A while ago, while I was doing some readings, I found a result that worked for ‘‘an open dense set of nonlinearities’’. Putting aside what this really means, I found out that to establish this fact, one had to effectively reconstruct a differential operator from its eigenvalues. I found it rather fascinating, and took a shallow dive into this topic.

Unfortunately, a lot of these were not getting into my head, and so I volunteered to do a talk series in my university’s unofficial mathematics Discord chat. But to do so would require me to prepare notes, which I did, but they were not to my standard, and so I am redoing them, this time in an online format here, so that I can finally do the talks and finally learn inverse spectral theory.

Yes, instead of doing my research, I am making notes online, in order to fix my offline notes, in order to give a talk, in order to learn a topic that would probably not be useful ever for my work. This is procrastination done at the highest level, and I advise the reader not to follow in my footsteps.

The goal of this series is to study one (and only one) family of ordinary differential equations $$-y^{\prime\prime}+ q(x)y = \lambda y \text{ for $x\in[0,1]$,}$$ subject to the simplest possible boundary conditions $$y(0) = y(1) = 0.$$

Here $q(x)$ lies in $L^2_{\mathbb{R}}([0,1])$, the space of real, square-integrable functions in $[0,1]$, and $\lambda$ is a complex number. Even for this simplest possible case, there are many interesting questions that we can ask. For instance:

  • Having fixed $q$, for which $\lambda\in\mathbb{C}$ will the above ODE have solutions? (These $\lambda$ will be called the eigenvalues of $q$.)
  • How are the eigenvalues distributed in $\mathbb{C}$?

These are questions of spectral-theoretic flavour. Having established these, we can also ask questions like

  • For what sequences $\lambda_1, \lambda_2,\dots$ is there a $q$ wiith the $\lambda_n$ as eigenvalues?
  • Given the eigenvalues, can we say anything about the potential $q$?
  • When do two $q$’s give rise to the same spectrum?

These are questions of inverse spectral-theoretic flavour. Hopefully we will be able to answer some of these questions throughout the series.

I will now give a rough sketch of the plan for the talks and notes. All of these are subject to change. We will be very closely following Pöschel and Trubowitz’s text Inverse Spectral Theory, to the point where I feel like I am just lifting from the text.

Just like how matrices have characteristic polynomials, which we can use to find the eigenvalues of the matrix, we have a ‘‘characteristic polynomial’’ for our differential operator here, which will turn out to be a solution to a different, but related ODE. Therefore, the first few parts of the notes will be a recap of basic ODE theory: we will introduce the celebrated method of power series, and use it to construct ‘‘fundamental’’ solutions to our ODE. Thereafter, we will derive some useful results about the solutions, such as the asymptotics of the solutions (i.e., can we approximate, with controlled error, the solutions with simpler functions?), as well as some basic relations that these fundamental solutions satisfy.

Inverse Spectral Theory Part 01: Power Series Methods for ODEs

Inverse Spectral Theory Part 02: Properties of the Fundamental Solutions

After that, we will go on to look at the spectral theory proper. We will show that for our problem above, the eigenvalues are in fact real and increase to $\infty$. Then, we will iteratively improve our estimates, to obtain the rather precise asymptotic estimate $$\lambda_n = n^2\pi^2 + \int_0^1 q(x) \, dx + d_n,$$ where the $d_n$ form a square-summable sequence. In the meantime, we will also establish some results which will prove useful in future parts.

With the two prerequisite parts done we will we moving on to inverse spectral theory proper. I’m not exactly sure what to expect here too, as I am learning it myself at the moment :P

Prerequisites wise, some familiarity with complex analysis, functional anlaysis, and a little measure theory will be needed. Taking a peek in later sections, I anticipate that some differential geometry will show up eventually too. As for the nonlinear functional analysis that shows up, I will try my best to elaborate on it more, but it will of course be helpful if you already know a little of it.