Skip to main content
Log in

Zeta Functions and Periodic Orbit Theory: A Review

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The article reviews the interrelationship of periodic orbit theory and the Selberg trace formula. Examples from recent work on quantization of chaos are surveyed. The review emphasizes the development in terms of Lie group representation theory and differential geometry. Finally, the formal connections to string theory are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • S. Agmon, Spectral theory of Schrödinger operators on Euclidean and non-Euclidean spaces, Comm. Pure App. Math. 39 (1986) Number S, Supp.

  • E. Artin, Ein mechanisches system mit quasi-ergodischen bahnen, Hamburg Math. Abh. 3 (1924) 170–175.

    Article  MATH  Google Scholar 

  • R. Aurich, E. B. Bogomolny and F. Steiner, Periodic orbits on the regular hyperbolic octagon, Physica D48 (1991) 91.

    MathSciNet  Google Scholar 

  • R. Aurich, C. Matthies, M. Sieber, and F. Steiner, A new rule for quantizing chaos (preprint, 1991); Novel rule for quantizing chaos, Phys. Rev. Lett. 68 (1992) 1629–1632.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Aurich, M. Sieber and F. Steiner, Quantum chaos of the Hadamard-Gutzwiller model, Phys. Rev. Lett. 61 (1988) 483–487.

    Article  MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Periodic orbits of a strongly chaotic system, Physica D32 (1988) 451–460.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Periodic-orbit sum rules for the Hadamard-Gutzwiller model, Physica D39 (1989) 169–193.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Energy-level statistics of the Hadamard-Gutzwiller ensemble, Physica D43 (1990) 155–180.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Exact theory for the quantum eigenstates of the Hadamard-Gutzwiller model (preprint, 1990); Exact theory for the quantum eigenstates of a strongly chaotic system, Physica D48 (1991) 445–470.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, From classical periodic orbits to the quantization of chaos, Proc. Roy. Soc. London Ser. A (to appear).

  • R. Aurich and F. Steiner, Staircase functions, spectral rigidity and a rule for quantizing chaos, Phys. Rev. A45 (1992) 583–592.

    MathSciNet  Google Scholar 

  • R. Aurich J. Bolte, C. Matthies, M. Sieber and F. Steiner, Crossing the entropy barrier of dynamical zeta functions (preprint, 1992).

  • R. Balian and C. Bloch, Asymptotic evaluation of the Green’s function for large quantum numbers, Ann. Phys. 63 (1971) 592–606; 69 (1972) 76-160; 85 (1974) 514-545.

    Article  MathSciNet  Google Scholar 

  • N. L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rep. 143 (1986) 109–240.

    Article  MathSciNet  Google Scholar 

  • A. Beardon, The exponent of convergence of Poincaré series, Proc. London Math. Soc. 18 (1968) 446–483.

    Article  MathSciNet  Google Scholar 

  • P. H. Berard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977) 249–276.

    Article  MathSciNet  MATH  Google Scholar 

  • M. V. Berry, Quantizing a classically ergodic system: Sinai’s billiard and the KKR method, Ann. Phys. 131 (1981) 163–216.

    Article  Google Scholar 

  • M. V. Berry, Riemann’s zeta function: a model for quantum chaos? in Quantum Chaos and Statistical Nuclear Physics, ed. T. H. Seligman and H. Nishioka (Lecure Notes in Physics, 263, Springer, Berlin, 1986), 1–17.

    Chapter  Google Scholar 

  • M. V. Berry, Spectral zeta functions for Aharonov-Bohm, quantum billiards, J. Phys. A19 (1986) 2281.

    Google Scholar 

  • M. V. Berry, Some quantum-to-classical asymptotics, in Chaos and Quantum Physics: Les Houches Lecture Series 52 (ed. M.-J. Giannoni, A. Voros, and J. Zinn-Justin, North-Holland, Amsterdam, 1991).

    Google Scholar 

  • M. V. Berry, N. L. Balazs, M. Tabor and A. Voros, Ann. Phys. 122 (1979) 26–63.

    Article  MathSciNet  Google Scholar 

  • M. V. Berry and J. P. Keating, A rule for quantizing chaos?, J. Phys. A23 (1990) 4839–4849.

    MathSciNet  Google Scholar 

  • M. V. Berry and J. P. Keating, A new asymptotic representation for ζ(1/2 + it) and quantum spectral determinants, Proc. Roy. Soc. London 437 (1992) 151–173.

    Article  MathSciNet  MATH  Google Scholar 

  • M. V. Berry and M. Tabor, Closed orbits and the regular bound spectrum, Proc. Roy. Soc. Lond. A 349 (1976) 101–123.

