On the spectral analysis of a differential operator with an involution and general boundary conditions
Eurasian mathematical journal, Tome 11 (2020) no. 2, pp. 30-39.

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

We study first-order differential operators with an involution and non-periodic boundary conditions. We exhibit their spectral properties such as the asymptotic estimates of their eigenvalues, eigenvectors and spectral projections. We also use these properties to estimate the groups generated by the differential operators we study. The results were obtained by using the method of similar operators.
@article{EMJ_2020_11_2_a3,
     author = {A. G. Baskakov and I. A. Krishtal and N. B. Uskova},
     title = {On the spectral analysis of a differential operator with an involution and general boundary conditions},
     journal = {Eurasian mathematical journal},
     pages = {30--39},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2020_11_2_a3/}
}
TY  - JOUR
AU  - A. G. Baskakov
AU  - I. A. Krishtal
AU  - N. B. Uskova
TI  - On the spectral analysis of a differential operator with an involution and general boundary conditions
JO  - Eurasian mathematical journal
PY  - 2020
SP  - 30
EP  - 39
VL  - 11
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2020_11_2_a3/
LA  - en
ID  - EMJ_2020_11_2_a3
ER  - 
%0 Journal Article
%A A. G. Baskakov
%A I. A. Krishtal
%A N. B. Uskova
%T On the spectral analysis of a differential operator with an involution and general boundary conditions
%J Eurasian mathematical journal
%D 2020
%P 30-39
%V 11
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2020_11_2_a3/
%G en
%F EMJ_2020_11_2_a3
A. G. Baskakov; I. A. Krishtal; N. B. Uskova. On the spectral analysis of a differential operator with an involution and general boundary conditions. Eurasian mathematical journal, Tome 11 (2020) no. 2, pp. 30-39. http://geodesic.mathdoc.fr/item/EMJ_2020_11_2_a3/

[1] A. G. Baskakov, “A theorem on splitting an operator, and some related questions in the analytic theory of perturbations”, Math. USSR-Izv., 28:3, 421–444 | MR | Zbl

[2] A. G. Baskakov, “Estimates for Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations”, Sb. Math., 205:8 (2015), 1049–1086 | MR

[3] A. G. Baskakov, “Methods of abstract harmonic analysis in the perturbation of linear operators”, Siberian Math. Journ., 24:1 (1983), 17–32 | MR

[4] A. G. Baskakov, “Spectral analysis of perturbed nonquasianalytic and spectral operators”, Izv. Math., 45:1 (1995), 1–31 | MR

[5] A. G. Baskakov, A. V. Derbushev, A. O. Scherbakov, “The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials”, Izv. Math., 75:3 (2011), 445–469 | MR | Zbl

[6] A. G. Baskakov, I. A. Krishtal, E. Yu. Romanova, “Spectral analysis of a differential operator with an involution”, J. Evolut. Equat., 17 (2017), 669–684 | MR | Zbl

[7] A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Linear differential operator with an involution as a generator of an operator group”, Operators and Matrices, 12:3 (2018), 723–756 | MR | Zbl

[8] A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Similarity techniques in the spectral analysis of perturbed operator matrices”, J. Math. Anal. Appl., 2019, no. 2, 930–960 | DOI | MR | Zbl

[9] A. G. Baskakov, D. M. Polyakov, “The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential”, Sb. Math., 208:1 (2017), 1–43 | MR | Zbl

[10] A. G. Baskakov, N. B. Uskova, “A generalized Fourier method for the system of first-order differential equations with an involution and a group of operators”, Diff. Equat., 54:2 (2018), 276–280 | MR | Zbl

[11] A. G. Baskakov, N. B. Uskova, “Fourier method for first order differential equations with an involution and for groups of operators”, Ufa Math. Journ., 10:3 (2018), 11–34 | MR

[12] A. G. Baskakov, N. B. Uskova, “Spectral properties of the differential operators with an involution and operator groups”, Diff. Equat., 54:9 (2018), 277–281 | MR | Zbl

[13] M. Sh. Burlutskaya, A. P. Khromov, “Dirac operator with a potential of special form and with the periodic boundary conditions”, Diff. Equat., 54:5 (2018), 586–595 | MR | Zbl

[14] M. Sh. Burlutskaya, A. P. Khromov, “Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution”, Comput. Math. and Math. Phys., 51:12 (2011), 2102–2114 | MR | Zbl

[15] M. Sh. Burlutskaya, A. P. Khromov, “Functional-differential operators with involution and Dirac operators with periodic boundary conditions”, Dokl. Math., 89:1 (2014), 8–10 | MR | Zbl

[16] M. Sh. Burlutskaya, A. P. Khromov, “Mixed problem for simplest hyperbolic first order equations with involution”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:1 (2014), 10–20 | MR | Zbl

[17] M. Sh. Burlutskaya, A. P. Khromov, “Mixed problems for first-order hyperbolic equations with involution”, Dokl. Math., 84:3 (2011), 783–786 | MR | Zbl

[18] A. Cabada, F. A.F. Tojo, “Existence results for a linear equations with reflection, non-constant coefficient and periodic boundary conditions”, J. Math. Anal. Appl., 412:1 (2014), 529–546 | MR | Zbl

[19] A. Cabada, F. A.F. Tojo, “Solutions and GreenTs function of the first order linear equation with reflection and initial conditions”, Boundary Value Problems, 99 (2014), 1–16 | MR

[20] P. Djakov, B. S. Mityagin, “Equiconvergence of spectral decomposition of 1D Dirac operator with regular boundary conditions”, J. Approx. Theory, 164:7 (2012), 879–927 | MR | Zbl

[21] I. Ts. Gohberg, M. G. Krein, An introduction to the theory of linear nonselfadjoint operators in Hilbert space, Amer. Math. Soc., Providence, RI, 1969 | MR

[22] R. E. Kalman, R. S. Bucy, “New results in linear filtering and prediction theory”, Trans. ASME. Ser. D.J. Basic Eng., 86 (1961), 95–108 | MR

[23] 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, 590781 | MR | Zbl

[24] L. V. Kritskov, A. M. Sarsenbi, “Riesz basis property of system of root function of second-order differential operator with involution”, Diff. Equat., 53:1 (2017), 35–48 | MR | Zbl

[25] D. Piao, “Periodic and almost periodic solutions for differential equations with reflection of the argument”, Nonlinear Anal., 57:4 (2004), 633–637 | MR | Zbl

[26] V. A. Pliss, Nonlocal problems of oscillations theory, Nauka, M., 1964 (in Russian) | MR

[27] V. A. Pliss, “Families of periodic solutions of systems of differential equations of second order without dissipation”, Differencial'nye Uravneniya, 1965, no. 1, 1428–1448 | MR | Zbl

[28] A. M. Savchuk, A. A. Shkalikov, “The Dirac operator with complex-valued summable potential”, Math. Notes, 96:5 (2014), 777–810 | MR | Zbl

[29] W. Watkins, “Asymptotic properties of differential equations with involutions”, Int. J. Pure. Appl. Math., 44:4 (2008), 485–492 | MR | Zbl

[30] J. Wiener, A. R. Aftabizadeh, “Boundary value problems for differential equations with reflection of the argument”, Intern. J. Math. Math. Sci., 8:1 (1985), 151–163 | MR | Zbl