Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

A brief Survey on the Selberg Trace formula (the compact case)

Document Type : Original Article

Author
Jafar Shaffaf
Department of Mathematics, Shahid Beheshti University, Tehran, Faculty member, Assistant professor.
10.30511/ttmp.2025.2048769.1045
Abstract
In this note, we aim to elucidate the fundamental mathematical concepts and mathematical ingredients underlying the Selberg trace formula. We explicitly present the formula for compact quotients and provide a brief overview of its interpretation in both mathematics and physics. The Selberg trace formula transcends the boundary of mathematics, establishing intriguing connections between classical mechanical entities such as volume, shape, and geodesics on a surface and quantum mechanical entities such as eigenvalues (frequencies), eigenfunctions, and resonances of the underlying geometry. A rudimentary understanding of complex analysis and hyperbolic geometry is assumed.
Keywords
Key words: Discrete group
Laplace spectrum
length spectrum
closed geodesic
Subjects
1. A. Selberg, J. Indian Math. Soc. 20, (1956)#

2. P. Buser, Geometry and spectra of compact Riemann sur
faces, Birkhauser Publication, (1992)#

3. D. Hejhal, The Selberg trace formula for PSL(2,R),
Springer Verlag Publication, (1976)#

4. S. Sarnak, Bull. Am. Math. Soc. 40, (2003)#

5. E. Lindenstrauss, Ann. Math. 163, (2006)#

6. S. Zelditch, Quantum ergodicity and mixing of eigen
functions, Encyclopedia of mathematical physics, Ox
ford, Vol. 1-5, (2006)#
 
Transactions in Theoretical and Mathematical Physics
Volume 2, Issue 2
Spring 2025
Pages 70-76

  • Receive Date 23 December 2024
  • Accept Date 18 May 2025
  • First Publish Date 18 May 2025
  • Publish Date 01 May 2025