The course is an introduction to the spectral theory of the difference operators connected with recurrence relations for rational approximants and orthogonal polynomials.

We start with the classical spectral theory of Jacobi matrices (discrete Schrodinger operators) and asymptotics of orthogonal polynomials. Then we touch spectral properties of difference operators generated by the three terms recurrence relations for the denominators (numerators) of the rational Pade approximants. The bulk of the course is devoted to the Hermite-Pade approximants for vector of analytic functions and multiple orthogonal polynomials with respect to various systems of measures: to their asymptotics, recurrence relations and spectral properties of the corresponding difference operators (of higher order or higher dimensionality).

Applications to the random matrices, integrable systems and special function theories will be briefly illuminated.