International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 3, Pages 519-532
doi:10.1155/S0161171298000738

Regularized sum for eigenfunctions of multi-point problem in the commensurable case

Abstract

Consider the eigenvalue problem which is given in the interval [0,π] by the differential equation −y″(x)+q(x)y(x)=λy(x);   0≤x≤π       (0,1) and multi-point conditions U1(y)=α1y(0)+α2y(π)+∑K=3nαKy(xKπ)=0,U2(y)=β1y(0)+β2y(π)+∑K=3nβKy(xKπ)=0,        (0,2) where q(x) is sufficiently smooth function defined in the interval [0,π]. We assume that the points X3,X4,…,Xn divide the interval [0,1] to commensurable parts and α1β2−α2β1≠0. Let λk,s=ρk,s2 be the eigenvalues of the problem (0.1)-(0.2) for which we shall assume that they are simple, where k,s, are positive integers and suppose that Hk,s(x,ξ) are the residue of Green's function G(x,ξ,ρ) for the problem (0.1)-(0.2) at the points ρk,s. The aim of this work is to calculate the regularized sum which is given by the form: ∑(k)∑(s)[ρk,sσHk,s(x,ξ)−Rk,s(σ,x,ξ,ρ)]=Sσ(x,ξ)       (0,3) The above summation can be represented by the coefficients of the asymptotic expansion of the function G(x,ξ,ρ) in negative powers of k. In series (0.3) σ is an integer, while Rk,s(σ,x,ξ,ρ) is a function of variables x,ξ, and defined in the square [0,π]x[0,π] which ensure the convergence of the series (0.3).