Selberg's zeta function

The Selberg zeta function is a function from the mathematical field of harmonic analysis . It is used to investigate the relationship between the eigenvalues ​​of the Laplace operator and the length spectrum of a hyperbolic surface.

definition

Let it be a hyperbolic surface or orbital fold . For a simple closed geodesic denote its length. The Selberg zeta function is meromorphic continuation of for by ${\displaystyle S=\Gamma \backslash H^{2}}$ ${\displaystyle \gamma \subset S}$ ${\displaystyle l(\gamma )}$ ${\displaystyle {\mathcal {Z}}_{\Gamma }\colon \mathbb {C} \to \mathbb {C} \cup \left\{\infty \right\}}$ ${\displaystyle \operatorname {Re} (s)\gg 0}$

${\displaystyle {\mathcal {Z}}_{\Gamma }(s)=\prod _{\gamma }\prod _{k=0}^{\infty }(1-e^{-l(\gamma )(s+k)})}$

given function.

zeropoint

The zeros of Selberg's zeta function are the numbers that make up the equation ${\displaystyle s_{j}}$

${\displaystyle s_{j}(1-s_{j})=\lambda _{j}}$

for one of the eigenvalues

${\displaystyle 0=\lambda _{0}<\lambda _{1}<\lambda _{2}<\ldots }$

of the Laplace operator to satisfy. ${\displaystyle S}$

Mayerscher transfer operator

For you have the identity ${\displaystyle \Gamma =SL(2,\mathbb {Z} )}$

${\displaystyle {\mathcal {Z}}_{\Gamma }(s)=\det(1-{\mathcal {L}}_{2}^{2})}$.

In this case, referred to the Mayer's transfer operator on the space of the functions on the open disk with the center and radius holomorphic and are continuous to its edge. It is defined by ${\displaystyle {\mathcal {L}}_{s}\colon V\to V}$ ${\displaystyle V}$ ${\displaystyle 1}$ ${\displaystyle {\tfrac {3}{2}}}$

${\displaystyle {\mathcal {L}}_{s}\phi (z)=\sum _{n=1}^{\infty }{\frac {1}{(z+n)^{2s}}}\,\phi \left({\frac {1}{z+n}}\right)}$.

literature

• U. Bunke, M. Olbrich: Selberg Zeta and Theta Functions. A differential operator approach. Vol. 83 of Mathematical Research (Akademie-Verlag, 1995).
• d'Hoker, E. and Phong, DH: Multiloop Amplitudes for the Bosonic Polyakov String. Nucl. Phys. B 269, 205-234, 1986.
• d'Hoker, E. and Phong, DH: On Determinants of Laplacians on Riemann Surfaces. Commun. Math. Phys. 104, 537-545, 1986.
• Fried, D .: Analytic Torsion and Closed Geodesics on Hyperbolic Manifolds. Invent. Math. 84, 523-540, 1986.
• Selberg, A .: Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series. J. Indian Math. Soc. 20, 47-87, 1956.
• Voros, A .: Spectral Functions, Special Functions and the Selberg Zeta Function. Commun. Math. Phys. 110, 439-465, 1987.

Web links

• Selberg Zeta Function (Math World)
• Don Zagier : New points of view on the Selberg zeta function
• Ulrich Bunke : Theta and Selberg zeta functions