[Math] Ways to calculate the spectrum of an operator

Question

Friends,

I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. For example, by applying directly the definitions, I am able to compute the spectrum of the orthogonal projection to be the set $\{0,I\}$.

But, in order to find the essential spectrum of some differential operators, say $L=\partial_{xx}+c\partial_{x}+F$ where $F$ is some linearized term of a nonlinear function $f(u)$, I would perform a Fourier transform $\mathfrak{F}(L)$ and calculate the eigenvalues of this operator. (I am not even sure if this is the correct way of doing it.)

Other ways are taking different kinds of transforms (which I have no idea; but by talking to some people, I sensed that taking a Laplace transform sometimes works, too!).

I think applying directly the definitions would not be possible in at lot of the cases. Can someone give me references for techniques of finding spectra of different operators, when they fail and work? At least, when I see some kind of operator, I would like to know that I have a sense of what to do.

Best regards,

Answer 1

There is a whole theory dedicated to this, so the short answer is: there are a lot. I can think of three:

1. Solving analytically the resolvent differential equation (i.e. the equation $Lu - \lambda u = v$). This tends to work when the geometrical domain is one-dimensional (Sturm-Liouville's theory) or when it is very symmetrical (separation of variables, polar coordinates ...).
2. Fourier transforming the resolvent equation to turn it into an algebraic equation. This works when the Fourier transform is available (which for me means that you are either on the torus or on the free space), and when the operator has constant coefficients. See here for an example.
3. Using the calculus of variations to determine the spectrum by some technique such as the Minimax principle. This is what one usually does to determine the spectrum of elliptic operators on bounded domains and on compact Riemannian manifolds.