    MathSciNet  Google Scholar 

  • M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. Lond. A356 (1977) 375–394.

    Google Scholar 

  • R. L. Bishop and B. O’Neill, Manifold of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Blümel and U. Smilansky, Random-matrix description of chaotic scattering, Phys. Rev. Let. 64 (1990) 241.

    Article  MATH  Google Scholar 

  • E. B. Bogomolny, Wave functions of quantum systems, Physica D31 (1988) 169–189.

    MathSciNet  Google Scholar 

  • E. B. Bogomolny, Semiclassical quantization of multidimensional systems, Comm. Atom. Mol. Phys. 25 (1990) 67.

    Google Scholar 

  • E. B. Bogomolny, On dynamical zeta function, Chaos 2 (1992) 5–13.

    Article  MathSciNet  MATH  Google Scholar 

  • L. A. Bunimovich, On ergodic properties of nowhere dispersing billiards,Comm. Math. Phys. 65 (1979) 295–312; see also ZhETF 89 (1985) 1452 and J. Func. Anal. Appl. 8 (1974) 254.

    Article  MathSciNet  MATH  Google Scholar 

  • L. A. Bunimovich and Ya. G. Sinai, Markov partitions for dispersed billiards, Comm. Math. Phys. 78 (1980) 247–280.

    Article  MathSciNet  MATH  Google Scholar 

  • L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980) 479–497.

    Article  MathSciNet  Google Scholar 

  • I. Chavel, Eigenvalues in Riemannian Geometry (Academic Press, New York, 1984).

    MATH  Google Scholar 

  • S. Chen, Entropy of geodesic flow and exponent of convergence of some Dirichlet series, Math. Ann. 255 (1981) 97–103.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Chen and A. Manning, The convergence of zeta functions for certain geodesic flows depends on their pressure, Math. Zeit. 176 (1981) 379–382.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Christiansen and P. Cvitanovic, Periodic orbit quantization of the anisotropic Kepler problem, Chaos 2 (1992) 61–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Y. Colin de Verdiere, Pseudo-Laplacians, Ann. Inst. Fourier 32 (1982) 275–286; 33 (1983) 87-113.

    Article  MathSciNet  Google Scholar 

  • P. Cvitanović, Periodic orbit theory in classical and quantum mechanics, Chaos 2 (1992) 1–4.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Cvitanović and B. Eckhardt, Periodic-orbit quantization of chaotic systems, Phys. Rev. Lett. 63 (1989) 823–826.

    Article  MathSciNet  Google Scholar 

  • P. Cvitanović and B. Eckhardt, Symmetry decomposition of chaotic dynamics (preprint, 1992).

  • E. B. Davies, Heat Kernels and Spectral Theory (Cambridge University Press, Cambridge, 1989).

    Book  MATH  Google Scholar 

  • E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, Proc. London Math. Soc. 57 (1988) 182–208.

    Article  MathSciNet  MATH  Google Scholar 

  • D. L. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly negative curvature, Ann. scient. Ec. Norm. Sup. 10 (1977) 133–152.

    MathSciNet  MATH  Google Scholar 

  • D. L. DeGeorge and N. R. Wallach, Limit formulas for multiplicities in L2(ΓG), Ann. Math. 107 (1978) 133–150.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Deitmar, The Selberg trace formula and the Ruelle zeta function for compact hyperbolics, Abh. Math. Sem. Univ. Hamburg 59 (1989) 101–106.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Dodziuk and B. Randol, Lower gounds for λ1 on a finite-volume hyperbolic manifold, J. Diff. Geom. 24 (1986) 133–139.

    MathSciNet  MATH  Google Scholar 

  • J. Dodziuk, T. Pignataro, B. Randol, and D. Sullivan, Estimating small eigenvalues of Riemann surfaces, Contem. Math. 64 (1987) 93–121.

    Article  MathSciNet  Google Scholar 

  • E. D’Hoker and D. Phong, Multiloop amplitudes for the bosonic Polyakov string, Nucl. Phys. B269 (1986) 205–234.

    Article  MathSciNet  Google Scholar 

  • H. Donnelly, On the analytic torsion and eta invariant for negatively curved manifolds, Am. J. Math. 101 (1979) 1365–1379.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Doron and U. Smilansky, Chaotic spectroscopy, Phys. Rev. Lett. 68 (1992) 1255–1258; and Chaos 2 (1992) 117-124.

    Article  Google Scholar 

  • E. Doron and U. Smilansky, Semiclassical quantization of chaotic billiards: a scattering theory approach (preprint, 1991).

  • J. Duistermaat, J. Kolk and V. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Inven. math. 52 (1979) 27–93.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Eberlein and B. O’Neill, Visibility manifolds, Pac. J. Math. 46 (1973) 45–109.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Eckhardt, Quantum mechanics of classically non-integrable systems, Phys. Rep. 163 (1988) 205–297.

    Article  MathSciNet  Google Scholar 

  • B. Eckhardt, Irregular scattering, Physica D33 (1988) 89–98.

    MathSciNet  Google Scholar 

  • B. Eckhardt, Periodic orbit theory, in Lecture notes for the International School of Physics ”Enrico Fermi” on Quantum Chaos (1991).

  • B. Eckhardt and E. Aurell, Convergence of the semi-classical periodic orbit expansion, Eu-rophys. Lett. 9 (1989) 509-512.

    Google Scholar 

  • B. Eckhardt, S. Fishman, K. Muller and D. Wintgen, Semiclassical matrix elements from periodic orbits (preprint, 1991).

  • B. Eckhardt and G. Russberg, Resummation of classical and semiclassical periodic orbit formulas (preprint).

  • B. Eckhardt and D. Wintgen, Symbolic description of periodic orbits for the quadratic Zeeman effect, J. Phys. B23 (1990) 355–363.

    Google Scholar 

  • B. Eckhardt and D. Wintgen, Indices in classical mechanics, J. Phys. A24 (1991) 4335–4348.

    MathSciNet  Google Scholar 

  • J. Elstrodt, Die resolvente zum eigenwertproblem der automorphen formen in der hyperbolischen ebene, Math. Ann. 203 (1973) 295–330; Math. Z. 132 (1973) 99-134; Math. Ann. 208 (1974) 99-132.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Elstrodt, Die Selbergsche spurformel fur kompakte Riemannsche Flachen, Jber. d. Dt. Math.-Verein. 83 (1981) 45–77.

    MathSciNet  MATH  Google Scholar 

  • J. Elstrodt, F. Grunewald and J. Mennike, Discontinuous groups on three-dimensional hyperbolic space: analytical theory and arithmetic applications, Russ. Math. Sur. 38:1 (1983) 137–168.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Elstrodt, F. Grunewald, and J. Mennicke, Arithmetic applications of the hyperbolic lattice point theorem, Proc. London Math. Soc. 57 (1988) 239–283.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Elstrodt, F. Grunewald and J. Mennicke, Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Inven. math. 101 (1990) 641–685.

    Article  MathSciNet  MATH  Google Scholar 

  • L. D. Fadeev, Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on the Lobacevskii plane, Trans. Moscow Math. Soc. 17 (1967) 357–396.

    Google Scholar 

  • J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. f. Reine ang. Math. 293/294 (1977) 143–203.

    MathSciNet  Google Scholar 

  • J. Fay, Analytic torsion and Prym differentials, in Riemann Surfaces and Related Topics (Princeton University Press, Princeton, 1980) 107–122.

    Google Scholar 

  • J. Fay, Perturbation of analytic torsion on Riemann surfaces, AMS Mem. (to appear).

  • D. Fried, The zeta functions of Ruelle and Selberg I, Ann. scient. Ec. Norm. Sup. 19(1986) 491–517.

    MathSciNet  MATH  Google Scholar 

  • D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Inv. math. 84 (1986) 523–540.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Fried, Fuchsian groups and Reidemeister torsion, Contemp. Math. 53 (1986) 141–163.

    Article  MathSciNet  Google Scholar 

  • D. Fried, Lefschetz formulas for flows, Contemp. Math. 58 (1987) 19–69.

    Article  MathSciNet  Google Scholar 

  • D. Fried, Torsion and closed geodesics on complex hyperbolic manifolds, Inv. math. 91 (1988) 31–51.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Gangolli, Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968) 151–192.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Gangolli, The length spectra of some compact manifolds of negative curvature, J. Diff. Geom. 12 (1977) 403–424.

    MathSciNet  MATH  Google Scholar 

  • R. Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Ill. J. Math. 21 (1977) 1–41.

    MathSciNet  MATH  Google Scholar 

  • R. Gangolli and G. Warner, On Selberg’s trace formula, J. Math. Soc. Japan 27 (1975) 328–343.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Gangolli and G. Warner, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980) 1–44.

    MathSciNet  Google Scholar 

  • P. Gaspard, Scattering and resonances: classical and quantum dynamics, in Proc. of Inter. School of Physics ”Enrico Fermi” Course CXIX-Quantum Chaos (1991).

