Zeta functions and periodic orbit theory: A review

DOI10.1007/BF03323131zbMath0785.11033MaRDI QIDQ2366158

Norman E. Hurt

Publication date: 29 June 1993

Published in: Results in Mathematics (Search for Journal in Brave)

zbMATH Keywords

symmetric spacesSelberg trace formulaperiodic orbitstring theorydifferential geometryzeta functionsLie group representationcomprehensive reviewquantization of chaos

Mathematics Subject Classification ID

Spectral problems; spectral geometry; scattering theory on manifolds (58J50) Discrete subgroups of Lie groups (22E40) Other Dirichlet series and zeta functions (11M41) Differential geometry of symmetric spaces (53C35) Determinants and determinant bundles, analytic torsion (58J52) Spectral theory; trace formulas (e.g., that of Selberg) (11F72) Spectral theory; eigenvalue problems on manifolds (58C40) Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) (30F35)

Cites Work

• The arithmetic and geometry of some hyperbolic three manifolds
• Periodic-orbit sum rules for the Hadamard-Gutzwiller model
• The Selberg trace formula and the Ruelle zeta function for compact hyperbolics
• On cusp forms for co-finite subgroups of PSL(2,\({\mathbb{R}})\)
• Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen
• Nonclassical eigenvalue asymptotics
• An analogue of the prime number theorem for closed orbits of Axiom A flows
• Some quantum operators with discrete spectrum but classically continuous spectrum
• An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions
• Zeta function of Selberg's type for compact quotient of SU(n,1) (n\(\geq 2)\)
• Some applications of thermodynamic formalism to manifolds with constant negative curvature
• Exact theory for the quantum eigenstates of a strongly chaotic system
• Energy-level statistics of the Hadamard-Gutzwiller ensemble
• Classical and quantum mechanics of a strongly chaotic billiard system
• On the spectrum of the Hecke groups
• Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
• Zeta functions of Selberg's type associated with homogeneous vector bundles
• Lower bounds for \(\lambda _ 1\) on a finite-volume hyperbolic manifold
• Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces
• Meromorphic extensions of generalised zeta functions
• The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups
• Determinants of Laplacians
• Analytic torsion and closed geodesics on hyperbolic manifolds
• Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces
• Spectral functions, special functions and the Selberg zeta function
• The Laplace operator on a hyperbolic manifold. I: Spectral and scattering theory
• Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
• Uniform distribution of eigenfunctions on compact hyperbolic surfaces
• Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms
• Extremals of determinants of Laplacians
• Locating resonances for axiom A dynamical systems
• Torsion and closed geodesics on complex hyperbolic manifolds
• On the periodic orbits of a strongly chaotic system
• Irregular scattering
• Eta invariants of Dirac operators on locally symmetric manifolds
• On the ergodic properties of nowhere dispersing billiards
• Spectra of compact locally symmetric manifolds of negative curvature
• Entropy of geodesic flow and exponent of convergence of some Dirichlet series
• The convergence of zeta functions for certain geodesic flows depends on their pressure
• The density at infinity of a discrete group of hyperbolic motions
• Die Selbergsche Spurformel für kompakte Riemannsche Flächen. (The Selberg trace formula for compact Riemann surfaces)
• Markov partitions for dispersed billiards
• Statistical properties of Lorentz gas with periodic configuration of scatterers
• Pseudo-laplaciens. I
• The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces
• Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces
• Chaos in classical and quantum mechanics
• The spectrum of Fermat curves
• A note on the Selberg zeta function for compact quotients of hyperbolic spaces
• Spectral limits for hyperbolic surfaces. I
• On Selberg's trace formula
• Zeta-functions for expanding maps and Anosov flows
• The limit set of a Fuchsian group
• On the wave equation on a compact Riemannian manifold without conjugate points
• The Selberg trace formula and the Riemann zeta-function
• Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one
• The length spectra of some compact manifolds of negative curvature
• Limit formulas for multiplicities in \(L^2(\Gamma\setminus G)\)
• Closed geodesics and the \(\eta\)-invariant
• A nonarithmetic derivation of the Selberg trace formula
• Trace formula for compact \(\Gamma \setminus PSL_ 2({\mathbb{R}})\) and the equidistribution theory of closed geodesics
