Generalized Jacobian Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients

This paper is devoted to the matrix representation of ordinary symmetric differential operators with polynomial coefficients on the whole axis. We prove that in this case, generalized Jacobian matrices appear. We examine the problem of defect indexes for ordinary differential operators and generalized Jacobian matrices corresponding to them in the spaces $\mathcal{L}^2(-\infty,+\infty)$ and $l^2$, respectively, and analyze the spectra of self-adjoint extensions of these operators (if they exist). This method allows one to detect new classes of entire differential operators of minimal type (in the sense of M. G. Krein) with certain defect numbers. In this case, the defect numbers of these operators can be not only less or equal, but also greater than the order of the corresponding differential expressions. In particular, we construct examples of entire differential operators of minimal type that are generated by irregular differential expressions.