  • P. Gaspard, Dynamical chaos and many-body quantum systems (preprint).

  • P. Gaspard and S. A. Rice, Scattering from a classically chaotic repellor, J. Chem. Phys. 90 (1989) 2225–2241.

    Article  MathSciNet  Google Scholar 

  • P. Gaspard and S. A. Rice, Semiclassical quantization of the scattering from a classically chaotic repellor, J. Chem. Phys. 90 (1989) 2242–2254.

    Article  MathSciNet  Google Scholar 

  • P. Gaspard and S. A. Rice, Exact quantization of the scattering from a classically chaotic repellor, J. Chem. Phys. 90 (1989) 2255–2262.

    Article  MathSciNet  Google Scholar 

  • S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ec. Norm. Super. 11 (1978) 471–542.

    MathSciNet  MATH  Google Scholar 

  • I. M. Gel’fand, Automorphic functions and the theory of representations, Proc. Int. Cong. Math. Stockholm, 1962.

    Google Scholar 

  • L. Greenberg, Fundamental polygons for Fuchsian groups, J. Analyse Math. 18 (1967) 99–105.

    Article  MathSciNet  MATH  Google Scholar 

  • M. C. Gutzwilier, Phase-integral approximations in momentum space and the bound states of an atom, J. Math. Phys. 8 (1967) 1979-2000; see also, ibid. 10 (1969) 1004-1020; 11 (1970) 1791-1806; 14 (1973) 139; 18 (1977) 806.

  • M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12 (1971) 343–358.

    Article  Google Scholar 

  • M. C. Gutzwiller, Classical quantization of a Hamiltonian with ergodic behavior, Phys. Rev. Lett. 45 (1980) 150–153.

    Article  Google Scholar 

  • M. C. Gutzwiller, Stochastic behavior in quantum scattering, Physica 7D (1983) 341–355.

    MathSciNet  Google Scholar 

  • M. C. Gutzwiller, The geometry of quantum chaos, Physica Scripta T9 (1985) 184–192.

    Article  MathSciNet  MATH  Google Scholar 

  • M. C. Gutzwiller, Physics and Selberg’s trace formula, Cont. Math. 53 (1986) 215–251.

    Article  MathSciNet  Google Scholar 

  • M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).

    MATH  Google Scholar 

  • D. Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J. 43 (1976) 441–482.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. I, LNM 548; Vol. II, LNM 1001 (Springer, 1976, 1983).

  • D. A. Hejhal, Eigenvalues of the Laplacian for PSL(2,Z): some new results and computational techniques, in International Symposium in Memory of Hua Loo-Keng”, Vol. 1 (Springer, 1991) 59-102.

  • D. A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. AMS 97 (1992).

  • S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).

    MATH  Google Scholar 

  • H. Huber, Uber eine neue klasse automorpher funktionen und ein gitterpunktproblem in der hyperbolischen ebene, Comm. Math. Helv. 30 (1956) 20–62.

    Article  Google Scholar 

  • H. Huber, Zur analytischen theorie hyperbolisher raumformen und bewegungsgruppen, Math. Ann. (1959) 1–26; 142 (1961) 385-398; 143 (1961) 463-464.

    Google Scholar 

  • M. Huxley, Introduction to Kloostermania, in Banach Center Publ. 17,(ed. H. Iwaniec, Warsaw, 1985).

  • C. Jung, Poincare map for scattering states, J. Phys. A19 (1986) 1345–1353.

    Google Scholar 

  • J. Keating, The Riemann zeta-function and quantum chaology, in Proc.’ Enrico Fermi’ Summer School CXIX on Quantum Chaos (ed. G. Casati et al., 1991).

  • J. Keating, The semiclassical sum rule and Riemann’s zeta function, in Quantum Chaos (ed. H. Cerdeira et al., World Scientific, Singapore, 1991).

  • J. P. Keating, Periodic orbit resummation and the quantization of chaos, Proc. Roy. Soc. London A436 (1992) 99–108.

    MathSciNet  Google Scholar 

  • J. Keating, The semiclassical functional equation, Chaos 2 (1992) 15–17.

    Article  MathSciNet  MATH  Google Scholar 

  • J. P. Keating and M. V. Berry, False singularities in partial sums over closed orbits, J. Phys. A20 (1987) L1139–L1141.

    MathSciNet  Google Scholar 

  • J. A. C. Kolk, The Selberg trace formula and asymptotic behavior of spectra (thesis, Rijksuniversiteit Utrecht, 1977).

