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Good morning, welcome to Lecture 18, which is titled the Fourier operator, and you probably heard of the Fourier transform.

Well, it's exactly the same. The Fourier transform is an operator, and in fact, for instance, you can calculate its spectrum.

The word transform somehow hides that fact. We had this before with the BLT theorem, which will also play a role today.

It's the theorem on densely defined bounded linear transformations, meaning bounded linear operators.

So the Fourier operator will be today's topic, and in fact, this is not for its own sake, although you probably already met that in the experimental part of the lectures, right?

You had Fourier transforms and momentum space and stuff like this? Did you meet that? Yes? No? You did not? Okay, even better.

So there is a purpose to this, and the purpose is that with this lecture, we start a new part of the course.

We start a systematic exploration of so-called Schrodinger operators.

And these Schrodinger operators are operators of the following form. They're of interest as Hamiltonians of the system,

but of course, for the abstract theory, it doesn't matter whether you call that then the Hamilton or the energy observable of the system.

But the idea is that if you look at operators with a certain domain, it's always about operators on L2 or some subdomain or R to the D.

A Schrodinger operator is an operator that acts on its domain like this, that the operator applied to a function psi yields the following function,

minus h bar squared by 2m Laplace operator on psi at x plus v of x times psi of x, where this here, as I just said, is the Laplacian.

This is d by, well, is d in the first direction squared plus d in the second direction squared plus plus plus d in the dth direction,

the dth slot, I should say, squared. That's the Laplace operator.

And this here is what one usually calls the potential. It's just a name. So that's a function from Rd into the real numbers.

And the significance of these Schrodinger operators is that they define the right hand side of the Schrodinger equation.

Or if we express it in terms of our axioms, these are the operators that for particles that live on some Rd, so the typical case would be d equals 3,

that particles that live on a three dimensional real physical space, so that quantum mechanics looks at the square integrable functions on there,

and that the generator of the time translation, so the generator of the unitary time evolution, is of this form.

And we saw something like this before, I think, in one dimension. This is, of course, the momentum operator squared by 2m plus the potential.

So this really looks like the classical Hamiltonian, but made into an operator. We talked about quantization and what it could mean and not mean.

Well, it's in a sense it defines quantum mechanical problems in R3, on R3, that also where the particle in question fields this potential.

So this is of very general interest. And textbooks, of course, discuss a whole number of situations where this V of x is given.

And then you discuss that case in particular. We also did that. I took V of x to be m omega squared x squared, that was the harmonic oscillator.

You can take this V of x, and you did this in the experimental part, I know. You took this to be a central potential, given by something like minus some constant,

1 over x1 squared plus x2 squared plus x3 squared square root. So 1 over R, if you wish.

It's a central potential, a very particular central potential. It's the Coulomb potential, or the Kepler potential, but you applied it in the electrostatics, so it's the Coulomb potential.

These are all special examples, and of course the solution of the harmonic oscillator, so for instance the eigenvectors, vary wildly if you change the potential.

The eigenvectors of the harmonic oscillator are very different from the eigenvectors of, say, the hydrogen atom, and so on.

So this is what this is. But the question is, can we study this in a more systematic manner, or do we really have to go example by example?

Well, of course, physics often demands you not to consider the problem in its general position, as the mathematician says. That's the problem in general position.

What can you say about this in general? Maybe making assumptions on the potential, and so on.

In physics, very often we consider a problem in special position. I'm interested in the hydrogen atom, and then I don't care about the general theory.

But if we really claim to understand something about quantum mechanics, we need to study this in higher generality, in greater generality.

And that's what we're trying to do in the next few lectures. And the Fourier operator is an indispensable tool in conducting that study. That's it.

Okay, so the Fourier operator is an indispensable tool in conducting this study.

And it's also very useful, sometimes maybe indispensable, in special problems. So this is not only a tool for this general theory.

So let's start with the definition. So ultimately, we wish to define the Fourier operator on all of L2, but initially we cannot, and explicitly we cannot do it different than this.

We look at the Fourier operator on Schwarz space.

So you recall that we defined Schwarz space S of R for the real line, and we now extend the previous definition, which we now had sufficient opportunity to practice, extend the previous definition of Schwarz space on the real line to Schwarz space S on Rd.

This is as simple as it precisely the same idea. A Schwarz function is a function that decays sufficiently rapidly if you go to infinity, and also has nice properties under differentiation.

Precisely speaking, a function that is at least smooth. That's the set from which we pick Schwarz functions. We impose two times a countable infinity of conditions. We impose that the supremum, if you look at any point x now on Rd,

we take the absolute value of an arbitrary power x alpha of x and an arbitrary derivative del beta acting on the function, evaluating at x, and we require this be less than infinity.

So this looks exactly like what I wrote before with the slight difference. I now say this must be true for all alpha and beta that come from n 0 to the dth power.

What is that supposed to mean? Well, and what is supposed to mean if x is in Rd, what do I mean by the alpha power of an Rd element, and the alpha itself is a d-tuple of non-negative integers?

Well, very simply, this is just very convenient notation. What we mean by x to the alpha, again x in Rd, and alpha in n 0 cross cross, so Cartesian product with n 0, d times.

What we mean by this, the x has components x1, x2, x3 up to xd, right, and we take the first component of x and raise it to the alpha first power, because the alpha is a d-tuple,

and we take the second component of x and raise it to the alpha second power, times dot dot dot dot, we take the dth component and raise it to the alpha dth power.