Spectral properties of some tridiagonal matrices

In the talk, we survey a few recent works on spectral properties of tridiagonal matrices — including certain our results. In particular, we compare several viewpoints to generalised Sylvester–Kac matrices, as well as to other matrices whose spectra are “linear” (i.e., spectral points constitute arithmetic progressions). In connection with this question, we present a nice property tridiagonal matrices with zero diagonals, give some examples of Leonard pairs and a certain related algebraic structure. We also briefly touch upon matrices whose spectra are simple and “quadratic”.
The talk is based on a joint work with Mikhail Tyaglov.