  • P. D. Lax and R. S. Phillips, Scattering Theory for Automorphic Functions, Ann. Math. Studies No. 87 (Princeton University Press, Princeton, 1976).

    Google Scholar 

  • P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in euclidean and non-euclidean spaces, J. Fnct. Anal. 46 (1982) 280–350.

    Article  MathSciNet  MATH  Google Scholar 

  • R. G. Littlejohn, Semiclassical structure of trace formulas (preprint).

  • N. Mandouvalos, The theory of Eisenstein series for Kleinian groups, Cont. Math. 53 (1986) 357–370.

    Article  MathSciNet  Google Scholar 

  • N. Mandouvalos, Scattering operator, Eisenstein series, inner product formula and Maass-Selberg relations for Kleinian groups, Mem. AMS 78 (1989) No. 400.

  • N. Mandouvalos, Spectral theory and Eisenstein series for Kleinian groups, Proc. London Math. Soc. 57 (1988) 209–238.

    Article  MathSciNet  MATH  Google Scholar 

  • G. A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, J. Fnct. Anal. Appl. 3 (1969) 335–336.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Matthies and F. Steiner, Selberg’s zeta function and the quantization of chaos, Phys. Rev. A44 (1991) R7877–R7880.

    MathSciNet  Google Scholar 

  • D. H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z), Bull. AMS 25 (1991) 55–60.

    Article  MATH  Google Scholar 

  • H. P. McKean, An upper bound for the spectrum of Δ on a manifold of negative curvature, J. Diff. Geom. 4 (1970) 359–366.

    MathSciNet  MATH  Google Scholar 

  • H. P. McKean, Selberg’s trace formula as applied to a compact Riemann surface, Comm. Pure and Appl. Math. 25 (1972) 225–246.

    Article  MathSciNet  Google Scholar 

  • R. Melrose and R. Mazzeo, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987) 260–310.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Miatello, The Minakshisundaram-Pleijel coefficients for the vector valued heat kernel on compact locally symmetric spaces of negative curvature, Trans. Amer. Math. Soc. 260 (1980) 1–33.

    MathSciNet  MATH  Google Scholar 

  • W. H. Miller, Classical-limit Green’s function (fixed energy propagator) and classical quantization of nonseparable systems, J. Chem. Phys. 56 (1972) 38–45.

    Article  Google Scholar 

  • J. J. Millson, Closed geodesics and the η-invariant, Ann. Math. 108 (1978) 1–39.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Milnor, Infinite cyclic coverings, Conf. on the Topology of Manifolds (Prindle, Weber and Schmidt, Boston, 1968) 115–133.

    Google Scholar 

  • H. Moscovici and R. J. Stanton, Eta invariants of Dirac operators on locally symmetric manifolds, Invent, math. 95 (1989) 629–666.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Moscovici and R. J. Stanton, R-torsion and zeta functions for locally symmetric manifolds, Invent, math. 105 (1991) 185–216.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Fnct. Anal. 80 (1988) 148–211.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions,Isr. J. Math. 45 (1983) 41–52.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. Math. 118 (1983) 573–591.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Parry and M. Pollicott, The Chebotarov theorem for Galois coverings of Axiom A flows, Ergod. Th. Dyn. Sys. 6 (1986) 133–148.

    MathSciNet  MATH  Google Scholar 

  • S. J. Patterson, A lattice-point problem in hyperbolic space, Mathematica 22 (1975) 81–88.

    MathSciNet  MATH  Google Scholar 

  • S. J. Patterson, The limit set of a Fuchsian group, Acta math. 136 (1976) 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  • S. J. Patterson, The Laplacian operator on a Riemann surface, Comp. Math. 31 (1975) 83–107; 32 (1976) 71-112; 33 (1976) 227-259.

    MathSciNet  MATH  Google Scholar 

  • S. J. Patterson, Spectral theory and Fuchsian groups, Math. Proc. Camb. Phil. Soc. 81 (1977) 59–75.

    Article  MathSciNet  MATH  Google Scholar 

  • S. J. Patterson, The Selberg zeta function of a Kleinian group, in Number Theory, Trace Formulas and Discrete Groups (Academic Press, New York, 1987) 409–442.

    Google Scholar 

  • P. A. Perry, The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory, J. Func. Anal. 75 (1987) 161–187.

    Article  MathSciNet  MATH  Google Scholar 

  • P. A. Perry, The Laplace operator on a hyperbolic manifold II. Eisenstein series and the scattering matrix, J. reine angew. Math. 398 (1989) 67–91.

