Polyakov, D. M. Formula for regularized trace of a second order differential operator with involution. (English. Russian original) Zbl 1460.34078 J. Math. Sci., New York 251, No. 5, 748-759 (2020); translation from Probl. Mat. Anal. 106, 169-178 (2020). The author finds a formula for the regularized trace and obtains an estimate for the deviations of spectral projections of the differential operator \(L:D(L)\subset L_2 [0,1]\rightarrow L_2 [0,1]\) with involution \[ (Ly)(x)=-y^{\prime\prime}(x)-p(x)y(x)-q(x)y(1-x), \quad x\in [0,1] \] where \(p(x), q(x)\in L_2[0,1]\) are complex functions and \[ D(L)= \{ y\in W_2 ^2 [0,1]: y^{(j)} (0)=y^{(j)} (1),j=0,1\}. \] Reviewer: Erdogan Sen (Tekirdağ) Cited in 3 Documents MSC: 34K08 Spectral theory of functional-differential operators Keywords:regularized trace; asymptotics of eigenvalues PDFBibTeX XMLCite \textit{D. M. Polyakov}, J. Math. Sci., New York 251, No. 5, 748--759 (2020; Zbl 1460.34078); translation from Probl. Mat. Anal. 106, 169--178 (2020) Full Text: DOI References: [1] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), New York, NY: Springer, New York, NY [2] Kurdyumov, VP; Khromov, AP, Riesz bases formed by root functions of a functionaldifferential equation with a reflection operator, Differ. Equ., 44, 2, 203-212 (2008) · Zbl 1187.34083 [3] 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 ID 590781 (2012). · Zbl 1259.34084 [4] Baskakov, AG; Krishtal, IA; Romanova, EY, Spectral analysis of a differential operator with an involution, J. Evol. Equ., 17, 2, 669-684 (2017) · Zbl 1481.47059 [5] A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,” Differ. Equ.46, No. 4, 509-514 (2010). · Zbl 1214.34082 [6] A. M. Sarsenbi and A. A. Tengaeva, “On the basis properties of root functions of two generalized eigenvalue problems,” Differ. Equ.48, No. 2, 306-308 (2012). · Zbl 1280.34084 [7] Sadybekov, MA; Sarsenbi, AM, Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution, Differ. Equ., 48, 8, 1112-1118 (2012) · Zbl 1268.47053 [8] V. P. Kurdyumov, “On Riesz bases of eigenfunciton 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). · Zbl 1353.34105 [9] Kritskov, LV; Sarsenbi, AM, Spectral properties of a nonlocal problem for a secondorder differential equation with an involution, Differ. Equ., 51, 8, 984-990 (2015) · Zbl 1331.34161 [10] Kritskov, LV; Sarsenbi, AM, Riesz basis property of system of root functions of second-order differential operator with involution, Differ. Equ., 53, 1, 33-46 (2017) · Zbl 1367.34085 [11] Vladykina, VE; Shkalikov, AA, Regular ordinary differential operators with involution, Math. Notes, 106, 5, 674-687 (2019) · Zbl 1443.34064 [12] Vladykina, VE; Shkalikov, AA, Spectral properties of ordinary differential operators with involution, Dokl. Math., 99, 1, 5-10 (2019) · Zbl 1418.34156 [13] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators Am. Math. Soc., Providence, RI (1969). · Zbl 0181.13503 [14] Sadovnichii, VA; Podolskii, VE, Traces of operators, Russ. Math. Surv., 61, 5, 885-953 (2006) · Zbl 1157.47013 [15] Baskakov, AG, The method of similar operators and formulas for regularized traces, Sov. Math., 28, 3, 1-13 (1984) · Zbl 0561.47031 [16] Baskakov, AG; Polyakov, DM, 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 [17] Braeutigam, IN; Polyakov, DM, On the asymptotics of eigenvalues of a third-order differential operator, St. Petersbg. Math. J., 31, 4, 585-606 (2020) · Zbl 07222199 [18] Polyakov, DM, Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions, St. Petersbg. Math. J., 27, 5, 789-811 (2016) · Zbl 1360.34171 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.