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 calculation of a first regularized trace of one integro-differential operator with the main part of the Sturm-Liouville type on a segment with punctured points at integral perturbation of “transmission” conditions. The Sturm-Liouville operator −y″(x)+q(x)y(x)+γ∫0πy(t)dt=λy(x) given on the segments πn(k−1)<x<πnk,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 are continuous on [0, π], the first derivatives of which have jumps at the points x=πnk are solutions. The value of jumps is expressed by the formula y′(πkn−0)=y′(πkn+0)−βk∫0πy(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.