Invariant Imbedding Technique Research Articles

Invariant imbedding and the band theory of solids

The mathematical theory of invariant imbedding has been described in numerous papers by Bellman and Kalaba and their co-workers [l]. The invariant imbedding technique has been applied to wave mechanics [2], and we will show here that one-dimensional band theory can also be analyzed by means of the technique. The band theory formulation is an extension of the work by Davies. We will be concerned with two problems in this paper. We will first show that the Kronig-Penney model of a rectangular barrier can be analyzed by invariant imbedding and that the same restrictions are reached on the energy levels of the electron. We will then show that the invariant imbedding technique gives a computational form which can be solved for any one-dimensional potential variation. The equations to be solved to determine the energy levels are numerical equations and can be solved by means of a high speed digital computer. There are many extensions of the work which we shall not describe in detail here. For example, the potential function can be modified by random changes in height and periodicity. A discussion of the use of invariant imbedding in problems with random properties will be found in the book by Adams and Denman [3]. The band structure of ordered and disordered alloys can be studied by using the method described in this paper.

Invariant imbedding technique and age-dependent birth and death processes

Invariant imbedding technique and age-dependent birth and death processes

Invariant imbedding and the numerical integration of boundary-value problems for unstable linear systems of ordinary differential equations

Abstract : In such diverse areas as radiative transfer in planetary atmospheres and optimal guidance and control, two-point boundary-value problems for unstable systems arise, greatly complicating the numerical solution. An invariant imbedding technique is presented which is useful in overcoming these frequently encountered instabilities, and the results of some numerical experiments are presented. (Author)

Invariant imbedding and scattering of light in a one-dimensional medium with a moving boundary

Abstract : It is shown how the invariant-imbedding technique is used for the derivation of integral equations governing the reflection and the transmission coefficients of radiation in a one-dimensional medium with a moving boundary. When formulating the source function, account is taken by the relaxation of photon emission under the assumption that during an elementary act of scattering a photon remains in a state of absorption for a certain duration of temporal capture. The Schuster problem for a moving one-dimensional medium is exactly treated. When formulating the transfer equations, the influence of the Doppler effect caused by the macroscopic differential motions is considered as a principal agency of the change of frequency on scattering.

Invariant imbedding and time-dependent scattering of light in a one-dimensional medium

Abstract : By means of the invariant-imbedding technique, the integral equations for the reflection and transmission coefficients of radiation in a one dimensional medium are obtained, allowing for the release of absorbed energy with a random time delay. Furthermore, the reflected and transmitted intensities for the fluorescence problem in a one-dimensional medium are expressed in terms of these coefficients, assuming no distribution of emitting sources in the medium.