General form of the Green's function regular at infinity for the homogeneous Sturm-Liouville matrix operator

The standard Fourier transform method is used to analyze the expression of the Green's function regular at infinity for the Sturm-Liouville matrix operator in the important case of position independent parameters. A quadratic eigenvalue and eigenvector problem appears naturally. The classification of the former problem solutions allows to obtain the Green's function in a compact general form. Different physical problems were analyzed and the corresponding Green's function for various elementary excitations in less studied systems was predicted also.