Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các vấn đề Regge cho các chuỗi, mở rộng eigenfunction vô điều kiện hội tụ, và các cơ sở vô điều kiện của exponentials trong L ^ 2 (-T, T). | J. OPERATOR THEORY 14 1985 67-85 Copyright by INCREST 1985 THE REGGE PROBLEM FOR STRINGS UNCONDITIONALLY CONVERGENT EIGENFUNCTION EXPANSIONS AND UNCONDITIONAL BASES OF EXPONENTIALS IN L -T T s. V. HRLlẫCEV A String is the interval 0 oo carrying a non-negative measure ám. The function m x dm evaluates the mass of the string supported by 0 x The o - point X 0 is assumed to be a point of growth of m . m x 0 for X 0. It is supposed also that the string is obtained from the classical homogeneous string corresponding to Lebesgue measure dx by a finite perturbation. The latter means that dm dx on a co for some a 4-00. In what follows a inf a dm dx on a 4-oo . Given a 0 let L2 0 a dm denote the Hilbert space of all m-measurable functions f with a x 2dm x 4-00. 0- Every string determines the formal differential operator f dy dmdx defined on-the class Do of functions onR oo 4-oo such that 0 f 0 x for X 0 0 0 x gO dm i ld X 0 68 s. V. HRUSCEV d2f with g satisfying g e 2 0 a dm for every a 0. Clearly - g for dmdx such an f The symbols f x and f x denote the right-hand and left-hand derivatives of f respectively. Fix a ỳ gm and let ơ m be the set of all complex numbers k such that the equation 1 d 7 k y y- ũ - -ữ y a iky a -- 0 dmdx has a non-zero solution J a x k . In fact the set ơ m does not depend on the parameter a when a am and coincides with the zero-set of an entire function. It can be shown and we will do it later that o m is disposed in the open upper half-plane c . The spectrum ơ m is always symmetric with respect to the imaginary axis because ya x k0 is a non-zero solution of 1 corresponding to k - kữ provided k0 e The spectra which occur in the eigenfunction problem 1 are described by Arov s theorem 1 Theorem 1. A closed countable subset Ơ of c symmetric with respect to the imaginary axis coincides with the spectrum of a problem 1 if and only if a is the zero-set for an entire function F of exponential type with ị l 4 X2 -1 F x i 2dx 4-00 ị 1 4- X2 -1 log 1F x .