    MathSciNet  MATH  Google Scholar 

  • P. A. Perry, The Selberg zeta function and a local trace formula for Kleinian groups, J. reine ang. Math. 410 (1990) 116–152.

    MathSciNet  MATH  Google Scholar 

  • R. S. Phillips and P. Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta math. 155 (1985) 173–241.

    Article  MathSciNet  MATH  Google Scholar 

  • R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure and App. Math. 38 (1985) 853–866.

    Article  MathSciNet  MATH  Google Scholar 

  • R. S. Phillips and P. Sarnak, On cusp forms for cofinite subgroups of PSL(2, R), Inven. Math. 80 (1985) 339–364.

    Article  MathSciNet  MATH  Google Scholar 

  • R. S. Phillips and P. Sarnak, On the spectrum of the Hecke groups, Duke Math. J. 52 (1985) 211–221.

    Article  MathSciNet  MATH  Google Scholar 

  • R. S. Phillips and P. Sarnak, The spectrum of Fermat curves, Geom. Func. Anal. 1 (1991) 80–146.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Pignataro, Hausdorff dimension, spectral theory and applications to the quantization of geodesic flows on surfaces of constant negative curvature, (thesis, Princeton University, 1984).

  • T. Pignataro and D. Sullivan, Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces, Comm. Math. Phys. 104 (1986) 529–535.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Pollicott, Mermorphic extensions of generalized zeta functions, Invent. math. 85 (1986) 147–164.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Pollicott, Some applications of thermodynamic formalism to manifolds with negative curvature, Adv. Math. 85 (1991) 161–192.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. 103B (1981) 207–210.

    MathSciNet  Google Scholar 

  • A. Preismann, Quelques proprietes globales des espaces de Riemann, Comm. Math. Helv. 15 (1943) 175–216.

    Article  Google Scholar 

  • B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. AMS 80 (1974) 996–1000.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Randol, On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces, Trans. Amer. Math. Soc. 201 (1975) 241–246.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. AMS 233 (1977) 241–247.

    Article  MathSciNet  Google Scholar 

  • B. Randol, The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator, Trans. Amer. Math. Soc. 236 (1978) 209–223.

    MathSciNet  MATH  Google Scholar 

  • D. Ray and I. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 (1973) 154–177.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145–210.

    Article  MathSciNet  MATH  Google Scholar 

  • S. A. Rice, P. Gaspard, and K. Nakamura, Signatures of chaos in quantum dynamics and the controllability of evolution in a quantum system, in Advances in Classical Trajectories, Vol. 1 (JAI Press, 1991) 215-313.

  • K. Richter and D. Wintgen, Calculations of planetary atom states, J. Phys. B24 (1991) L565–L571.

    Google Scholar 

  • J. M. Robbins, Discrete symmetries in periodic-orbit theory, Phys. Rev. A40 (1989) 2128–2136.

    MathSciNet  Google Scholar 

  • J. M. Robbins, Maslov indices in the Gutzwiller trace formula, Nonlinearity 4 (1991) 343–363.

    Article  MathSciNet  MATH  Google Scholar 

  • J. M. Robbins, Winding number formula for Maslov indices, Chaos 2 (1992) 145–147.

    Article  MathSciNet  MATH  Google Scholar 

  • J. M. Robbins, S.C. Creagh and R. G. Littlejohn, Complex periodic orbits in the rotational spectrum of molecules: the example of SF6, Phys. Rev. A39 1989) 2838–2854.

    Google Scholar 

  • J. M. Robbins and M. V. Berry, The geometric phase for chaotic systems (preprint).

  • W. Roelcke, Das eigenwertproblem der automorphen formen in der hyperbolischen ebene, Math. Ann. 167-168 (1966/67) 292-337; 168 (1967) 261–324.

    Article  MathSciNet  Google Scholar 

  • D. Ruelle, Zeta functions and statistical mechanics, Asterisque 40 (1976) 167–176.

    MathSciNet  Google Scholar 

  • D. Ruelle, Zeta functions for expanding maps and Anosov flows, Inv. math. 34 (1976) 231–242.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Ruelle, Statistical Mechanics, Thermodynamic Formalism (Addison-Wesley, Reading, 1978).

    Google Scholar 

  • D. Ruelle, Locating resonances for axiom A dynamical systems, J. Stat. Phys. 44 (1986) 281–292.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Sarnak, Class numbers of indefinite binary quadratic forms I, II, J. Numb. Thy. 15 (1982) 229–247; 21 (1985) 333-346.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Sarnak, The arithmetic and geometry of some hyperbolic three manifolds, Acta Math. 151 (1983) 253–295.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987) 113–120.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Sarnak, On cusp forms, Contemp. Math. 53 (1986) 393–407.

