Geodesic
Regular Ordinary Differential Operators with Involution
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 643-659.

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The main results of the paper are related to the study of differential operators of the form $$ Ly = y^{(n)}(-x) + \sum_{k=1}^n p_k(x) y^{(n-k)}(-x) + \sum_{k=1}^n q_k(x) y^{(n-k)}(x),\qquad \ x\in [-1,1], $$ with boundary conditions of general form concentrated at the endpoints of a closed interval. Two equivalent definitions of the regularity of boundary conditions for the operator $L$ are given, and a theorem on the unconditional basis property with brackets of the generalized eigenfunctions of the operator $L$ in the case of regular boundary conditions is proved.
Keywords: operators with involution, regular differential operators, basis property of eigenfunctions of operators, Riesz bases. 
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V. E. Vladykina; A. A. Shkalikov. Regular Ordinary Differential Operators with Involution. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 643-659. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a0/

[1] V. E. Vladykina, A. A. Shkalikov, “Spektralnye svoistva obyknovennykh differentsialnykh operatorov s involyutsiei”, Dokl. AN, 484:1 (2019), 12–17 | DOI | Zbl

[2] A. A. Kopzhassarova, A. L. Lukashov, A. M. Sarsenbi, “Spectral properties of non-self-adjoint perturbations for a spectral problem with involution”, Abstr. Appl. Anal., 2012 (2012), Art. ID 590781 | DOI | MR

[3] V. P. Kurdyumov, A. P. Khromov, “O bazisakh Rissa iz sobstvennykh i prisoedinennykh funktsii funktsionalno-differentsialnogo uravneniya s operatorom otrazheniya”, Differents. uravneniya, 44:2 (2008), 196–204 | MR | Zbl

[4] T. Sh. Kalmenov, U. A. Iskakova, “Kriterii silnoi razreshimosti smeshannoi zadachi Koshi dlya uravneniya Laplasa”, Dokl. AN, 414:2 (2007), 168–171 | MR | Zbl

[5] A. M. Sarsenbi, “Bezuslovnye bazisy, svyazannye s neklassicheskim differentsialnym operatorom vtorogo poryadka”, Differents. uravneniya, 46:4 (2010), 506–511 | MR | Zbl

[6] A. Kopzhassarova, A. Sarsenbi, “Basis properties of eigenfunctions of second-order differential operators with involution”, Abstr. Appl. Anal., 2012 (2012), Art. ID 576843 | DOI | MR

[7] M. A. Sadybekov, A. M. Sarsenbi, “Kriterii bazisnosti sistemy sobstvennykh funktsii operatora kratnogo differentsirovaniya s involyutsiei”, Differents. uravneniya, 48:8 (2012), 1126–1132 | MR | Zbl

[8] V. P. Kurdyumov, “O bazisakh Rissa iz sobstvennykh funktsii differentsialnogo operatora vtorogo poryadka s involyutsiei i integralnymi kraevymi usloviyami”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:4 (2015), 392–405 | DOI | Zbl

[9] L. V. Kritskov, A. M. Sarsenbi, “Spektralnye svoistva odnoi nelokalnoi zadachi dlya differentsialnogo uravneniya vtorogo poryadka s involyutsiei”, Differents. uravneniya, 51:8 (2015), 990–996 | MR | Zbl

[10] A. A. Shkalikov, “Kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii s parametrom v granichnykh usloviyakh”, Tr. sem. im. I. G. Petrovskogo, 9 (2007), 190–229 | MR

[11] B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[12] G. D. Birkhoff, “On the asymptotic character of the solution of certain linear differential equations containing a parameter”, Trans. Amer. Math. Soc., 9:2 (1908), 219–231 | DOI | MR

[13] G. D. Birkhoff, “Boundary value and expansion problem of ordinary linear differential equations”, Trans. Amer. Math. Soc., 9:4 (1908), 373–395 | DOI | MR | Zbl

[14] M. A. Naimark, Lineinye differentsialnye operatory, Nauka, M., 1969 | MR | Zbl

[15] A. A. Shkalikov, “O bazisnosti sobstvennykh funktsii obyknovennogo differentsialnogo operatora”, UMN, 34:5 (209) (1979), 235–236 | MR | Zbl

[16] A. A. Shkalikov, “Vozmuscheniya samosopryazhennykh i normalnykh operatorov s diskretnym spektrom”, UMN, 71:5 (431) (2016), 113–174 | DOI | MR | Zbl

[17] N. Danford, Dzh. Shvarts, Lineinye operatory. T. 3. Spektralnye operatory, Mir, M., 1974 | MR | Zbl

[18] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR | Zbl

[19] A. S. Markus, Vvedenie v spektralnuyu teoriyu polinomialnykh operatornykh puchkov, Shtiintsa, Kishinev, 1986 | MR | Zbl

[20] A. K. Motovilov, A. A. Shkalikov, “Sokhranenie svoistva bezuslovnoi bazisnosti pri nesamosopryazhennykh vozmuscheniyakh samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 53:3 (2019), 45–60 | DOI

[21] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1995 | MR | Zbl

[22] I. D. Evzerov, P. E. Sobolevskii, “Drobnye stepeni obyknovennykh differentsialnykh operatorov”, Differents. uravneniya, 9:2 (1973), 228–240 | MR

[23] I. D. Evzerov, “Oblasti opredeleniya drobnykh stepenei obyknovennykh differentsialnykh operatorov v prostranstvakh $L_p$”, Matem. zametki, 21:4 (1977), 509–518 | MR | Zbl

[24] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, P. E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR | Zbl

[25] D. V. Treschev, A. A. Shkalikov, “O gamiltonovosti lineinykh dinamicheskikh sistem v gilbertovom prostranstve”, Matem. zametki, 101:6 (2017), 911–918 | DOI | MR