Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds
Authors:
Dmitry Jakobson and Iosif Polterovich
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 71-77
MSC (2000):
Primary 58J50; Secondary 35P20, 37C30, 81Q50
DOI:
https://doi.org/10.1090/S1079-6762-05-00149-6
Published electronically:
September 23, 2005
MathSciNet review:
2176067
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Abstract: We announce asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in the local Weyl’s law on Riemannian manifolds. In the negatively curved case, methods of thermodynamic formalism are applied to improve the estimates. Our results develop and extend the unpublished thesis of A. Karnaukh. We discuss some ideas of the proofs; for complete proofs see our extended paper on the subject.
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ABS R. Aurich, J. Bolte, F. Steiner. Universal signatures of quantum chaos. Phys. Rev. Lett. 73 (1994), no. 10, 1356–1359.
Berard P. Berard. On the wave equation on a compact riemannian manifold without conjugate points. Math. Z. 155 (1977), 249–276.
Berry M. V. Berry. Semiclassical theory of spectral rigidity. Proc. R. Soc. Lond. A 400 (1985), 229–251.
Bogomolny E. Bogomolny. Smoothed wavefunctions of chaotic quantum systems. Physica D 31 (1988), 169–189.
BS E. Bogomolny, C. Schmit. Semiclassical computations of energy levels. Nonlinearity 6 (1993), 523–547.
Bowen R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470. Springer, 1975.
BR R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), no. 3, 181–202.
CdV Y. Colin de Verdière. Spectre conjoint d’opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable. Math. Z. 171 (1980), no. 1, 51–73.
Don H. Donnelly. On the wave equation asymptotics of a compact negatively curved surface. Invent. Math. 45 (1978), 115–137.
DG J. Duistermaat and V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Math. 29 (1975), 39–75.
Gelfand I. M. Gelfand and G. E. Shilov. Generalized functions, vol. 1, Academic Press, New York, 1964.
Hejhal D. Hejhal. The Selberg trace formula for $PSL(2,\textbf {R})$, Vol. I. Lecture Notes in Math. 548, Springer, 1976.
H L. Hörmander. The spectral function of an elliptic operator. Acta Math. 121 (1968), 193–218.
Ivrii V. Ivrii. Precise spectral asymptotics for elliptic operators. Lecture Notes in Math. 1100, Springer, 1984.
JP D. Jakobson, I. Polterovich. Estimates from below for the spectral function and for the remainder in local Weyl’s law, math.SP/0505400.
K A. Karnaukh. Spectral count on compact negatively curved surfaces. Ph.D. thesis under the supervision of P. Sarnak, Princeton University (1996), 1–48.
LiS X. Li and P. Sarnak. Number variance for $SL(2,\textbf {Z})/ \textbf {H}$. Preprint 2004.
LuS W. Luo and P. Sarnak Number variance for arithmetic hyperbolic surfaces. Comm. Math. Phys. 161 (1994), 419–432.
Par W. Parry. Equilibrium states and weighted uniform distribution of closed orbits. Dynamical Systems (College Park, MD 1986-87), Lecture Notes in Math. 1342, 617–625. Springer, 1988.
PP W. Parry and M. Pollicott. Zeta functions and closed orbit structure for hyperbolic systems. Astérisque, 187-188 (1990), 1–256.
TP Y. Petridis and J. Toth. The remainder in Weyl’s law for Heisenberg manifolds. J. Diff. Geom. 60 (2002), no. 3, 455–483.
PR R. Phillips and Z. Rudnick. The circle problem in the hyperbolic plane. J. Funct. Anal. 121 (1994), no. 1, 78–116.
Randol2 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.
Randol B. Randol. A Dirichlet series of eigenvalue type with applications to asymptotic estimates. Bull. London Math. Soc. 13 (1981), 309–315.
RS M. Rubinstein and P. Sarnak. Chebyshev’s bias. Experiment. Math. 3 (1994), no. 3, 173–197.
SV Y. Safarov and D. Vassiliev. The asymptotic distribution of eigenvalues of partial differential operators. Translations of Mathematical Monographs, 155. AMS, 1997.
Sinai2 Ya. Sinai. Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27 (1972), 4(166), 21–64. English transl., Russian Math. Surveys 27 (1972), 21–69.
Shubin M. Shubin. Pseudodifferential operators and spectral theory, Springer-Verlag, 1987.
SZ C. Sogge and S. Zelditch. Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114 (2002), no. 3, 387–437.
Tsang K.-M. Tsang. Counting lattice points in the sphere. Bull. London Math. Soc. 32 (2000), no. 6, 679–688.
Vol A. Volovoy. The Hamilton flow conditions associated with Weyl’s conjecture, Ann. Global Anal. Geom. 8, No. 2 (1990), 127–136.
Zelditch S. Zelditch, Lectures on wave invariants, in: Spectral theory and geometry, edited by B. Davies and Y. Safarov, LMS lecture note series 273, Cambridge University Press, 1999.
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Additional Information
Dmitry Jakobson
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A 2K6, Canada
Email:
jakobson@math.mcgill.ca
Iosif Polterovich
Affiliation:
Département de mathématiques et de statistique, Université de Montréal CP 6128 Succ. Centre-Ville, Montréal QC H3C 3J7, Canada
Email:
iossif@dms.umontreal.ca
Keywords:
Weyl’s law,
spectral function,
wave kernel,
negative curvature,
Anosov flow,
thermodynamic formalism
Received by editor(s):
June 7, 2005
Published electronically:
September 23, 2005
Additional Notes:
The first author was supported by NSERC, FQRNT, Alfred P. Sloan Foundation fellowship and Dawson fellowship. The second author was supported by NSERC and FQRNT
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
