How Does the Selberg Trace Formula Connect Eigenvalues and Manifold Properties?

[tpm]

Could someone explain the 'Selberg Trace formula' concept??

for example let be the Laplacian in curved Space-time:

\Delta \Psi = E_{n} \Psi

My question is is there a relationship between the set of eigenvalues E(n) and a certain charasteristic of the SUrface (length, Areal or so on) due to Selberg Trace ?..thanks.

 

[Chris Hillman]

Trace formula in a paragraph? I think not!

Well, someone like Terry Tao can probably explain the gist in a paragraph, but I hardly dare try that myself. I'll say this much: it makes a big difference whether or not your manifold is compact and Riemannian.

See survey articles like those in these books:

Bert-Wolfgang Schulze and Hans Triebel (editors).
Surveys on analysis, geometry, and mathematical physics.
Teubner, 1990.

Steven Zelditch
Selberg trace formulae, and equidistribution theorems for closed geodesics and Laplace eigenfunctions : finite area surfaces
American Mathematical Society, 1992

Sources listed at
http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm