Polyakov, D. M. Nonlocal perturbation of a periodic problem for a second-order differential operator. (English. Russian original) Zbl 1462.34118 Differ. Equ. 57, No. 1, 11-18 (2021); translation from Differ. Uravn. 57, No. 1, 14-21 (2021). The paper under review deals with the differential operator \(L:D(L)\subset L_2 (0,1) \rightarrow L_2 (0, 1)\) of the form \[ (Lu)(x)\equiv-u^{\prime\prime}+q(x)u(x),\ x\in(0,1) \] with the periodic and nonlocal boundary conditions \[ u(0)=u(1),\quad u^{\prime} (0)=u^{\prime} (1)+ \int^{1}_{0} \overline{p(x)} u(x) dx. \] Here \(D(L)\subset W^{2}_2 (0,1)\) and the functions \(p\) and \(q\) are complex-valued functions in \(L_2(0,1)\).The author obtains asymptotic formulas for the eigenvalues of the operator \(L\) as well as establishing a number of assertions on the estimates of deviations of spectral projections. These estimates permit one to obtain a new result on the Bari basis property for subspaces of the space \(L_2(0, 1)\). Reviewer: Erdogan Sen (Tekirdağ) Cited in 4 Documents MSC: 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 34D10 Perturbations of ordinary differential equations Keywords:periodic problem; asymptotics of eigenvalues; nonlocal perturbations PDFBibTeX XMLCite \textit{D. M. Polyakov}, Differ. Equ. 57, No. 1, 11--18 (2021; Zbl 1462.34118); translation from Differ. Uravn. 57, No. 1, 14--21 (2021) Full Text: DOI References: [1] Feller, W., The parabolic differential equations and the associated semi-groups of transformations, Ann. Math., 55, 4, 468-519 (1952) · Zbl 0047.09303 [2] Feller, W., Diffusion processes in one dimension, Trans. Am. Math. Soc., 77, 1-30 (1954) · Zbl 0059.11601 [3] Shkalikov, A. A., On the basis property of eigenfunctions of ordinary differential equations with integral boundary conditions, Moscow Univ. Math. Bull., 37, 6, 10-20 (1982) · Zbl 0565.34020 [4] Baskakov, A. G.; Katsaran, T. K., Spectral analysis of integro-differential operators with nonlocal boundary conditions, Differ. Equations, 24, 8, 934-941 (1988) · Zbl 0671.47040 [5] Makin, A. S., On a nonlocal perturbation of a periodic eigenvalue problem, Differ. Equations, 42, 4, 599-602 (2006) · Zbl 1132.34332 [6] Sadybekov, M. A.; Imanbaev, N. S., On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition, Differ. Equations, 48, 6, 896-900 (2012) · Zbl 1272.34115 [7] Imanbaev, N. S.; Sadybekov, M. A., On spectral properties of a periodic problem with an integral perturbation of the boundary condition, Eurasian Math. J., 4, 3, 53-62 (2013) · Zbl 1336.35260 [8] Sadybekov, M. A.; Imanbaev, N. S., A regular differential operator with perturbed boundary condition, Math. Notes, 101, 5, 878-887 (2017) · Zbl 1391.34136 [9] Skubachevskii, A. L.; Steblov, G. M., On the spectrum of differential operators with a domain nondense in \(L_2(0,1)\), Dokl. Akad. Nauk SSSR, 321, 6, 1158-1163 (1991) [10] Gokhberg, I. Ts.; Krein, M. G., Introduction to the Theory of Linear Nonself-Adjoint Operators in a Hilbert Space (1969), Providence, RI: Am. Math. Soc., Providence, RI [11] Krall, A. M., The development of general differential and general differential-boundary systems, Rocky Mountain J. Math., 5, 4, 493-542 (1975) · Zbl 0322.34009 [12] Baskakov, A. G.; Polyakov, D. M., The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential, Sb. Math., 208, 1, 1-43 (2017) · Zbl 1441.47054 [13] Lomov, I. S.; Chernov, V. V., Study of spectral properties of a loaded second-order differential operator, Differ. Equations, 51, 7, 857-861 (2015) · Zbl 1335.34136 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.