Geodesic
On multiple completeness of the root functions of ordinary differential polynomial pencil with constant coefficients
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 340-361.

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In the space of square integrable functions on a finite segment we consider a class of polynomial pencils of $n$th-order ordinary differential operators with constant coefficients and two-point boundary-value conditions (at the edges of the segment). We suppose that roots of the characteristic equation of pencils of this class are simple and nonzero. We establish sufficient conditions for $m$-multiple completeness ($1\le m\le n$) of the system of root functions of pencils from this class in the space of square integrable functions on this segment. 
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V. S. Rykhlov. On multiple completeness of the root functions of ordinary differential polynomial pencil with constant coefficients. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 340-361. http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a6/

[1] A. I. Vagabov, Expansions into Fourier Series with Respect to Main Functions of Differential Operators and Their Applications, PhD Thesis, Moscow, 1988 (in Russian)

[2] A. I. Vagabov, Introduction to the Spectral Theory of Differential Operators, Izd-vo Rost. un-ta, Rostov-na-Donu, 1994 (in Russian)

[3] M. G. Gasymov, A. M. Magerramov, “On multiple completeness of a system of eigenfunctions and adjoined functions of one class of differential operators”, Dokl. AN Azerb. SSR, 30:12 (1974), 9–12 (in Russian) | MR | Zbl

[4] M. V. Keldysh, “On eigenvalues and eigenfunctions of some classes of nonself-adjoint equations”, Dokl. AN SSSR, 77:1 (1951), 11–14 (in Russian) | Zbl

[5] M. V. Keldysh, “On completeness of eigenfunctions of some classes of nonself-adjoint linear operators”, Usp. mat. nauk, 26:4 (1971), 15–41 (in Russian) | MR | Zbl

[6] M. A. Naimark, Linear Differential Operators, Nauka, Moscow, 1969 (in Russian) | MR

[7] V. S. Rykhlov, “Multiple completeness of eigenfunctions of ordinary differential polynomial pencil”, Investigations in the Theory of Operators, BNTs UrO AN SSSR, Ufa, 1988, 128–140 (in Russian)

[8] V. S. Rykhlov, “On completeness of eigenfunctions of quadratic pencils of ordinary differential operators”, Izv. vuzov. Ser. Mat., 1992, no. 2, 35–44 (in Russian) | MR | Zbl

[9] V. S. Rykhlov, “On properties of eigenfunctions of ordinary differential quadratic pencil of second order”, Integr. Transforms and Special Functions, Inform. Bull., 2, no. 1, Nauchno-issledovat. gruppa mezhdun. zhurnala “Integral Transforms and Special Functions” i VTs RAN, Moscow, 2001, 85–103 (in Russian)

[10] V. S. Rykhlov, “On double completeness of eigenfunctions of one quadratic pencil of second-order differential operators”, Oper. Theory, Diff. Equ., and Function Theory, Proc. Math. Inst. Acad. Sci. Ukraine, 6, no. 1, 2009, 237–249 (in Russian) | Zbl

[11] V. S. Rykhlov, “On completeness of root functions of simplest strongly irregular differential operators with two-term, two-point boundary-value conditions”, Dokl. RAN, 428:6 (2009), 740–743 (in Russian) | MR | Zbl

[12] V. S. Rykhlov, “On completeness of eigenfunctions of one class of pencils of differential operators with constant coefficients”, Izv. vuzov. Ser. Mat., 2009, no. 6, 42–53 (in Russian) | MR | Zbl

[13] V. S. Rykhlov, “On properties of eigenfunctions of one quadratic pencil of differential operators”, Izv. Sarat. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 9:1 (2009), 31–44 (in Russian)

[14] V. S. Rykhlov, “On multiple completeness of root functions of one class of pencils of differential operators”, Izv. Sarat. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 10:2 (2010), 24–34 (in Russian)

[15] V. S. Rykhlov, “On completeness of root functions of polynomial pencils of ordinary differential operators with constant coefficients”, Tavr. vestn. inform. i mat., 2015, no. 1(26), 69–86 (in Russian)

[16] V. S. Rykhlov, “Multiple completeness of root functions of some irregular pencils of differential operators”, Tavr. vestn. inform. i mat., 2016, no. 1(31), 87–103 (in Russian)

[17] V. S. Rykhlov, O. V. Blinkova, “On multiple completeness of root functions of one class of pencils of differential operators with constant coefficients”, Izv. Sarat. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 14:4, pt. 2 (2014), 574–584 (in Russian) | Zbl

[18] V. S. Rykhlov, O. V. Parfilova, “On multiple completeness of root functions of pencils of differential operators with constant coefficients”, Izv. Sarat. un-ta. Nov. ser. Ser. Mat. Mekh. Inform., 11:4 (2011), 45–58 (in Russian)

[19] S. A. Tikhomirov, Finite-Dimensional Perturbations of Integral Volterra Operators in the Space of Vector Functions, PhD Thesis, Saratov, 1987 (in Russian)

[20] A. P. Khromov, Finite-Dimensional Perturbations of Volterra Operators, Doctoral Thesis, Novosibirsk, 1973 (in Russian)

[21] A. P. Khromov, “On generating functions of Volterra operators”, Mat. sb., 102(144):3 (1977), 457–472 (in Russian) | MR | Zbl

[22] A. A. Shkalikov, “On completeness of eigenfunctions and adjoined functions of ordinary differential operators with irregular boundary-value conditions”, Funkts. analiz i ego prilozh., 10:4 (1976), 69–80 (in Russian) | MR | Zbl

[23] A. A. Shkalikov, “Boundary-value problems for ordinary differential equations with a parameter in boundary-value conditions”, Tr. sem. im. I. G. Petrovskogo, 9, 1983, 190–229 (in Russian) | Zbl

[24] Eberhard W., “Zur Vollständigkeit des Biorthogonalsystems von Eigenfunktionen irregulärer Eigenwertprobleme”, Math. Z., 146:3 (1976), 213–221 | DOI | MR

[25] Freiling G., “Zur Vollständigkeit des Systems der Eigenfunktionen und Hauptfunktionen irregulärer Operator büschel”, Math. Z., 188:1 (1984), 55–68 | DOI | MR | Zbl

[26] Freiling G., “Über die mehrfache Vollständigkeit des Systems der Eigenfunktionen und assoziierten Funktionen irregulärer Operatorenbüschel in $L_2[0,1]$”, ZAMM Z. Angew. Math. Mech., 65:5 (1985), 336–338 | MR

[27] Rykhlov V. S., “Multiple completeness of the root functions for a certain class of pencils of ordinary differential operators with constant coefficients”, Results Math., 68:3–4 (2015), 427–440 | DOI | MR | Zbl

[28] Rykhlov V. S., “Multiple completeness of the root functions for a certain class of pencils of ordinary differential operators”, Results Math., 72:1–2 (2017), 281–301 | DOI | MR | Zbl