The first regularized trace of integro-differential Sturm-Liouville operator on the segment with punctured points at integral perturbation of transmission conditions

Abstract

The paper is devoted to calculating a first regularized trace of one integro-differential operator with the main part of the Sturm-Liouville type on the segment with punctured points at integral perturbation of "transmission" conditions. The Sturm-Liouville operator −y′′(x) + q(x)y(x) + γ∫^π_0 y(t)dt = λy(x) given on the segments π/n(k − 1) < x < π/n k, k = 1, n; n ≥ 2 is considered. Boundary conditions of the Dirichlet type: y(0) = 0, y(π) = 0 are given on the left-hand and right-hand ends of the segment [0, π]. The functions continuous on [0, π] , the first derivatives of which have jumps at the points x = π/n k, are solutions. The value of jumps is expressed by the formula y′(πk/n − 0) = y′(πk/n + 0) − β_k ∫^π_0 y(t)dt, k = 1, n − 1. The basic result of the paper is the exact formula of the first regularized trace of the considered differential operator.