, derivative or differential equations here, it is just a system of periodic linear equations.
Thanks in advance
Best Answer
Thank you for the paper link. It's right down my alley. The fact that the matrix is three band connects it to ordinary differential operator of second order by $$\frac{d}{dx}(a(x)\frac{df(x)}{dx})+b(x)f(x)\approx a(x_{k-1})(f(x_k)-f(x_{k-1}))+a(x_{k+1})(f(x_k)-f(x_{k+1}))+b(x_k)f(x_k).$$ Where $a(x)$ and $b(x)$ should be discretized with some care. The eigenvalues $\rho$ correspond to eigenvalues of the monodoromy of the differential operator. In general $2n-1$ band matrix corresponds to $n$th order differential operator/equation. I've developed a whole theory connecting discrete/difference equations and continuous/differential equations Floquet multipliers. There is a beautiful connection to continued fractions. Here are my PhD thesis and a paper on the subject. Please, do not hesitate to ask more questions on the topic.
https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkYXZlZGNsdWJ8Z3g6MTczNjY5NzM5YTA4YmUzYw
https://www.academia.edu/35236460/Parameter_continuation_of_the_circular_domain_boundary_value_map