Eigenvalues of almost Tridiagonal matrix

Question

i would like to find a (closed) formula for the eigenvalues of the following matrix A: $$A=\begin{pmatrix} a_n^{-2}+a_1^{-2} & -a_1^{-2} & 0 & \cdots & 0 & -a_n^{-2} \\ -a_1^{-2} & a_1^{-2}+a_2^{-2} & -a_2^{-2} & \cdots & 0 & 0 \\ 0 & -a_2^{-2} & a_2^{-2}+a_3^{-2} & -a_3^{-2} & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & -a_{n-2}^{-2} & a_{n-2}^{-2}+a_{n-1}^{-2} & -a_{n-1}^{-2} \\ -a_n^{-2} & 0 & \cdots & 0 & -a_{n-1}^{-2} & a_{n-1}^{-2}+a_n^{-2} \end{pmatrix}$$ where let us set $1<a_i, \forall i=1,\dots ,n$ for the beginning. I know the formula for $a_i=1, \forall i$ due to a discret fourier transform, but now i would like to formulate a formula for the more general case. I would be really happy if some of you could help me.