Ways to calculate the spectrum of an operator

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,

Solution 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.

Solution 2:

Giuseppe Negro already mentioned it, but I wanted to go into a bit more detail with his first point, the Sturm-Liouville methods.

A strategy which is used very often in practice is: Calculate the Greens function or respectively the fundamental solution for $L-\lambda \mathbf{1}$ (you can actually allow $G$ to be a distribution), this induces a resolvent function $R_\lambda(f)=\int G_\lambda(x,y)f(y)dy$ depending on $\lambda$. This means, that every singular point of the resolvent function (i.e. $G$ does not exist, the integral diverges or something) are good candidates for your spectrum.

In order to get there, you often use methods like the Fourier Transform and/or separation methods or, in may cases, you can just look up Greens functions. For example, you can obtain the spectrum of $-\Delta$ simply from knowing, that the Greens function of the Helmholtz Operator $\Delta+k^2$ is $\frac{-e^{-ikr}}{4\pi r}$. To prove that you have actually obtained the full spectrum, approximation methods are often very useful, i.e. you use that $\lambda\in\sigma(A)$ if and only if there exists a normalized sequence for which $(A-\lambda)\varphi_n\rightarrow 0$ (for $A$ selfadjoint).

Good luck, spectral theory is wonderful!

EDIT: This just came to my mind:

One more very elegant and intuitive strategy to calculate the spectrum I am currently working on involves Rigged Hilbert Spaces. It is the strategy which is usually used by the physicists to calculate spectra of Schrödinger operators. The idea is the following: You just try to calculate an eigenfunction or eigenfunctional to you differential operator without caring for domains or such stuff, e.g. $e^{\pm ikx}$ as "eigenfunction" of $-\Delta$, although this fuction is obviously not in $L^2(\Omega)$ if $\Omega$ isnt bounded. However, it obviously can be interpreted as a distribution, therefore you can interpret this result in the RHS setting. The tricky part now is to construct an RHS where you can prove, that each eigenfunctional belongs to a $\lambda\in\sigma(A)$ (the other direction is provided by the nuclear spectral theorem), a so-called "tight rigging". For special cases, this can be done using approximation methods. However, the general case has not been investigated as far as I know.