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So good morning, welcome to lecture number eight. Today we'll turn to the definition of spectra of operators

and I'll give the bare initial definitions for a spectrum in general and for the decomposition of a spectrum

in the particularly important case of a self-adjoint operator. And then in lecture 9 we will continue to investigate the spectrum

and prove a number of very important theorems which we're going to make use of.

However, today after I gave the initial definitions I'll show you how to calculate the point part of a spectrum

by perturbation theory in case you can't do it exactly. This is something that in the experimental part of the integrated course will be needed.

So this is the service to that course, but to that part of the course, but it fits in perfectly well here.

So the significance of spectra that we're going to define is that from the axioms of quantum mechanics,

which I presented in the first lecture, you know that the possible measurement values.

So if you perform a measurement in quantum mechanics, which in the theory,

a particular measurement is encoded in a self adjoint operator,

the question is what are the possible outcomes of a measurement?

And the answer is from the axiom, possible measurement values are those in the spectrum of an operator.

Are those in the so-called spectrum sigma of a of an observable,

and you know that an observable a is a self adjoint operator in quantum mechanics.

Now the spectrum will decompose into a continuous part and a point part,

and we'll talk more precisely about this at least for self adjoint operators,

and a common task in almost any quantum mechanics problem that you want to solve is to determine the spectrum

of one or a determined spectra of one or several operators in the game,

usually always of the Hamiltonian because the Hamiltonian, so the energy operator,

yields the time evolution of the system that follows from another axiom, namely the one of the two dynamical axioms

about the unitary evolution. You need to understand the Hamiltonian.

Well, you exponentiate the Hamiltonian, but in order to practically calculate that,

it is best to determine its spectrum. So it is a very common task to determine a spectra,

and more often than not, it is not possible exactly. So more often than not,

and this even refers to the problems that the textbooks choose to discuss.

Those are already highly select problems so that they can be calculated well.

But even there, more often than not, determination of the spectrum

is not analytically possible. So the spectrum will be defined,

Of course, you can go arbitrarily close to it by numerical methods, maybe, and so on.

But analytically, so real calculation by hand, very often that doesn't work.

And then resorts to perturbation theory, where you represent the operator whose spectrum

you want to calculate as a sum of a part where you can analytically calculate and a small

perturbation of that part.

If your operator lends itself to that kind of perturbation decomposition, you're in the

game.

And we will do this today.

We will discuss perturbation theory for the point spectrum.

So the spectrum consists of a continuous part and a point part, and they're not even disjointed.

But for the point spectrum, we can discuss perturbation theory at this stage already,

and we're going to do so.

Okay.

So first section, resolvent map and spectrum.

What is the spectrum of an operator?

And rather than looking at the spectrum immediately, we consider the following.

Let A be a closed operator.

It's just clever to already restrict to closed operators.

Let A be a closed operator.

Then the resolvent map of A is the map is called R sub A, R like resolvent, and it starts