Spectral Theory of the Heat Operator on S1
as an Example of What to Expect in General
The Laplacian Δ on S1
, , is, for functions, f(θ) →
and for
1-forms f(θ)dθ→
dθ.
Eigenfunctions for Δ on
functions are thus solutions of =λf, namely
λ=0 f(θ)=constant
λ=+n2
f(θ)= n
+
(n>0) .
(note
the + sign here: Δ= gives this)
Actually, it is easier to do -valued
functions in which case
λ=+n2
f(θ)= n
n≠0 (n<0 o.k.)
It is standard Fourier analysis that any function in C∞ (or L2 even ) has an expansion in these eigenfunctions
which is L2 convergent to f and, if f is a C∞ function , is point-wise convergent
( uniformly, indeed) .
Here (
all n, including 0)
It is easy to see that solves
the first term gives ,
the second .
This will clearly work in general:
If Δx f(x,
t )=λf(x ,t), then e-λt will solve .
Thus the spectral theory of Δ gives solutions of the heat equation.
The heat equation is an “evolution equation”: it arises in a
situation where the natural thing to do is specify an “initial condition” G(θ) and look for f(θ,t) , t>0 which solves the heat equation for
t>0 and which has .
Clearly, the natural choice for this is to expand G(θ)= in Fourier expansion and let f(θ,t)=
.
Note that f(θ,t) =
=
Thus f(θ,t) is “convolution with a kernel”.
In the Riemann manifold case G(-θ) does not make sense: We have no way to take differences of
points. The representation to be used is
the one in the S1 case of the form f(θ,t)=
where the expression
becomes simply a function K(t,x,y) and the
integral formula has the form
f(y,t)= .
Then K is called the heat kernel, and it is given by an expression
.
We have been looking at this in terms of analyzing the heat equation when we know (as we do for S1) the eigenvalues and eigenfunctions of the Laplacian.
But what we really want to do is run this backwards: we want to understand Δ via understanding the heat equation. For this, we think of the operator Ht that takes the initial condition G(θ) to f(θ,t) , t fixed.
The eigenfunctions for this ( in our S1 example)
are einx , n
(including n=0) and these have eigenvalues e-0t
( 0 is a multiplicity 1 eigenvalue ), and
n≠0 where
n2
, n≠0 , is a multiplicity 2 eigenvalue. Note that the eigenvalues depend on t, but
the eigenfunctions do not. This has to
be the case, because the operators of "running the heat" for time t
and time s commute: both orders of composition give running the heat for time
t+s = s+t !
The question arises of why we might want to analyze Δ by
working with the apparently more complicated heat operators .
The answer is that (in our previous notation) G(θ)→f(θ,t) is a compact operator!
Its eigenvalues are (t>0) , which are clearly bounded above (
and go to 0 , which is what makes the operator compact, not just bounded) ,
while Δ has eigenvalues
n2, which are unbounded so Δ is not
only non-compact, it is not even bounded ( in the L2 sense ) .
The compactness of the heat flow map Ht:G(x)→f(x,t) , t fixed, is associated to two different but related phenomena. First, if f comes from G by convolution with a C∞ kernel, then Ht is "infinitely smoothing" in the sense that f(x,t) is C∞ even if G is only L2, say. Moreover, by differentiation under the integral sign and the Cauchy-Schwarz inequality, the L2 norm of f(x,t) ( in x ) and its first k derivatives are "estimated (from above)" by the L2 norm of G, i.e., are ≤ Ck (L2 norm of G). Then Arzela-Ascoli together with Sobolev embedding (which says that C0 norm is estimated by L2 norm of derivative of order ≤k, k large enough ) gives compactness of the Ht-image of the L2 unit ball ( in G).
On the level of Fourier analysis, we note that convolution gives product of Fourier coefficients, i.e. kth coefficient of convolution f,g = (kth coefficient of f ) (kth coefficient of g) all f,g . Since the heat kernel (t fixed ) will turn out to be C∞ , it has kth Fourier coefficients o(k-l) any l , and hence so does the convolution of the kernel with G; hence the convolution is C∞.
This makes it clear, at least on general terms, how the "infinitely smoothing"
property of Ht is related to the rapid decay of the eigenvalues .