A stable method to solve the Matrix Sturm-Liouville Equation for inhomogeneous layered structures

The study of elementary excitations in layered structures can be addressed via a Matrix Sturm-Liouville Equation of motion. A stable numerical method to solve this ubiquitous system of coupled second-order differential equations is presented. Its straightforward algorithm, especially useful for inhomogeneous layered structures, combines a local exponential approximation for the associated transfer matrix with the Hybrid matrix method. The latter is a well-known numerical stable method which is also useful to solve several boundary problems involving structures with planar, cylindrical or spherical geometry. Two numerical examples are considered and compared with the results reported previously to demonstrate the validity of the method.