Spectral Theory - What Is Spectral Theory, Roughly Speaking?

What Is Spectral Theory, Roughly Speaking?

In functional analysis and linear algebra the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist.

This topic is easiest to describe by introducing the bra-ket notation of Dirac for operators. As an example, a very particular linear operator L might be written as a dyadic product:

in terms of the "bra" and the "ket" . A function is described by a ket as . The function defined on the coordinates is denoted as:

and the magnitude of by:

where the notation '*' denotes a complex conjugate. This inner product choice defines a very specific inner product space, restricting the generality of the arguments that follow.

The effect of upon a function is then described as:

expressing the result that the effect of on is to produce a new function multiplied by the inner product represented by .

A more general linear operator might be expressed as:

where the are scalars and the are a basis and the a reciprocal basis for the space. The relation between the basis and the reciprocal basis is described, in part, by:

If such a formalism applies, the are eigenvalues of and the functions are eigenfunctions of . The eigenvalues are in the spectrum of .

Some natural questions are: under what circumstances does this formalism work, and for what operators are expansions in series of other operators like this possible? Can any function be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite dimensional spaces and finite dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.