EXPANSION OF ANALYTIC-FUNCTIONS IN SERIES OF FLOQUET SOLUTIONS OF 1ST-ORDER DIFFERENTIAL-SYSTEMS

In this paper we study boundary eigenvalue problems for first order sy stems of ordinary differential equations of the form zy'(z) = (lambdaA 1(z) + A0(z)) y(z), y(ze2pii) = e(2piinu)y(z) for z is-an-element-of S (log), where S is a ring region around zero, S(log) denotes the Rieman n surface of the logarithm over S, the coefficient matrix functions A1 (z) and A0(z) are holomorphic on S, and nu is a complex number. The ei genfunctions of this eigenvalue problem are the Floquet solutions of t he differential system with nu as characteristic exponent. For an open subset S0 of S, the notion of A1-convexity of the pair (S0, S) is int roduced. For A1-convex pairs (S0, S) it is shown that the expansion in to eigenfunctions and associated functions of holomorphic functions on S(log), satisfying the monodromy condition y(ze2pii) = e(2piinu)y(z), converges regularly on S0log and is unique. If S is a pointed neighbo urhood of 0 and A1(z) is holomorphic in S or {0}, it is shown that the re is a pointed neighbourhood S0 of 0 such that (S0, S) is A1-convex. It follows from the results of this paper that many expansions of anal ytic functions in terms of special functions can be considered as eige nfunction expansions of this kind.