This post will be of a different nature to my other posts: I want to talk about a speculative idea that: (i) I had while sitting in a talk; (ii) I haven’t worked through at all; (iii) involves higher power mathematics than what I have at my disposal right now; and (iv) may be stupid, wrong, too naive, or already well-known.
These kind of ideas occur to theorists all the time and, due to resource constraints (i.e., time, energy, money, etc.), seldom get mentioned, except to, say, a conference speaker, and are quickly forgotten by both parties by the time the conference is over.
However, with access to this new medium I hope to broadcast these ideas to more people, some of whom may be experts on the maths required, and maybe, just maybe, someone will be inspired to check it out.
Dumb questions are especially encouraged in these posts because, after all, I’m posting about a dumb question!
Please read on for my first speculative idea post.
1. Nodal domains
Recently Uzi Smilansky gave a talk to the quantum dynamics group at Royal Holloway entitled “Can one count the shape of a drum?” (see here for the paper the talk was based on). This is heady stuff! Uzy is a real physicist of the highest caliber and he is able to understand subtle problems with powerful physical arguments that, as yet, cannot be studied at a mathematical level of rigour.
Uzy is responsible for a kind of revolution in our understanding of the simple (Schrödinger) wave equation: whereas a lot of research energy has been spent on understanding the eigenvalue distribution of the 2D time-independent Schrödinger wave equation
a great deal of interesting open questions remain!
Uzy has a new approach to the question, first put by Victor Kac: “can one hear the shape of a drum?” What does this mean? Suppose you knew the all of the eigenvalues of the Schrödinger equation (or, as originally posed, the simple wave equation) for a particle in an arbitrarily shaped 2D box. Could you reconstruct the shape of the box knowing just the eigenvalues? The answer, is no. Although, the simplest counterexamples are nonconvex. In fact, one of the reasons the question is interesting is precisely because it could so easily have been yes: there are large families of domains which can be uniquely reconstructed from knowing the spectrum of the Schrödinger equation in that domain.
To understand Uzy’s contribution we need to define a nodal domain. Let be an eigenfunction of (1). A nodal domain is a connected region where is positive or negative. The boundaries between nodal domains are given by the zeros of , i.e., by the (non simply) connected curve
Write for the number of nodal domains for the th eigenfunction. Uzy’s idea is as follows: suppose you know both and , then you can reconstruct the domain. He presents a great deal of evidence for this conjecture and has gone on to study the properties of nodal domains in great detail.
In his talk, Uzy mentioned a problem: how do you count nodal domains? This seems like a stupid question when you first encounter it; after all, don’t you just write down the eigenfunction and count the nodal domains? Well, yes and no: the problem is that you need to count 2D connected regions and there is no elegant mathematical way to represent this count short of running a 2D search algorithm. Contrast this with the situation in 1D where nodal domains can be easily counted by instead counting the zeros of and these, in turn, can be studied via the poles of , upon which the machinery of complex analysis can be brought to bear. Additionally, nodal domains are pretty sensitive to boundary conditions: they easily split if the slightest perturbation is added. Thus they are not very robust. Arguably eigenvalues are similarly non-robust but they aren’t that sensitive to the shape of the perturbation: under a perturbation all that can happen is that degeneracies get lifted. Of course, you might well argue that this holds true for the distribution , and to this I have no real reply other than we’d like to have a robust way to count them which somehow accounts for perturbations. This obsession with perturbations is because when you solve these things numerically, or approximate them theoretically, the errors induced can be modelled as perturbations. One would like to have a way to track how errors propagate into .
2. Introducing the genus
I think it was Michael Nielsen who once said to me: “if the answer is ugly, then maybe the question was wrong” (the original source of this quote, and accuracy, is unknown to me). I think that the difficulty in counting nodal domains that I mentioned in the previous section could be a symptom of the fact the question asked is slightly wrong. I really don’t want to sound like I’m criticising the whole nodal domain approach, which is a truly wonderful idea, not least because Uzy has gotten actual results. Rather, I would like to suggest a modification of the question which might be easier to study but which might lose some of the expressive power of nodal domains.
The idea is based one of the fundamental ideas involved in the study of algebraic curves such as the elliptic curve
and represents a two-dimensional surface in . (See here for some example visualisations.)
If is the equation of the boundary between nodal domains for an eigenfunction then we can, as above, regard as defining an equation over , or defining a two-dimensional surface in . The actual boundary is then the intersection of this surface with the two-dimensional plane defined by and .
Given that we can interpret nodal boundaries in this way, i.e. as 2D surfaces, then we can try ask questions about these surfaces which should be topologically robust. The first question which comes to mind, inspired by the fact that all closed 2D surfaces have been classified up to homeomorphism, is to ask what is the genus of the (projective version of the) equation defined by . (By projectively equivalent I mean the equation when interpreted over the complex numbers with infinity adjoined, so the space becomes closed.) This quantity arises from the fact that every 2D surface is homeomorphic to a sphere with handles.
So that’s it: the idea is to study not the sequence of eigenvalues and numbers of nodal domains, but rather to study of eigenvalues and corresponding genuses of the surface defined by . (Note that it’s probably important that (1) be defined over all to begin with, and we mimic say, boxes, with infinite potentials .)
Presumably (I guess!) the genus is easier to understand in the presence of perturbations than the number of nodal domains. The problem is I have no idea how to compute the genus in terms of the original differential operator, so it’s not really any gain whatsoever. (Indeed, it may be impossible to calculate practically or even approximate!)
Any comments or suggestions or contributations, as always, are most welcome!