Defect indices of $J_m$-matrices and of differential operators with polynomial coefficients

The problem of the defect indices of the symmetric operator $C$ which acts on the space $l_2$ and is generated by a regular Hermitian $J_m$-matrix with increasing elements is investigated. The asymptotics, as $k\to\infty$, of the eigenvectors $U=(u_0,u_1,\dots,u_k,\dots)$ of the operator $C^*$ which correspond to the nonreal eigenvalues are obtained. The results are applied to ordinary differential operators with polynomial coefficients defined on the entire $x$-axis.
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Bibliography: 15 titles.