Operator Theory for Electromagnetics : An Introduction

This text discusses electromagnetics from the view of operator theory, in a manner more commonly seen in textbooks of quantum mechanics. It includes a self-contained introduction to operator theory, presenting definitions and theorems, plus proofs of the theorems when these are simple or enlightening.

while There Are Many Excellent Books Available On Fundamental And Applied Electromagnetics, Most Introduce Operator Concepts In An Ad Hoc Manner, And Few Discuss The Subject Within The General Framework Of Operator Theory. This Is In Contrast To Quantum Theory, Where The Use Of Operators And Concepts From Functional Analysis Is Common. However, Casting Electromagnetic Problems In Terms Of Operator Theory Produces Useful Insights Into The Mathematical Properties And Physical Characteristics Of Solutions. For Instance, The Commonly Used Modal Expansion Of Fields In Waveguides Are Immediately Justified Upon Identifying The Differential Operators As Being Of The Appropriate Sturm-liouville Type. As Another Example, Existence, Uniqueness And Solvability Of Integral Formulations Can Often Be Settled By Appealing To The Theory Of Fredholm Operators. Many Other Examples That Illustrate The Value Of Abstracting Problems To An Operator Level Are Provided. Although The Book Focuses On Mathematical Fundamentals, It Is Written From The Perspective Of Engineers And Applied Scientists Working In Electromagnetics.
the Book Begins With A Review Of Electromagnetic Theory, Including A Discussion Of Singular Integral Operators Commonly Encountered In Applications. It Then Turns To A Self-contained Introduction To Operator Theory, Including Basic Functional Analysis, Linear Operators, Green¿s Functions And Green¿s Operators, Spectral Theory, And Sturm-liouville Operators. The Discussion Is At An Introductory Mathematical Level, Presenting Definitions And Theorems, As Well As Proofs Of The Theorems When These Are Particularly Simple Or Enlightening. The Tools Developed In This First Part Of The Book Are Then Applied To Problems In Classical Electromagnetic Theory: Boundary-value Problems And Potential Theory, Transmission Lines, Waves In Layered Media, Scattering Problems In Waveguides, And Electromagnetic Cavities.

The purpose of this book is to describe methods for solving problems in applied electromagnetic theory using basic concepts from functional anal­ ysis and the theory of operators. Although the book focuses on certain mathematical fundamentals, it is written from an applications perspective for engineers and applied scientists working in this area. Part I is intended to be a somewhat self-contained introduction to op­ erator theory and functional analysis, especially those elements necessary for application to problems in electromagnetics. The goal of Part I is to ex­ plain and synthesize these topics in a logical manner. Examples principally geared toward electromagnetics are provided. With the exception of Chapter 1, which serves as a review of basic electromagnetic theory, Part I presents definitions and theorems along with associated discussion and examples. This style was chosen because it allows one to readily identify the main concepts in a particular section. A proof is provided for all theorems whose proof is simple and straightforward. A proof is also provided for theorems that require a slightly more elaborate proof, yet one that is especially enlightening, being either constructive or illustrative. Generally. theorems are stated but not proved in cases where either the proof is too involved or the details of the proof would take one too far afield of the topic at hand, such as requiring additional lemmas that are not clearly useful in applications. The coverage of electromagnetics in this chapter is somewhat brief, especially the physical aspects of the theory, since it is assumed that the reader is familiar with basic field theory at an undergraduate or beginning graduate level.