nLab Selberg trace formula

Idea

The Selberg trace formula (Selberg 1956) is an expression for certain sums of eigenvalues of the Laplace operator on a compact hyperbolic Riemann surface (e.g. Bump, theorem 19).

The formula 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.

Related concepts

• Riemann hypothesis

• number theory and physics

• Arthur-Selberg trace formula

• Duistermaat-Guillemin trace formula

• zeta function of a Riemann surface

• Selberg zeta function

References

• 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:

• Jens Bolte, Hans-Michael Stiepan, The Selberg trace formula for Dirac operators (arXiv:math-ph/0607010)

Discussion via supersymmetric quantum mechanics:

• Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas I. Compact Lie groups, Part I. Compact Lie groups. J. High Energ. Phys. 2024 26 (2024) [arXiv:2112.07942, doi:10.1007/JHEP06(2024)026]

• Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas II. Selberg trace formula [arXiv:2306.13636]

review:

• Leon A. Takhtajan: Supersymmetry and trace formulas: Selberg trace formula, talk at Integrable Systems and Field Theory, Paris (2023) [pdf]