Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with the nearest neighbours only. However, if such a system is arranged to a circle (or endowed with suitable periodical conditions on the boundary) one gets a slightly different matrix. In my case, it is a matrix of the following form: $$ A=\begin{pmatrix} a_{1} & 1 & 0 & \ldots & 0 & 0 & 1\\ 1 & a_{2} & 1 & \ldots & 0 & 0 & 0\\ 0 & 1 & a_{3} & \ldots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & \ldots & 1 & a_{n-1} & 1\\ 1 & 0 & 0 & \ldots & 0 & 1 & a_{n}\\ \end{pmatrix}. $$

Thus, $A$ is a special tridiagonal matrix with two additional 1's on positions $(1,n)$ and $(n,1)$. So far, I did not find any literature devoted to the study of mathematical properties of these matrices. I am interested in spectral properties of these matrices, in particular. Do you known some books, papers, etc.? Many thanks.