• Crossing the entropy barrier of dynamical zeta functions
• \(R\)-torsion and zeta functions for locally symmetric manifolds
• A formula for the logarithmic derivative of Selberg's zeta function
• Fundamental polygons for Fuchsian groups
• Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces
• An upper bound to the spectrum of \(\Delta\) on a manifold of negative curvature
• Spherical transforms on semisimple Lie groups
• Applications of ergodic theory to the investigation of manifolds of negative curvature
• R-torsion and the Laplacian on Riemannian manifolds
• Visibility manifolds
• Analytic torsion for complex manifolds
• Periodic orbits on the regular hyperbolic octagon
• Quelques propriétés globales des espaces de Riemann
• Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I
• The Chebotarov theorem for Galois coverings of Axiom A flows
• Discontinuous groups on three-dimensional hyperbolic space: analytical theory and arithmetic applications
• A rule for quantizing chaos?
• SELBERG'S TRACE FORMULA FOR THE HECKE OPERATOR GENERATED BY AN INVOLUTION, AND THE EIGENVALUES OF THE LAPLACE-BELTRAMI OPERATOR ON THE FUNDAMENTAL DOMAIN OF THE MODULAR GROUP $ PSL(2,\mathbf{Z})$
• Maslov indices in the Gutzwiller trace formula
• The thermodynamic formalism approach to Selberg’s zeta function for 𝑃𝑆𝐿(2,𝐙)
• The Selberg zeta function and a local trace formula for Kleinian groups.
• Refined asymptotic expansion for the partition function of unbounded quantum billiards
• Spectral theory of schrödinger operators on euclidean and on non-euclidean spaces
• The zeta functions of Ruelle and Selberg. I
• Poincare map for scattering states
• Spectral zeta functions for Aharonov-Bohm quantum billiards
• Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups
• Cusp forms and Hecke groups.
• Unstable periodic orbits and semiclassical quantisation
• Spectral Theory and Eisenstein Series for Kleinian Groups
• Scattering operator, Eisenstein series, inner product formula and “Maass-Selberg” relations for Kleinian groups
• The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix.
• On the Analytic Torsion and Eta Invariant for Negatively Curved Manifolds
• The Minakshisundaram-Pleijel Coefficients for the Vector Valued Heat Kernel on Compact Locally Symmetric Spaces of Negative Curvature
• Selberg trace formulae and equidistribution theorems for closed geodesics and Laplace eigenfunctions: finite area surfaces
• Periodic orbit resummation and the quantization of chaos
• Indices in classical mechanics
• A new asymptotic representation for ζ(½ + i t ) and quantum spectral determinants
• Semiclassical quantization of multidimensional systems
• From classical periodic orbits to the quantization of chaos
• Semiclassical quantization of chaotic billiards: a scattering theory approach
• Symmetry decomposition of chaotic dynamics
• On the Analytic Continuation of the Minakshisundaram-Pleijel Zeta Function for Compact Riemann Surfaces
• A lattice‐point problem in hyperbolic space
• Small eigenvalues of the Laplace operator on compact Riemann surfaces
• A Computational Technique for Evaluating L(1, χ) and the Class Number of a Real Quadratic Field
• On the Selberg trace formula in the case of compact quotient
• Fourier coefficients of the resolvent for a Fuchsian group.
• Scattering Theory for Automorphic Functions. (AM-87)
• Length spectrum for compact locally symmetric spaces of strictly negative curvature
• Spectral theory and Fuchsian groups
• The Riemann Hypothesis for Selberg's Zeta-Function and the Asymptotic Behavior of Eigenvalues of the Laplace Operator
• A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$
• Indistinguishability for quantum particles: spin, statistics and the geometric phase
• The Geometry of Quantum Chaos
• Random-matrix description of chaotic scattering: Semiclassical approach
• Quantum eigenvalues from classical periodic orbits
• Quantization of chaos
• Novel rule for quantizing chaos
• The semiclassical functional equation
• On dynamical zeta function
• Winding number formula for Maslov indices
• Periodic orbit theory in classical and quantum mechanics
• Quantization of chaotic systems
• Applications of periodic‐orbit theory
• Periodic orbit quantization of the anisotropic Kepler problem
• Arithmetic Applications of the Hyperbolic Lattice Point Theorem
• The weyl theorem and the deformation of discrete groups
• Level clustering in the regular spectrum
• The Exponent of Convergence of Poincaré Series
• Semiclassical structure of trace formulas
• On Some Exponential Sums
• Class numbers of indefinite binary quadratic forms
• Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Zeta functions and periodic orbit theory: A review