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.R. P.-S. acknowledges CONACyT support. R.P.-A. also acknowledges CONACyT support through grant 208108 and hospitality at ICMM-CSIC, Madrid, Spain. V.R.V. acknowledges the Spanish Ministerio de Economía y Competitividad for support through grant MAT2012-38045-C04-04