6.2 Kernel of Operator
We now turn to the heat equation. As usual, we define
and consider . Our objective is to understand
where is the kernel of . We do not make any claims about how this relates to an index (or not), since we are working on a non-compact manifold and life is tough.
We first consider , which is explicitly
We again perform separation of variables. We write our potential solution as . Our boundary conditions are then
Note that the first equation constrains for , and the second constrains for . So we have a single constraint for each .
To find the fundamental solution for , we have to find a fundamental solution for the operator
Up to the , this is just the classical heat equation.
For the boundary condition when we can simply write down the fundamental solution to be
This is easily seen to vanish when or .
When , we need to do something more complicated. People good at solving differential equations will find the kernel to be
where is the complementary error function
Equipped with these, we can then write the heat kernel of as
The formula for is basically the same, except the boundary conditions are swapped. So we find the difference to be
where we set .
Integrating first over then over , we get
Recall that in the compact case, this expression is identically equal to the index of the operator . In this case, we have
Writing , we see that in fact exponentially as .
To understand this better, we define the function
where of course we sum over eigenvalues of with multiplicity. One should think of this as some sort of Dirichlet -function for the spectrum. To relate this to , we first observe that
So after integration by parts once, we find that, at least formally,
This is in fact literally true when is big enough, since Lemma 1.1 tells us how quickly the eigenvalues grow, and for all .
If we assume that has an asymptotic expansion
which we will show in the next section by devious means, then we can explicitly do the integral to get
where is holomorphic for . Taking to be large enough, we know that admits a meromorphic continuation to the whole plane, and the value at is
This is the final result we seek.