International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 3, Pages 519-532

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

S. A. Saleh

Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

Received 3 January 1995; Revised 22 May 1995

Copyright © 1998 S. A. Saleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Consider the eigenvalue problem which is given in the interval [0,π] by the differential equation y(x)+q(x)y(x)=λy(x);0xπ(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 α1β2α2β10. 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).