Geodesic
Generalized Jacobian Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 91-102.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: regular and irregular differential expression, differential operator, generalized Jacobian matrices, defect index, integer operators of minimal type. 
@article{INTO_2018_152_a7,
     author = {K. A. Mirzoev and N. N. Konechnaja and T. A. Safonova and R. N. Tagirova},
     title = {Generalized {Jacobian} {Matrices} and {Spectral} {Analysis} of {Differential} {Operators} with {Polynomial} {Coefficients}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {91--102},
     publisher = {mathdoc},
     volume = {152},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_152_a7/}
}
TY  - JOUR
AU  - K. A. Mirzoev
AU  - N. N. Konechnaja
AU  - T. A. Safonova
AU  - R. N. Tagirova
TI  - Generalized Jacobian Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2018
SP  - 91
EP  - 102
VL  - 152
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2018_152_a7/
LA  - ru
ID  - INTO_2018_152_a7
ER  - 
%0 Journal Article
%A K. A. Mirzoev
%A N. N. Konechnaja
%A T. A. Safonova
%A R. N. Tagirova
%T Generalized Jacobian Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2018
%P 91-102
%V 152
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2018_152_a7/
%G ru
%F INTO_2018_152_a7
K. A. Mirzoev; N. N. Konechnaja; T. A. Safonova; R. N. Tagirova. Generalized Jacobian Matrices and Spectral Analysis of Differential Operators with Polynomial Coefficients. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 91-102. http://geodesic.mathdoc.fr/item/INTO_2018_152_a7/

[1] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR

[2] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965

[3] Danford N. Shvarts Dzh. T., Lineinye operatory: Spektralnaya teoriya, Mir, M., 1966

[4] Kostyuchenko A. G., Mirzoev K. A., “Obobschennye yakobievy matritsy i indeksy defekta obyknovennykh differentsialnykh operatorov s polinomialnymi koeffitsientami”, Funkts. anal. prilozh., 33:1 (1999), 30–45 | DOI | MR | Zbl

[5] Kostyuchenko A. G., Mirzoev K. A., “Priznaki vpolne neopredelennosti yakobievykh matrits s matrichnymi elementami”, Funkts. anal. prilozh., 35:4 (2001), 32–37 | DOI | MR | Zbl

[6] Krein M. G., “Osnovnye polozheniya teorii predstavleniya ermitovykh operatorov s indeksom defekta $(m,m)$”, Ukr. mat. zh., 2 (1949), 3–66 | Zbl

[7] Krein M. G., “Beskonechnye $J$-matritsy i matrichnaya problema momentov”, Dokl. AN SSSR, 69:3 (1949), 125–128 | Zbl

[8] Chistyakov A. L., “Indeksy defekta $J_m$-matrits i differentsialnykh operatorov s polinomialnymi koeffitsientami”, Mat. sb., 85 (127):4 (8) (1971), 474–503 | Zbl

[9] Chistyakov A. L., “Indeksy defektov simmetricheskikh operatorov v pryamoi summe gilbertovykh prostranstv, I”, Vestn. MGU. Ser. mat. mekh., 1969, no. 3, 5–21

[10] Chistyakov A. L., “Indeksy defektov simmetricheskikh operatorov v pryamoi summe gilbertovykh prostranstv, II”, Vestn. MGU. Ser. mat. mekh., 1969, no. 4, 3–5