The Selberg trace formua (Selberg 56) is an expression for certain sums of eigenvalues of the Laplace operator on a compact hyperbolic Riemann surface (recalled e.g. as Bump, theorem 19).
It was introduced as a nonabelian generalization of the Poisson summation formula (e.g.Voros-Balasz 86, p. 169), thought of as a relation between the eigenvalues of the Laplacian and the lengths of closed geodesics.
It motivates (e.g. Bump, p.18) the definition of the Selberg zeta function of a Riemann surface.
Atle Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Journal of the Indian Mathematical Society 20 (1956) 47-87.
Joshua Friedman, The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations (arXiv:math/0410067)
Wikipedia, Selberg trace formula
Bump, theorem 19 in Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$ (pdf)
A. Voros and N.L. Balasz, Chaos on the pseudosphere, Physics Reports 143 no. 3 (1986)
For Dirac operators:
Last revised on May 22, 2019 at 11:33:51. See the history of this page for a list of all contributions to it.