Minakshisundaram And The Birth Of Geometric Spectral Asymptotics: By prof. S. A. Fulling.
Those who do not know what “geometric spectral asymptotics” is should think in terms of this elementary example: A vibrating string of length L has normal modes (eigen functions) with frequencies (in units where the speed of sounds 1) where runs though all the positive integers. Note that
Is essentially the famous Riemann zeta function. The eigenfunctions satisfy the equation and boundary condition.
If we changed the boundary condition to (which would be appropriate for sound waves in a pipe with open ends), the normal modes would become and the case n=0 would need to be included. Although we started with a wave problem, exactly the same eignefunctions solve the problem of heat condition in a bar of length L. The Green function (integral kernel) that gives the temperature distribution u(x) at time t in terms of the intial temperature f(y) at time 0 is
On the other hand, by the method of images (and the well known solution of the heat problem on the whole real line) one can see that
Where the + sign applies to the Neumann (organ pipe) boundary condition and the –to the Dirichlet (string) case. Now examine the integral at small t; it can be shown that only the first term in (2) makes a significant contribution except when x is close to an endpoint, whereupon the second or third term must be taken onto account and one gets.
Therefore, from a knowledge of the small time asymptotics (3) of K we can read off the size of the interval, L, and also (from the second term) which boundary condition is in force. Working in another direction, comparing (3) with (1) one could get some information about the eigenvalues if one did not know them already; for example, the fact that expressions (3) for the Neumann and Dirichlet cases differ by exactly 1 reflects the presence of the extra eigenvalue, λ0=0 in the Neumann case.