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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 Δ_x 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 H_t  that takes the initial condition G(θ) to f(θ,t) , t fixed. 

The eigenfunctions for this ( in our S¹ example) are e^inx  , 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 H_t: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 H_t 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 ≤ C_k (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 H_t-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. k^th coefficient of convolution f,g = (k^th coefficient of f ) (k^th coefficient of g) all f,g .  Since the heat kernel (t fixed ) will turn out to be C^∞ , it has k^th  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 H_t is related to the rapid decay of the eigenvalues .