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).
