Theory of Single and Multiple Interfaces

Abstract

Primed in Singapore by General Priming Services Pie. Lid. There is unfortunately a rather widespread prejudice against Green functions which is totally unjustified. Perhaps the trouble is that most of us learnt Quantum Mechanics from textbooks written in terms of wavefunctions. The Green function is a very transparent concept which can be viewed as a propagator or as a response function and which is equally clear and meaningful all the way from classical physics to ordinary quantum mechanics down to the most intricate manybody quantum field theory. Not only that. It is also a very useful object to do practical calculations. It is simply not true that a Green function calculation is more complicated and one could find many test cases which bear this out. Yet we find sometimes this prejudice among very respectable colleagues who insist on this view. We respectfully submit that they have not tried. Theory of Single andIn writing this book we have also been guided by comments, questions and response from many colleagues. This has helped us to detect the points where readers are likely to find obscurities and want clarification. The didactic tone of some passages, the explicit derivations almost step by step of some of the key results, the reflections on some elementary questions and the choice of some very simple examples selected to bear out basic points are our response to this very stimulating input we have received from many We always seem to manage to stumble upon our results in the clumsiest possible way and then it is very difficult to recommend in good faith any of the lousy papers we have written. These are for experts in the field, who can see their way through the increasing pollution in the current scientific literature, but not for those who are looking for an introduction to the field and want to learn the subject in order to use it in their own work. It is the latter that we have in mind as potential readers. So, having been doing SGFM calculations for a number of years we felt it was time to try and collect together our thoughts on the matter, enjoying the rare pleasure of pausing to meditate, and making an effort to put all these things together in one volume so we can give one reference when we are asked. If someone feels disappointed then this will only reflect our inability to do better. But we have tried.In order to avoid interruptions in a continuous line of argument, which we feel is a more appropriate style for a book, we have listed the references by chapters at the end. We hope to have done justice to other authors and present our sincere apologies if someone has been unintentionally left out.It is a pleasure to express our gratefulness to Ignacio Reguera for helping us in the preparation of the text with his bottomless magician's bagful of tricks.This book owes a great deal to the work of some of our coworkers. M. Carmen Muoz has contributed very substantially to the practical development of the SGFM method for discrete systems. The whole of Chapter 7 would have been impossible without the...

Electronic Properties of Fibonacci Quasi-Periodic Heterostructures

“…The Fibonacci heterostructures involve many inequivalent interfaces. We shall use a version of the surface Green's function matching (SGFM) method [41], specially adapted to deal with an arbitrary number N of inequivalent interfaces [42]. The electron states are obtained from the peaks in the imaginary part of the trace of the interface projection of the Green's function of the matched systemG G S [42].…”

General form of the Green’s function regular at infinity for the homogeneous Sturm–Liouville matrix operator

“…As far as we know, there are no cases where P + Y = 0 as a consequence of P being Hermitian and P = 0. For sagittal elastic waves [1,11] Y is the transpose of P, therefore, the TFD is included. For Dimmock Hamiltonian [1,11] Y = P, and P + Y = 2P.…”

“…The same happens for the Kohn-Luttinger model (see, for example, Refs. [1,11], the work of W. Pötz, W. Porod and D.K. Ferry [17] and citations therein) and for any Envelope Functions Model.…”

“…For example, using (25) we obtained the Green's function given in Ref. [11] for the sagittal elastic modes.…”

“…In the models SMM2 and SMM3, d is a parameter only valid for those models and is a variable representing an eigenvalue. For example, if we compare (2) with the one-dimension Schrödinger equation, with potential V (z) = 0, we see that is 2m [2,11] which involves the Green's functions of the constituent media. A theorem on arbitrary extended pseudo media [16] confirms that the final Green's function of a matched system is independent of the arbitrary extended pseudo media assigned to each constituent medium.…”