Relations between transfer matrices and numerical stability analysis to avoid the $\mathrm{\Omega }d$ problem

The transfer matrix method is usually employed to study problems described by $N$ equations of matrix Sturm-Liouville (MSL) kind. In some cases a numerical degradation (the so called $\mathrm{\Omega }d$ problem) appears thus impairing the performance of the method. We present here a procedure that can overcome this problem in the case of multilayer systems having piecewise constant coefficients. This is performed by studying the relations between the associated transfer matrix and other transfer matrix variants. In this way it was possible to obtain the matrices which can overcome the $\mathrm{\Omega }d$ problem in the general case and then in problems which are particular cases of the general one. In this framework different strategies are put forward to solve different boundary condition problems by means of these numerically stable matrices. Numerical and analytic examples are presented to show that these stable variants are more adequate than other matrix methods to overcome the $\mathrm{\Omega }d$ problem. Due to the ubiquity of the MSL system, these results can be applied to the study of many elementary excitations in multilayer structures.