Discrete Spectrum of the Jacobi Matrix Related to Recurrence Relations with Periodic Coefficients

Abstract

In this note, we study the discrete spectrum of the Jacobi matrix corresponding to polynomials defined by recurrence relations with periodic coefficients. As examples, we consider

(a) the case where the period N of coefficients of recurrence relations equals 3 (as a particular case, we consider “parametric” Chebyshev polynomials introduced by the authors earlier);

(b) elementary N-symmetric Chebyshev polynomials (N = 3, 4, 5), which were introduced by the authors in the study of the “composite model of a generalized oscillator.”