Abstract
By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.
Highlights
Reflection and refraction problems are considered with such approximations and discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with
A countable infinite system of boundary conditions has been obtained and its first few low-order approximations are derived for this equation in case of sufficiently smooth potentials
The information about the deviations of the real potential from the chosen asymptotic potentials on both sides of the “imaginary” boundary surface is involved in the boundary conditions in form of the moments of these deviations
Summary
When the transitions between the different media are sufficiently abrupt, but still smooth, one can often solve such problems approximately by generalized boundary conditions We will demonstrate this here for one of the simplest second-order differential equations in physics, the Sturm-Liouville equation. A relatively new procedure to derive generalized boundary conditions is to involve generalized functions and, in particular, the step function and all its derivatives which are the delta function with its derivatives This method is appropriate in classical electrodynamics for the derivation of generalized boundary conditions for media with spatial dispersion [7] (including, e.g., optical activity) and for correct derivations in presence of transition layers. XII and in our paper [12] appear under this last name with one of its special cases as natural optical activity)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
