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:

[tex] \Delta \Psi = E_{n} \Psi [/tex]

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

 

[tacman]

The Selberg Trace formula is a powerful mathematical tool used in the study of spectral theory and geometry. It was first introduced by the Norwegian mathematician Atle Selberg in the 1950s and has since been applied in various fields such as number theory, representation theory, and physics.

In simple terms, the Selberg Trace formula relates the eigenvalues of a Laplace operator on a manifold to the geometric and topological properties of that manifold. In the example given, the Laplace operator is the differential operator used to study the behavior of waves on a curved space-time.

The Selberg Trace formula states that the sum of the eigenvalues of the Laplace operator is equal to the trace of the heat kernel, which is a function that describes the propagation of heat on the manifold. This means that the eigenvalues can be determined by studying the heat kernel and the geometric and topological properties of the manifold.

To address the question about the relationship between eigenvalues and characteristics of the surface, the Selberg Trace formula does not directly provide such a relationship. However, it does allow for the calculation of the eigenvalues, which in turn can be used to study the geometric and topological properties of the surface. So while there is not a direct relationship, the Selberg Trace formula does provide a way to indirectly study the characteristics of the surface through the eigenvalues of the Laplace operator.