We consider a two-term even order operator with involution in the domain given by periodic or antiperiodic boundary conditions. We obtain asymptotics of eigenvalues and the regularized trace formula.
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References
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York (1996).
A. Cabada, F. A. F. Tojo, Differential Equations with Involutions, Atlantis Press, Amsterdam (2015).
V. P. Kurdyumov and A. P. Khromov, “Riesz bases formed by root functions of a functionaldifferential equation with a reflection operator,” Differ. Equ. 44, No. 2, 203–212 (2008).
A. A. Kopzhassarova, A. L. Lukashov, and A. M. Sarsenbi, “Spectral properties of nonself-adjoint perturbations for a spectral problem with involution,” Abstr. Appl. Anal. 2012, Article No. 590781 (2012).
A. G. Baskakov, I. A. Krishtal, and E. Yu. Romanova, “Spectral analysis of a differential operator with an involution,” J. Evol. Equ. 17, No. 2, 669–684 (2017).
V. P. Kurdyumov, “On Riesz bases of eigenfunction of 2-nd order differential operator with involution and integral boundary conditions” [in Russian], Izv. Sarat. Univ., Ser. Mat. Mekh. Inform. 15, No. 4, 392–405 (2015).
A. A. Kopzhassarova and A. M. Sarsenbi, “Basis properties of eigenfunctions of second-order differential operators with involution,” Abstr. Appl. Anal. 2012, Article No. 576843 (2012).
L. V. Kritskov and V. L. Ioffe, “Spectral properties of the Cauchy problem for a second-order operator with involution,” Differ. Equ. 57, No. 1, 1–10 (2021).
M. A. Sadybekov and A. M. Sarsenbi, “Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution,” Differ. Equ. 48, No. 8, 1112–1118 (2012).
D. Nurakhmetov, S. Jumabayev, A. Aniyarov, and R. Kussainov, “Symmetric properties of eigenvalues and eigenfunctions of uniform beams,” Symmetry 12, No. 12, Article No. 2097 (2020).
V. E. Vladykina and A. A. Shkalikov, “Spectral properties of ordinary differential operators with involution,” Doklady Math. 99, No. 1, 5–10 (2019).
V. E. Vladykina and A. A. Shkalikov, “Regular ordinary differential operators with involution,,” Math. Notes 106, No. 5, 674–687 (2019).
A. G. Baskakov, “The method of similar operators and formulas for regularized traces,” Sov. Math. 28, No. 3, 1–13 (1984).
A. G. Baskakov, I. A. Krishtal, and N. B. Uskova, “Similarity techniques in the spectral analysis of perturbed operator matrices,” J. Math. Anal. Appl. 477, No. 2, 930–960 (2019).
A. G. Baskakov and D. M. Polyakov, “The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential,” Sb. Math. 208, No. 1, 1–43 (2017).
I. N. Braeutigam and D. M. Polyakov, “On the asymptotics of eigenvalues of a third-order differential operator,” St. Petersbg. Math. J. 31, No. 4, 585–606 (2020).
D. M. Polyakov, “Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions,” St. Petersbg. Math. J. 27, No. 5, 789–811 (2016).
D. M. Polyakov, “Formula for regularized trace of a second order differential operator with involution,” J. Math. Sci. 251, No. 5, 748–759 (2020).
D. M. Polyakov, “A one-dimensional Schrödinger operator with square-integrable potential,” Sib. Math. J. 59, No. 3, 470–485 (2018).
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Translated from Problemy Matematicheskogo Analiza 113, 2022, pp. 89-100.
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Polyakov, D.M. Spectral Asymptotics of Two-Term Even Order Operators with Involution. J Math Sci 260, 806–819 (2022). https://doi.org/10.1007/s10958-022-05729-8
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DOI: https://doi.org/10.1007/s10958-022-05729-8