Composite model for generalized oscillator. I

We study a realization of given generalized oscillator by the system of $N$ generalized oscillators of other type. Considered generalized oscillator connected with given system of orthogonal polynomials defined by the three terms recurrent relations and related Jacobi matrix $J$. The case $N=2$ was considered in the previous work. In this case measure participated in orthogonality relation for polynomials is symmetric under rotation on the angle $\pi$. In present work we discuss the case $N=3$. We showed that such problem admits the solution only in two cases. The first one is realized when the Jacobi matrix, connected with given (“composite”) generalized oscillator has the block-diagonal form and consists from the similar $3\times 3$ blocks. More interesting is the second possible case in which Jacobi matrix has not block-diagonal form. For this Jacobi matrix we construct the related system of orthogonal polynomials. This system of orthogonal polynomials splits into three series, connected with Chebyshev polynomials of the second kind. The solution of the classical moment problem related to this Jacobi matrix is the main result of this work. The related measure is symmetric under rotation on the angle $2\pi/3$. Bibl. – 6 titles.