Formula for Regularized Trace of a Second Order Differential Operator with Involution
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
FORMULA FOR REGULARIZED TRACE OF A SECOND ORDER DIFFERENTIAL OPERATOR WITH INVOLUTION D. M. Polyakov Southern Mathematical Institute Vladikavkaz Scientific Center RAS 22, Markus St., Vladikavkaz 362027, Russia Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia [email protected]
UDC 517.927
We study a second order differential operator with involution and periodic boundary conditions. We prove a formula for the regularized trace, study the asymptotic behavior of eigenvalues, and obtain an estimate for deviations of spectral projections. Bibliography: 18 titles.
1
Introduction and the Main Results
We consider the second order differential operator L : D(L) ⊂ L2 [0, 1] → L2 [0, 1] with involution (Ly)(x) = −y (x) − p(x)y(x) − q(x)y(1 − x),
x ∈ [0, 1],
where the coefficients p and q are complex and belong to the space L2 [0, 1]. The domain of the operator L is given by the periodic boundary conditions D(L) = {y ∈ W22 [0, 1] : y (j) (0) = y (j) (1), j = 0, 1}. The operator L contains an involution of independent variable. We recall that an involution on a set X is a function ϕ such that ϕ(ϕ(x)) = x on X. In this paper, we consider only the involution of the reflection ϕ(x) = 1 − x on the segment [0, 1]. Differential equations with involution form a special class of linear functional–differential equations. Equations with involution arise in many physical systems. For example, delay differential equations, ecological population equations, retarded reaction diffusion equations are particular cases of differential equations with involution (cf. [1]). The spectral aspects connected with the asymptotic behavior of eigenvalues, the basis property of root functions, justification of the Fourier method for operators with involution have been well studied for first order differential operators (cf. [2]–[4] and the references therein). But Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 169–178. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0748
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spectral properties of second order differential operators have not been studied in detail. The study of such operators is mainly devoted to the basis property of root functions of the operators = −y (−x), Ly
= −y (x) + αy (−x) Ly
(1.1)
on the segment [−1, 1] with some α ∈ (−1, 1) (cf. [5]–[9]). A criterion for the unconditional basis property of operators of the form (1.1) with perturbation (in particular, containing an involution) is obtained in [10]. The notion of the regularity of boundary conditions is introduced in [11] for ordinary differential operators of an arbitrary order with boundary conditions of general form. Moreover, the theorem on the unconditional basis property with brackets of root functions is proved in [11]. A general classification of arbitrary operators depending on the type of boundary conditions can be found in [12]. In this paper, we continue to study spectral properties of second order differential operators wit
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