Resolvent Asymtotics of a Periodic Operator with Two Distant Perturbations on the Axis

Abstract

The paper considers a one-dimensional second-order differential operator with periodic coefficients on the real axis. This operator is perturbed using two functions that are real, finite and continuous. The disturbation function carriers are positioned at a large distance from each other. This work studies behavior of the considered operator resolvent exposed to the increasing distance between these perturbation function carriers. First two terms are obtained of the formal asymptotic representation of the resolvent of a perturbed one-dimensional periodic second-order differential operator. A rather complex structure of the first term of the asymptotic representation of the resolvent of the second-order differential operator under study with two diverging functions is revealed. Structure complexity of the asymptotics first term lies in the fact that it consists of the sum of three absolutely equal terms, each of them is localized in a certain section of the real axis. The first two terms of the formal asymptotic representation of the resolvent of the operator under study are constructed. The method used to determine results of this work makes it possible to construct not only the first terms of the asymptotic representation of the resolvent of a differential operator with two compactly diverging functions, but also all its subsequent terms

Please cite this article in English as:

Golovina A.M. Resolvent asymtotics of a periodic operator with two distant perturbations on the axis. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 2 (113), pp. 22--34 (in Russ.). EDN: HJBCVH

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