The Inverse Spectral Problem for Rst Order Systems on the Half Line

On the half line 0; 1) we study rst order diierential operators of the form B 1 i d dx + Q(x); where B := B 1 0 0 ?B 2 ; B 1 ; B 2 2 M(n; C) are self{adjoint positive deenite matrices and Q : R + ! M(2n; C); R + := 0; 1); is a continuous self{adjoint oo{diagonal matrix function. We determine the self{adjoint boundary conditions for these operators. We prove that for each such boundary value problem there exists a unique matrix spectral function and a generalized Fourier transform which diagonalizes the corresponding operator in L 2 (R; C). We give necessary and suucient conditions for a matrix function to be the spectral measure of a matrix potential Q. Moreover we present a procedure based on a Gelfand-Levitan type equation for the determination of Q from .