Variance of number of lattice points in random narrow elliptic strip

 Access to full text

 Full (PDF)

1. [Ber] M.V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. Lond., Vol. A 400, 1985, pp. 229-251. Zbl0875.35061MR805089
2. [BT] M.V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. Lond., Vol. A 356, 1977, pp. 375-394. Zbl1119.81395
3. [BCDL] P.M. Bleher, Zh. Cheng, F.J. Dyson and J.L. Lebowitz, Distribution of the error term of number of lattice points inside a shifted circle, Commun. Math. Phys., Vol. 154, 1993, pp. 433-469. Zbl0781.11038MR1224087
4. [BD1] P.M. Bleher and F.J. Dyson, The variance of the error function in the shifted circle problem is a wild function of the shift, Commun. Math. Phys. (to appear). Zbl0808.11058MR1266060
5. [BD2] P.M. Bleher and F.J. Dyson, Mean square value of exponential sums related to representation of integers as sum of two squares, Acta Arithm. (to appear). Zbl0915.11048MR1302508
6. [BDL] P.M. Bleher, F.J. Dyson and J.L. Lebowitz, Non-gaussian energy level statistics for some integrable systems, Phys. Rev. Lett., Vol. 71, 1993, pp. 3047-3050. Zbl1050.81530MR1246066
7. [BL] P.M. Bleher and J.L. Lebowitz, Energy-level statistics of model quantum systems: universality and scaling in a lattice-point problem, J. Stat. Phys., Vol. 74, 1994, pp. 167-217. Zbl0946.11520MR1257819
8. [Ble] P.M. Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. Journ., Vol. 67, 1992, pp. 461-481. Zbl0762.11031MR1181309
9. [CCG] G. Casati, B.V. Chirikov and I. Guarneri, Energy-level statistics of integrable quantum systems, Phys. Rev. Lett., Vol. 54, 1985, pp. 1350-1353. MR780344
10. [CL] Zh. Cheng and J.L. Lebowitz, Statistics of energy levels in integrable quantum systems, Phys. Rev., Vol. A44, 1991, pp. R3399-3402. 
11. [CLM] Zh. Cheng, L. Lebowitz and P. Major, On the number of lattice points between two enlarged and randomly shifted copies of an oval, Preprint, Rutgers University, 1993. 
12. [Gu] M.C. Gutzwiller, Chaos in classical and quantum mechanics, Springer-Verlag, N. Y. e.a., 1990. Zbl0727.70029MR1077246
13. [HL] G.H. Hardy and J.E. Littlewood, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, Proc. London Math. Soc., Vol. 13, 1914, p. 174. Zbl45.0389.02JFM45.0389.02
14. [LS] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Preprint, Princeton University, 1993. 
15. [Maj] P. Major, Poisson law for the number of lattice points in a random strip with finite area, Prob. Theory Related Fields, Vol. 92, 1992, pp. 423-464. Zbl0767.60010MR1169014
16. [OdA] A.M. Ozorio de Almeida, Hamiltonian systems: chaos and quantization, Cambridge Univ. Press, New York, 1988. Zbl0734.58002MR985103
17. [Sin] Ya.G. Sinai, Poisson distribution in a geometric problem, in Advances in Soviet Mathematics, Vol. 3, Ya G. SINAI Ed. (AMS, Providence, Rhode Island), 1991, pp. 199-214. Zbl0744.60016MR1118163
18. [Tab] M. Tabor, Chaos and integrabhility in nonlinear dynamics. An introduction, Wiley, New York, 1989. Zbl0682.58003MR1007309