    Article  MathSciNet  Google Scholar 

  • P. Sarnak, Special values of Selberg’s zeta function, Number Theory, Trace Formulas and Discrete Groups (Academic Press, New York, 1989) 457–465.

    Google Scholar 

  • P. Sarnak, Determinants of Laplacians; heights and finiteness (preprint, 1991).

  • R. Schoen, S. Wolpert and S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface, AMS Proc. Symp. Pure Math. 36 (1980) 279–285.

    Article  MathSciNet  Google Scholar 

  • A. Selberg, Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with application to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47–87.

    MathSciNet  MATH  Google Scholar 

  • A. Selberg, On the estimation of Fourier coefficients of modular forms, in Proc. Sympos. Pure Math. 8: Theory of numbers, pp 1-15 (Amer. Math. Soc, Providence, 1965).

  • M. Sieber, The hyperbola billiard: a model for the semiclassical quantization of chaotic systems (preprint, 1991).

  • M. Sieber, Applications of periodic-orbit theory, Chaos 2 (1992) 35–38.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Sieber and F. Steiner, Classical and quantum mechanics of a strongly chaotic billiard system, Physica D44 (1990) 248–266.

    MathSciNet  Google Scholar 

  • M. Sieber and F. Steiner, Quantum chaos in the hyperbola billiard, Phys. Lett. A148 (1990) 415–420.

    MathSciNet  Google Scholar 

  • M. Sieber and F. Steiner, Generalized periodic-orbit sum rules for strongly chaotic systems, Physics Lett. A144 (1990) 159–163

    Article  MathSciNet  Google Scholar 

  • M. Sieber and F. Steiner, On the quantization of chaos, Phys. Rev. Lett. 67 (1991) 1941–1944.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Phys. 146 (1983) 209–220.

    Article  MATH  Google Scholar 

  • B. Simon, Nonclassical eigenvalue asymptotics, J. Func. Anal. 53 (1983) 84–98.

    Article  MATH  Google Scholar 

  • Y. Sinai, The asymptotic behavior of the number of closed geodesics on a compact manifold of negative curvature, Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966) 1275–1296.

    MathSciNet  MATH  Google Scholar 

  • F. Steiner, Spectral sum rules for the circular Aharonov-Bohm quantum billiard, Fortschr. Phys 35 (1987) 87.

    Article  Google Scholar 

  • F. Steiner, On Selberg’s zeta function for compact Riemann surfaces, Phys. Lett B188 (1987) 447.

    Google Scholar 

  • F. Steiner and P. Trillenberg, Refined asymptotic expansion of the heat kernel for quantum billiards in unbounded regions, J. Math. Phys. 31 (1990) 1670.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, IHES Publ. Math. 50 (1979) 171–209.

    Article  MATH  Google Scholar 

  • D. Sullivan, Entropy, Hausdorff measures old and new, and the limit sets of geometrically finite Kleinian groups, Acta math. 153 (1983) 259–277.

    Article  Google Scholar 

  • D. Sullivan, Related aspects of positivity: λ-potential theory on manifolds, lowest eigen-states, Hausdorff geometry, renormalized Markoff processes…, (preprint, IHES, 1983).

  • G. Tanner, P. Scherer, E. B. Bogomonly, B. Eckhardt and D. Wintgen, Quantum eigenvalues from classical periodic orbits, Phys. Rev. Lett. 67 (1991) 2410–2413.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Tanner and D. Wintgen, Quantization of chaotic systems, Chaos 2 (1992) 53–59.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Terras, Harmonic Analysis on Symmetric Spaces and Applications (Springer-Verlag, New York, 1988).

    Book  MATH  Google Scholar 

  • P. Trombi and V. S. Varadarajan, Spherical transforms on semisimple Lie groups, Ann. Math. 94 (1971) 246–303.

    Article  MathSciNet  MATH  Google Scholar 

  • V. S. Varadarajan, The eigenvalue problem on negatively curved compact locally symmetric manifolds, Con. Math. 53 (1986) 449–462.

    Article  MathSciNet  Google Scholar 

  • A. B. Venkov, Expansion in automorphic eigenfunctions of the Laplace-Beltrami operator in classical symmetric spaces of rank one, and the Selberg trace formula, Proc. Steklov Inst. Math. 125 (1973).

