Duke Mathematical Journal

Semiclassical limits for the hyperbolic plane

Scott A. Wolpert

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the concentration properties of high-energy eigenfunctions for the Laplace-Beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions. At the center of our investigation is the microlocal lift of an eigenfunction to SL(2;ℝ) introduced by S. Zelditch. The microlocal lift is based on S. Helgason's Fourier transform and has a straightforward description in terms of the Lie algebra sl(2;ℝ). We begin with an elementary demonstration of Zelditch's exact differential equation for the microlocal lift. Further, for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity, we show that the limit of microlocal lifts is a geodesic-flow-invariant positive measure.

Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the Macdonald-Bessel functions. We find that with a scaling of auxiliary parameters the corresponding high-energy limit converges to the positive Dirac measure for the lift to SL(2;ℝ) of a single geodesic on the upper half-plane. In particular, at high energy the microlocal lift of a Macdonald-Bessel function is concentrated along the lift of a single geodesic. We further find that the convergence is uniform in parameters, and thus our considerations can be applied to study automorphic eigenfunctions, in particular, the Maass cusp forms for certain cofinite subgroups of SL(2;ℝ). A formula is presented for the microlocal lift of a cusp form as an integral of a family of Dirac measures and a measure given directly by sums of products of the classical Fourier coefficients of the cusp form. The formula establishes that questions on "quantum chaos" for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients. In particular, we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums. We extend the considerations to the case of the nonholomorphic Eisenstein series for the modular group. We find that a result of W. Luo and P. Sarnak is equivalent to a limit-sum formula involving the Riemann zeta-function and the elementary divisor function. The formula is suggestive of a formula of S. Ramanujan.

Article information

Duke Math. J., Volume 108, Number 3 (2001), 449-509.

First available in Project Euclid: 5 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11F30: Fourier coefficients of automorphic forms 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]


Wolpert, Scott A. Semiclassical limits for the hyperbolic plane. Duke Math. J. 108 (2001), no. 3, 449--509. doi:10.1215/S0012-7094-01-10833-8. https://projecteuclid.org/euclid.dmj/1091737181

