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II - Selberg's Trace Formula: An Introduction

Published online by Cambridge University Press:  05 January 2012

By
Jens Marklof
Edited by
Jens Bolte  and
Frank Steiner
Jens Marklof
Affiliation:
University of Bristol
Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
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Summary

The aim of this short lecture course is to develop Selberg's trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator –Δ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tangent bundle of M. The trace formula is currently the only available tool to analyze the fine structure of the spectrum of –Δ; no individual formulas for its eigenvalues are known. In the case of more general quantum systems, the role of Selberg's formula is taken over by the semiclassical Gutzwiller trace formula [11], [7].

We begin by reviewing the trace formulas for the simplest compact manifolds, the circle S1 (Section 1) and the sphere S2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a consequence of the Poisson summation formula. In the remaining sections we shall discuss the following topics: the Laplacian on the hyperbolic plane and isometries (Section 3); Green's functions (Section 4); Selberg's point pair invariants (Section 5); The ghost of the sphere (Section 6); Linear operators on hyperbolic surfaces (Section 7); A trace formula for hyperbolic cylinders and poles of the scattering matrix (Section 8); Back to general hyperbolic surfaces (Section 9); The spectrum of a compact surface, Selberg's pre-trace and trace formulas (Section 10); Heat kernel and Weyl's law (Section 11); Density of closed geodesics (Section 12); Trace of the resolvent (Section 13); Selberg's zeta function (Section 14); Suggestions for exercises and further reading (Section 15).

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Chapter
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Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology , pp. 83 - 120
Publisher: Cambridge University Press
Print publication year: 2011

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References

1. N.L., Balazs and A., Voros, Chaos on the pseudosphere, Phys. Rep. 143 (1986) 109–240.Google Scholar
2. M.V., Berry and C.J., Howls, High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon, Proc. Roy. Soc. London Ser. A 447 (1994) 527–555.Google Scholar
3. M.V., Berry and J.P., Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999) 236–266.Google Scholar
4. P., Buser, Geometry and spectra of compact Riemann surfaces, Progr. Math. 106, Birkhäuser Boston, Inc., Boston, MA, 1992.Google Scholar
5. A., Aigon-Dupuy, P., Buser and K.-D., Semmler, Hyperbolic geometry, this volume.
6. P., Cartier and A., Voros, Une nouvelle interprétation de la formule des traces de Selberg, The Grothendieck Festschrift, Vol. II, 1–67, Progr. Math. 87, Birkhäuser Boston, Boston, MA, 1990.Google Scholar
7. M., Combescure, J., Ralston and D., Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Comm. Math. Phys. 202 (1999) 463–480.Google Scholar
8. A., Comtet, On the Landau levels on the hyperbolic plane, Ann. Physics 173 (1987) 185–209.Google Scholar
9. J., Elstrodt, F., Grunewald and J., Mennicke, Groups acting on hyperbolic space, Springer-Verlag, Berlin, 1998.Google Scholar
10. I.M., Gelfand, M.I., Graev and I.I., Pyatetskii-Shapiro, Representation theory and automorphic functions, Academic Press, Inc., Boston, MA, 1990 (Reprint of the 1969 edition).
11. M.C., Gutzwiller, The semi-classical quantization of chaotic Hamiltonian systems. Chaos et physique quantique (Les Houches, 1989), 201–250, North-Holland, Amsterdam, 1991.Google Scholar
12. D.A., Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J. 43 (1976) 441–482.Google Scholar
13. D.A., Hejhal, The Selberg trace formula for PSL(2,ℝ), Vol. 1. Lecture Notes in Mathematics 548, Springer-Verlag, Berlin-New York, 1976.Google Scholar
14. D.A., Hejhal, The Selberg trace formula for PSL(2,ℝ), Vol. 2. Lecture Notes in Mathematics 1001, Springer-Verlag, Berlin-New York, 1983.Google Scholar
15. H., Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics 53, AMS, Providence, RI; Rev. Mat. Iberoamericana, Madrid, 2002.Google Scholar
16. A., Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47–87.Google Scholar
17. A., Terras, Harmonic analysis on symmetric spaces and applications I, Springer-Verlag, New York, 1985.Google Scholar
18. D.V., Widder, The Laplace Transform, Princeton Math. Series 6, Princeton University Press, Princeton, 1941.Google Scholar

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