Abstract
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 Jacobi matrices appear. We examine the problem of defect indexes for ordinary differential operators and generalized Jacobi matrices corresponding to them in the spaces L2(−∞,+∞) and l2, 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 than 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.
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References
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).
Yu. M. Berezansky, Expansions by Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
A. L. Chistyakov, “Defect indices of Jm-matrices and differential operators with polynomial coefficients,” Mat. Sb., 85 (127), No. 4 (8), 474–503 (1971).
A. L. Chistyakov, “Defect indices of symmetric operators in the direct sum of Hilbert spaces, I,” Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 3, 5–21 (1969).
A. L. Chistyakov, “Defect indices of symmetric operators in the direct sum of Hilbert spaces, II,” Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 4, 3–5 (1969).
N. Dunford and J. T. Schwartz, Linear Operators. Spectral Theory, Interscience, New York–London (1963).
A. G. Kostyuchenko and K. A. Mirzoev, “Generalized Jacobi matrices and defect indices of ordinary differential operators with polynomial coefficients,” Funkts. Anal. Prilozh., 33, No. 1, 30–45 (1999).
A. G. Kostyuchenko and K. A. Mirzoev, “Criteria of complete uncertainty of Jacobi matrices with matrix elements,” Funkts. Anal. Prilozh., 35, No. 4, 32–37 (2001).
M. G. Krein, “Foundations of the representation theory of Hermitian operators with defect index (m,m),” Ukr. Mat. Zh., 2, 3–66 (1949).
M. G. Krein, “Infinite J-matrices and the matrix problem of moments,” Dokl. Akad. Nauk SSSR, 69, No. 3, 125–128 (1949).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.
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Mirzoev, K.A., Konechnaya, N.N., Safonova, T.A. et al. Generalized Jacobi Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients. J Math Sci 252, 213–224 (2021). https://doi.org/10.1007/s10958-020-05154-9
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DOI: https://doi.org/10.1007/s10958-020-05154-9
Keywords and phrases
- regular differential expression
- irregular differential expression
- differential operator
- generalized Jacobi matrix
- defect index
- integer operators of minimal type