Export citation


  • \lccA. Alvarez-Parrilla, Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series, Trans. Amer. Math. Soc. 352 (2000), 5563–5582. MR CMP 1 675 233
  • \lccR. Aurich, E. B. Bogomolny, and F. Steiner, Periodic orbits on the regular hyperbolic octagon, Phys. D 48 (1991), 91–101. MR 92b:58173
  • \lccR. Aurich and F. Steiner, From classical periodic orbits to the quantization of chaos, Proc. Roy. Soc. London Ser. A 437 (1992), 693–714. MR 93k:81056
  • ––––, Statistical properties of highly excited quantum eigenstates of a strongly chaotic system, Phys. D 64 (1993), 185–214. MR 94e:81058
  • \lccN. L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rep. 143 (1986), 109–240. MR 88h:58070
  • \lccM. V. Berry, Quantum scars of classical closed orbits in phase space, Proc. Roy. Soc. London Ser. A 423 (1989), 219–231. MR 92c:81075
  • \lccA. Borel, Automorphic Forms on $\SL\sb 2(\textbf{R})$, Cambridge Tracts in Math. 130, Cambridge Univ. Press, Cambridge, 1997. MR 98j:11028
  • \lccR. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), 1–18. MR 57:12394
  • \lccY. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), 497–502. MR 87d:58145
  • \lccJ.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), 219–288. MR 84m:10015
  • ––––, “The nonvanishing of Rankin-Selberg zeta-functions at special points” in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984), Contemp. Math. 53, Amer. Math. Soc., Providence, 1986, 51–95. MR 88d:11047
  • \lccA. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. I (based, in part, on notes left by Harry Bateman), McGraw-Hill, New York, 1954. MR 15:868a
  • ––––, Tables of Integral Transforms, Vol. II (based, in part, on notes left by Harry Bateman), McGraw-Hill, New York, 1954. MR 16:468c
  • \lccA. Good, On various means involving the Fourier coefficients of cusp forms, Math. Z. 183 (1983), 95–129. MR 84g:10052
  • \lccI. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., Academic Press, Boston, 1994. MR 94g:00008
  • \lccG. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, New York, 1979. MR 81i:10002
  • \lccD. A. Hejhal, The Selberg Trace Formula for $\PSL(2, \textbf{R})$, Vol. 2, Lecture Notes in Math. 1001, Springer, Berlin, 1983. MR 86e:11040
  • ––––, “Eigenfunctions of the Laplacian, quantum chaos, and computation” in Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, France, 1995), École Polytech., Palaiseau, 1995, exp. no. 7. MR CMP 1 360 476
  • ––––, “On eigenfunctions of the Laplacian for Hecke triangle groups” in Emerging Applications of Number Theory (Minneapolis, 1996), IMA Vol. Math. Appl. 109, Springer, New York, 1999, 291–315. MR 2000f:11063
  • \lccD. A. Hejhal and B. N. Rackner, On the topography of Maass waveforms for $\PSL(2,\mathbf{Z})$, Experiment. Math. 1 (1992), 275–305. MR 95f:11037
  • \lccS. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progr. Math. 13, Birkhäuser, Boston, 1981. MR 83g:43009
  • ––––, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr. 39, Amer. Math. Soc., Providence, 1994. MR 96h:43009
  • \lccJ. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), 161–181. MR 95m:11048
  • \lccA. E. Ingham, Some asymptotic formulae in the theory of numbers, J. London Math. Soc. 2 (1927), 202–208.
  • \lccH. Iwaniec, Small eigenvalues of Laplacian for $\Gamma \sb 0(N)$, Acta Arith. 56 (1990), 65–82. MR 92h:11045
  • ––––, The spectral growth of automorphic $L$-functions, J. Reine Angew. Math. 428 (1992), 139–159. MR 93g:11049
  • ––––, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoam., Madrid, 1995. MR 96f:11078
  • \lccD. Jakobson, Equidistribution of cusp forms on $\PSL \sb 2(\mathbf{Z})\backslash \PSL \sb 2(\mathbf{R})$, Ann. Inst. Fourier (Grenoble) 47 (1997), 967–984. MR 99c:11063
  • \lccY. Katznelson, An Introduction to Harmonic Analysis, 2d ed., Dover, New York, 1976. MR 54:10976
  • \lccN. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture: Sums of Kloosterman sums (in Russian), Mat. Sb. (N.S.) 111 (153), no. 3 (1980), 334–383, 479. MR 81m:10053
  • \lccS. Lang, $\SL \sb 2(\textbf{R})$, Grad. Texts in Math. 105, Springer, New York, 1985. MR 86j:22018
  • \lccN. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972. MR 50:2568
  • \lccW. Z. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on $\PSL \sb 2(\mathbf{Z})\backslash \mathbf{H}\sp 2$, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207–237. MR 97f:11037
  • \lccH. M. Macdonald, Zeroes of the Bessel functions, Proc. London Math. Soc. 30 (1899), 165–179.
  • \lccT. Meurman, “On the order of the Maass $L$-function on the critical line” in Number Theory (Budapest, 1987), Vol. I, Colloq. Math. Soc. János Bolyai 51, North- Holland, Amsterdam, 1990, 325–354. MR 91m:11037
  • \lccJ.-P. Otal, Sur les fonctions propres du laplacien du disque hyperbolique, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 161–166. MR 99e:35161
  • \lccS. Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Math. 45 (1915), 81–84.
  • \lccZ. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195–213. MR 95m:11052
  • \lccP. Sarnak, “Arithmetic quantum chaos” in The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, 1995, 183–236. MR 96d:11059
  • \lccC. Schmit, “Quantum and classical properties of some billiards on the hyperbolic plane” in Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, 1991, 331–370. MR 94m:58176
  • \lccA. I. Šnirelman, Ergodic properties of eigenfunctions (in Russian), Uspekhi. Mat. Nauk 29, no. 6 (1974), 181–182. MR 53:6648
  • \lccA. Selberg, “On the estimation of Fourier coefficients of modular forms” in Theory of Numbers (Pasadena, Calif., 1963), Proc. Sympos. Pure Math. 8, Amer. Math. Soc., Providence, 1965, 1–15. MR 32:93
  • \lccA. Terras, Harmonic Analysis on Symmetric Spaces and Applications, I, Springer, New York, 1985. MR 87f:22010
  • \lccE. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2d ed., Oxford Univ. Press, New York, 1986. MR 88c:11049
  • \lccF. G. Tricomi, Integral Equations, Dover, New York, 1985. MR 86k:45001
  • \lccA. B. Venkov, Spectral Theory of Automorphic Functions and Its Applications, Math. Appl. (Soviet Ser.) 51, Kluwer, Dordrecht, 1990. MR 93a:11046
  • \lccD. V. Widder, The Laplace Transform, Princeton Math. Ser. 6, Princeton Univ. Press, Princeton, 1941. MR 3:232d
  • \lccS. A. Wolpert, Asymptotic relations among Fourier coefficients of automorphic eigenfunctions, to appear in Trans. Amer. Math. Soc.
  • \lccS. Zelditch, Pseudodifferential analysis on hyperbolic surfaces, J. Funct. Anal. 68 (1986), 72–105. MR 87j:58092
  • ––––, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919–941. MR 89d:58129
  • ––––, The averaging method and ergodic theory for pseudo-differential operators on compact hyperbolic surfaces, J. Funct. Anal. 82 (1989), 38–68. MR 91e:58194
  • ––––, Trace formula for compact $\Gamma \backslash \PSL \sb 2(\mathbb{R})$ and the equidistribution theory of closed geodesics, Duke Math. J. 59 (1989), 27–81. MR 91d:11056
  • ––––, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, J. Funct. Anal. 97 (1991), 1–49. MR 92h:11046
  • ––––, On the rate of quantum ergodicity, I: Upper bounds, Comm. Math. Phys. 160 (1994), 81–92. MR 95f:58084