DJ Strouse

My progress reports may have been dormant for a few weeks, but the search for Levinson’s theorem on graphs has not!  (My laptop was stolen last week so my recent time with computers has been necessarily precious and could not be wasted on blog ramblings.  More on this below.)

Explorations in Mathematica
As you may remember, the last we heard from our heroes, they were in search of a version of Levinson’s theorem (relating the number of bound states to the winding of the phase of scattered states) that could be applied to graphs.  Given that neither Andrew nor I were quite sure how this relationship between bound states and the phase shift would play out, we decided to do some experiments.  First, we cooked up equations for the simplest graphs we could think of – a single weighted edge and then (drumroll) a single weighted edge with a self-loop on one vertex.  It quickly became apparent that if we wanted to take a look at meatier graphs, we were going to need some help.  So next, I hacked together a series of programs in Mathematica that could take graphs, plug a tail on some arbitrary node, calculate the transcendental equation whose solutions correspond to the existence of bound states, calculate the phase shift of scattering states, and then plot the transcendental equations alongside the phase shift, all with manipulable graph parameters.  In this way, I could jiggle the weights on the graph, watch bound states come into and go out of existence, and simultaneously monitor the phase shift.  I’ve posted my current Mathematica notebook here, so you can make sense of the above jibba-jabba and do some explorations of your own.  (WARNING: The documentation is limited to my stream of consciousness as I code.  Also, I’ll probably kill the link when I need the server space, so you might have to email me for updated the notebook if the link is dead.)

This was a really fun approach to research that I hadn’t taken before.

1. Recognize a hand-wavy possible connection between two quantities.
2. Investigate (by hand) a few simple cases and try to get a flavor for the relationship.
3. Investigate (by computer) much more interesting cases and try to pin point the details of the relationship.
4. Prove the relationship rigorously.

It was also a welcome opportunity to polish off the increasingly dusty programming portions of my brain and to expand my Mathematica repertoire.  (Mathematica can be an incredibly powerful aid to theory work if you take the time to learn it.  Its visualization tools are especially nifty.)

Bound State Zoology
Those interested in taking a peak at my Mathematica notebook will need a quick intro to bound state zoology.  It turns out that there are at least three distinct species of bound states.

1. Confined bound states – these guys live only on the graph and have zero amplitude on the tail
2. (Standard) bound states – these guys “leak” onto the tail; that is, they have an exponentially decaying amplitude on the tail
3. Half-bound states – these bound/scattering chimera have a constant amplitude on the tail and exhibit some characteristics of bound states and some of scattering states

From our investigations, it looks like the first two types contribute one winding of the phase and the third type contributes (go figure) half a winding.

An Unexpected Hiatus or “The First Crime in Waterloo Since Wellington Spanked Napoleon”
Mid-mathematical adventure, we hit a snag.  Given the abundance of Mennonites, smiles, and unlocked doors in Waterloo, I simply assumed that Canada was crime-free and kept my backpack in an unlocked locker while working out at the university gym.  Little did I realize just how far the local citizenry would go to snag copies of my Mathematica notebooks and, alas, my laptop and wallet were stolen.  Unfortunately, I had not made backups of my latest research.  Every setback is of course an opportunity to learn and improve and this was no exception.

The lesson: sync your research to multiple computers and ideally a trustworthy server (Dropbox is my tool of choice for this)

The opportunity to improve: as any programmer knows, code you wrote a week ago always looks painfully clunky.  In rebuilding my Mathematica work from scratch, I was able to integrate plenty of tricks and lessons I’d learned along the way.  The result – a way sexier notebook.

Complex Analysis Boot Camp
In the last week, Andrew and I converged on the general flavor of the version of Levinson’s theorem that we think we can prove on graphs.  Thus, away goes the computer and out comes the physicist’s favorite tools – pen and paper.  In our first stabs at proving the theorem, Andrew and I took completely different approaches.  My caveman approach was to try to adapt a simple trick involving the spectral resolution of the identity that Marcel Wellner used to prove a version of Levinson’s theorem for continuous potentials back in the 1960s.  Andrew’s far more sophisticated and elegant approach was to apply the topological reasoning of complex analysis to our problem.  Since my caveman club began to look a little too primitive mid-way through my proof attempt, I decided to take this opportunity to learn a little complex analysis, one of the (many) major holes in my undergrad math education.

And sweet Feynman has it been a fun last couple days!  I picked up Tristan Needham’s Visual Complex Analysis from the UW library and this book has reminded me why I fell in love with math as a wee lad.  The book’s pedagogical approach is to teach math the way mathematicians actually think about it – visually.  Needham’s book is chock full of nifty pictures of Riemann spheres, conformal mappings, branches, and more.   (I’ll post a review sometime for those interested.)  Complex analysis is an absolutely beautiful subject when couched in geometric terms and is accessible to anyone with a bit of calculus under their belt.

This was also a change of approach to learning for me.  Usually, I get generally interested in a subject, pick a textbook, and read it cover to cover, doing the problems as I go.  This time, however, my explorations in complex analysis were motivated by a very specific application, so I dove right into the middle of the book to extract the particular pieces I needed.  Two great results:

1. I’m honing in on a proof of our version of Levinson’s theorem using tools from complex analysis.
2. I became so enamored with complex analysis and Needham’s book in particular that I’ve spent the whole weekend hopping from chapter to chapter learning all sorts of interesting things about the topological properties of complex functions.

I’ve noticed that I seem to learn much more quickly with an application in mind like this.  Further investigations are needed, but this might hint at a more efficient and productive approach to learning for me – first, find a particular problem that requires the tools you’re interested in and then, go off and learn them.  This is fun because every new topic I stumble upon in Needham’s book gives me new insights into the problem I’m working on.

I still want to take a crack at a proof using my original approach as well, as I might be able to get it to work with a little more insight from my adventures in complex analysis.  It would be really cool to prove our theorem using two entirely different approaches, each of which gives a unique insight into the problem.

• Progress Report: Beautifying Equations
• Progress Report: Scattering and Bound States of the Simplest Graph Imaginable
• Progress Report: IQC Project Defined