Lecture 16: Fourier Transform

tags: 224a

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So far in the course, we studied differential operators on bounded intervals. We began with the Laplacian

$-d^2/d{x}^2$ on

$[a,b]$, which we "solved" using Fourier series, and then generalized this to Schrodinger operators of type

$L=-d^2/d{x}^2+q(x)$ on

$[a,b]$. The generalization involved developing the theory of compact operators, and then observing that the Green's function

$(z-L{)}^{-1}$ (which is really just the resolvent at some point not in

$\sigma (L)$) is compact, so

$L$ inherits the (complete) set of eigenvectors of the compact case.

In the next three lectures we will generalize this to differential operators on

$\mathbb{R}$. The analogue of Fourier series is the Fourier transform, which will give us a spectral representation of the momentum and Laplacian operators on

$\mathbb{R}$. We will study Schrodinger operators on

$\mathbb{R}$ by observing that

$(z-L{)}^{-1}$ is bounded for a judicious choice of

$z$.

In this lecture we develop the basic theory of Schwartz spaces and the Fourier transform, covering the following topics. The treatment is very similar to Dimock 1.1.4 so I will not (re)write notes.

Schwartz Space
Fourier Transform
Multiplication and Differentiation
Inversion Theorem, Plancharel
Extension to a unitary operator on L^2

The punch line of this lecture was a spectral representation for the momentum operator

$A=-id/dx$ on the Schwartz space

$S(\mathbb{R})$:

$$\mathcal{F}A=M_x\mathcal{F}onS(\mathbb{R})$$
where

$\mathcal{F}$ is the

$L^2$ Fourier transform. In particular, this means that

$$\mathcal{F}A{\mathcal{F}}^{-1}=M_{x}onS(\mathbb{R})(\ast ).$$
where

$M_x$ is multiplication by

$x$.

For the purposes of spectral theory, however

$S(\mathbb{R})$ is not a convenient subspace of

$L^2$ to work with; the right one is the "maximal domain"

$$D(M_x):=\{f\in L^2(\mathbb{R}):M_{x}f\in L^2(\mathbb{R})\}.$$
In order to make sense of the identiy

$(\ast )$ on this extended domain, however, we need to extend

$A$ to

${\mathcal{F}}^{-1}D(M_x)$, and this is done by defining

$\hat{A}:{\mathcal{F}}^{-1}D(M_x)\to L^2(\mathbb{R})$ by

$\hat{A}f={\mathcal{F}}^{-1}{M}_x\mathcal{F}f$. The domain of

$\hat{A}$ thus obtained is called a Sobolev space and the resulting notion of derivative is called a weak derivative.

In the next lecture we will define the spectral theory of such operators and show how to do this for any self-adjoint unbounded operator (which we will also define).