  • A. B. Venkov, 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, Z), Russ. Math. Sur. 34 (1979) 79–153; v. Math. USSR Izv. 12 (1978) 448-462.A. B. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math. 4 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  • A. B. Venkov, V. L. Kalinin and L. D. Faddeev, A non arithmetic derivation of the Selberg trace formula, J. Sov. Math. 8 (1977) 171–199.

    Article  MATH  Google Scholar 

  • A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987) 439–465.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Voros, Unstable periodic orbits and semiclassical quantization, J. Phys. 21 (1988) 685.

    MathSciNet  MATH  Google Scholar 

  • M. Wakayama, Zeta functions of Selberg’s type for compact quotient of SU(n, 1)(n ≥ 2), Hiroshima Math. J. 14 (1984) 597–618.

    MathSciNet  Google Scholar 

  • M. Wakayama, Zeta functions of Selberg’s type associated with homogeneous vector bundles, Hiroshima Math. J. 15 (1985) 235–295.

    MathSciNet  MATH  Google Scholar 

  • M. Wakayama, A formula for the logarithmic derivative of Selberg’s zeta function, J. Math. Soc. Japan 41 (1989) 463–471.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Wakayama, A note on the Selberg zeta function for compact quotients of hyperbolic spaces, Hiroshima Math. J. 21 (1991) 539–555.

    MathSciNet  MATH  Google Scholar 

  • M. Wakayama, The relation between the η-invariant and the spin representation in terms of the Selberg zeta function, Adv. Studies in Pure Math. 21 (1991)1–16.

    MathSciNet  Google Scholar 

  • N. Wallach, An asymptotic formula of Gelfand and Gangolli for the spectrum of ΓG, J. Diff. Geom. 11 (1976) 91–101.

    MathSciNet  MATH  Google Scholar 

  • N. Wallach, On the Selberg trace formula in the case of compact quotient, Bull. Amer. Math. Soc. 82 (1976) 171–195.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Warner, Selberg’s trace formula for non-uniform lattices: the R-rank one case, Adv. in Math. Studies 6 (1979) 1–142.

    MathSciNet  Google Scholar 

  • R. L. Waterland et al. Classical-quantum correspondence in the presence of global chaos, Phys. Rev. Lett 61 (1988) 2733–2736.

    Article  MathSciNet  Google Scholar 

  • A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948) 204–207.

    Article  MathSciNet  MATH  Google Scholar 

  • H. C. Williams and J. Broere, A computational technique for evaluating L(1, χ) and the class number of a real quadratic field, Math. Comp. 30 (1976) 887–893.

    MathSciNet  MATH  Google Scholar 

  • A. M. Winkler, Cusp forms and Hecke groups, J.f. Reine angew. Math. 386 (1988) 187–204.

    MATH  Google Scholar 

  • D. Wintgen, Connection between long-range correlations in quantum spectra and classical periodic orbits, Phys. Rev. Lett. 58 (1987) 1589–1592.

    Article  Google Scholar 

  • D. Wintgen, Semiclassical path-integral quantization of nonintegrable Hamiltonian systems, Phys. Rev. Lett. 61 (1988) 1803–1806.

    Article  MathSciNet  Google Scholar 

  • D. Wintgen and A. Hönig, Irregular wave functions of a hydrogen atom in a uniform magnetic field, Phys. Rev. Let. 63 (1989) 1467.

    Article  Google Scholar 

  • D. Wintgen, K. Richter and G. Tanner, The semiclassical helium atom, Chaos 2 (1992) 19–33.

    Article  Google Scholar 

  • S. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. math. phys. 112 (1987) 283–315.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Wolpert, Spectral limits for hyperbolic surfaces (preprint).

  • D. Zagier, Eisenstein series and the Selberg trace formula, in Automorphic Forms, Representation Theory and Arithmetic (Springer-Verlag, Berlin, 1981) 305–355.

    Google Scholar 

  • S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987) 919–941.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Zelditch, Pseudo-differential operators, trace formulae and the geodesics integrals of automorphic forms, Duke Math. J. 56 (1988) 295–344.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Zelditch, Trace formulae for compact ΓPSL2(R) and the equidistribution theorems for closed geodesies and Laplace eigenfunctions, Duke Math. J. 59 (1989) 27–81.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Zelditch, Selberg trace formulae and equidistribution theorems for closed geodesies and Laplace eigenfunctions: finite area surfaces, Mem. AMS 465 (1992).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hurt, N.E. Zeta Functions and Periodic Orbit Theory: A Review. Results. Math. 23, 55–120 (1993). https://doi.org/10.1007/BF03323131

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03323131

Key Words

Navigation