REGULARIZED SUM FOR EIGENFUNCTIONS OF MULTI-POINT PROBLEM IN THE COMMENSURABLE CASE

S.A. SALEH

《International Journal of Mathematics and Mathematical Sciences》 1998卷 (1998/12) Pp. 519-532

https://doi.org/10.1155/S0161171298000738

摘要

Consider the eigenvalue problem which is given in the interval [0, π] by the differential equation –y^n(x)+q(x)y(x) =λy(x); 0≤x≤π (0.1) and multi-point conditions U_1(y) = α_1y(0)+ α_2y(π)+ Σ_(k=3)^nα_ky(x_kπ) = 0, U_2(y) =β_1y(0)+β_2y(π)+ Σ_(k=3)^nβ_ky(x_kπ) = 0, (0.2) where q(x) ix sufficiently smooth function defined in the interval [0, π]. We assume that the points X_3, X_4,...,X_n divide the interval [0,1] to commensurable parts and α_1β_2-α_2β_1 ≠ 0. Let λ_(k,s) = P_(k,s)^2 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 H_(k,s)(x,□) are the residue of Green's function G(x,□, p) for the problem (0.1)-(0.2) at the points P_(k.s) The aim of this work is to calculate the regularized sum which is given by the form: Σ_((k))Σ+((s))[p_(k,s)^(-σ)H_(k,s)(x,□)-R_(k,s)(σ,x,□,p)] = s_σ(x,□) (0.3) The above summation can be represented by the coelliciets of the asymptotic expansion of the function G(x,□, p) in negative powers of k. In series (0.3) σ is an integer, while R_(k,s)(σ,x,□,p) is a function of variables x,□ and defined in the square [0, π]x[0, π] which ensure the convergence of the series (0.3).

關鍵字

Regularized sum for eigenfunctions ； asymptotic formula ； Green's function ； differential operator