# American Institute of Mathematical Sciences

January  2016, 15(1): 9-39. doi: 10.3934/cpaa.2016.15.9

## Average error for spectral asymptotics on surfaces

 1 Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States

Received  June 2015 Revised  October 2015 Published  December 2015

Let $N(t)$ denote the eigenvalue counting function of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\widetilde N(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t)=\frac 1 t \int^t_0 D(s)\,ds$ for $D(t)=N(t)-\widetilde N(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.
Citation: Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure & Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9
##### References:
 [1] M. van den Berg and S. Srisatkunarajah, Heat flow and Brownian motion for a region in $\mathbb R^2$ with a polygonal boundary,, \emph{Probab. Theory Related Fields}, 86 (1990), 41. doi: 10.1007/BF01207512. Google Scholar [2] P. Bleher, Distribution of energy levels of a quantum free particle on a surface of revolution,, \emph{Duke Math. J.}, 74 (1994), 45. doi: 10.1215/S0012-7094-94-07403-6. Google Scholar [3] P. Buser, Geometry and Spectra of Compact Riemann Surfaces,, Birkhauser Boston, (1992). Google Scholar [4] P. B. Gilkey, Asymptotic Formulae in Spectral Geometry,, Chapman & Hall /CRC, (2004). Google Scholar [5] J. Fox, E. Greif, D. Kaplan and R. Strichartz, Spectrum of the Laplacian on Regular Polyhedral Surfaces,, in preparation., (). Google Scholar [6] V. Ivrii, Precise Spectral Asymptotics for Elliptic Operators,, Lecture Notes in Math \textbf{1100} (1984), 1100 (1984). Google Scholar [7] S. Jayakar and R. Strichartz, Average number of lattice points in a disk,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 1. Google Scholar [8] M. Kac, Can one hear the shape of a drum,, \emph{Amer. Math. Monthly}, 783 (1966), 1. Google Scholar [9] D. V. Kosygin, A. A. Minasov and Ya. G. Sinai, Statistical properties of the spectra of Laplace Beltrami operators on Liouville surfaces,, \emph{Russian Math. Surveys}, 48 (1993). doi: 10.1070/RM1993v048n04ABEH001052. Google Scholar [10] H. Lapointe, I. Potterovich and Yu. Safarov, Average growth of the spectral function on a Riemannian manifold,, \emph{Comm. P. D. E.}, 34 (2009), 581. doi: 10.1080/03605300802537453. Google Scholar [11] T. Murray and R. Strichartz, Numerical investigations of spectral asymptotics on surfaces,, in preparation., (). Google Scholar [12] P. Sarnak, Spectra of hyperbolic surfaces,, \emph{Bull. Amer. Math. Soc.}, 40 (2003), 441. doi: 10.1090/S0273-0979-03-00991-1. Google Scholar [13] C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (2014). doi: 10.1515/9781400850549. Google Scholar [14] R. Strichartz, Spectral asymptotics revisited,, \emph{J. Fourier Anal. Appl.}, 18 (2012), 626. doi: 10.1007/s00041-012-9216-7. Google Scholar [15] Yu. G. Safarov, Riesz means of the distribution function of the eigenvalues of an elliptic operator,, \emph{J. Sov. Math.}, 49 (1990), 1210. doi: 10.1007/BF02208718. Google Scholar [16] R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés,, \emph{Bull. Math. Soc. Fr.}, 91 (1963), 289. Google Scholar

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##### References:
 [1] M. van den Berg and S. Srisatkunarajah, Heat flow and Brownian motion for a region in $\mathbb R^2$ with a polygonal boundary,, \emph{Probab. Theory Related Fields}, 86 (1990), 41. doi: 10.1007/BF01207512. Google Scholar [2] P. Bleher, Distribution of energy levels of a quantum free particle on a surface of revolution,, \emph{Duke Math. J.}, 74 (1994), 45. doi: 10.1215/S0012-7094-94-07403-6. Google Scholar [3] P. Buser, Geometry and Spectra of Compact Riemann Surfaces,, Birkhauser Boston, (1992). Google Scholar [4] P. B. Gilkey, Asymptotic Formulae in Spectral Geometry,, Chapman & Hall /CRC, (2004). Google Scholar [5] J. Fox, E. Greif, D. Kaplan and R. Strichartz, Spectrum of the Laplacian on Regular Polyhedral Surfaces,, in preparation., (). Google Scholar [6] V. Ivrii, Precise Spectral Asymptotics for Elliptic Operators,, Lecture Notes in Math \textbf{1100} (1984), 1100 (1984). Google Scholar [7] S. Jayakar and R. Strichartz, Average number of lattice points in a disk,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 1. Google Scholar [8] M. Kac, Can one hear the shape of a drum,, \emph{Amer. Math. Monthly}, 783 (1966), 1. Google Scholar [9] D. V. Kosygin, A. A. Minasov and Ya. G. Sinai, Statistical properties of the spectra of Laplace Beltrami operators on Liouville surfaces,, \emph{Russian Math. Surveys}, 48 (1993). doi: 10.1070/RM1993v048n04ABEH001052. Google Scholar [10] H. Lapointe, I. Potterovich and Yu. Safarov, Average growth of the spectral function on a Riemannian manifold,, \emph{Comm. P. D. E.}, 34 (2009), 581. doi: 10.1080/03605300802537453. Google Scholar [11] T. Murray and R. Strichartz, Numerical investigations of spectral asymptotics on surfaces,, in preparation., (). Google Scholar [12] P. Sarnak, Spectra of hyperbolic surfaces,, \emph{Bull. Amer. Math. Soc.}, 40 (2003), 441. doi: 10.1090/S0273-0979-03-00991-1. Google Scholar [13] C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (2014). doi: 10.1515/9781400850549. Google Scholar [14] R. Strichartz, Spectral asymptotics revisited,, \emph{J. Fourier Anal. Appl.}, 18 (2012), 626. doi: 10.1007/s00041-012-9216-7. Google Scholar [15] Yu. G. Safarov, Riesz means of the distribution function of the eigenvalues of an elliptic operator,, \emph{J. Sov. Math.}, 49 (1990), 1210. doi: 10.1007/BF02208718. Google Scholar [16] R. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés,, \emph{Bull. Math. Soc. Fr.}, 91 (1963), 289. Google Scholar
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