Some years ago I speculated that it would nice if a certain mathematical object existed, and even nicer if it were to satisfy an ordinary differential equation of a special sort. I was motivated by a particular physical question, and it seemed very natural to me to imagine such an object… So natural that I was sure that it must already have been studied, the equation for it known. As a result, every so often I’d go down a rabbit hole of a literature dig, but not with much success because it isn’t entirely clear where best to look. Then I’d get involved with other projects and forget all about the matter.
Last year I began to think about it again because it might be useful in a method I was developing for a paper, went through the cycle of wondering, and looking for a while, then forgot all about it in thinking about other things.
Then, a little over a month ago at the end of March, while starting on a long flight across the continent, I started thinking about it again, and given that I did not have a connection to the internet to hand, took another approach: I got out a pencil and began mess around in my notebook and just derive what I thought the equation for this object should be, given certain properties it should have. One property is that it should in some circumstances reduce to a known powerful equation (often associated with the legendary 1975 work of Gel’fand and Dikii*) satisfied by the diagonal resolvent $latex {\widehat R}(E,x) {=}\langle x|({\cal H}-E)^{-1}|x\rangle$ of a Schrodinger Hamiltonian $latex {\cal H}=-\hbar^2\partial^2_x+u(x)$. It is:
$latex 4(u(x)-E){\widehat R}^2-2\hbar^2 {\widehat R}{\widehat R}^{\prime\prime}+\hbar^2({\widehat R}^\prime)^2 = 1\ .$
Here, $latex E$ is an energy of the Hamiltonian, in potential $latex u(x)$, and $latex x$ is a coordinate on the real line.
The object itself would be a generalisation of the diagonal resolvent $latex {\widehat R}(E,x)$, although non-diagonal in the energy, not the […] Click to continue reading this post →
