Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Mapping onto the plane of a class of concave billiards


Author: Richard L. Liboff
Journal: Quart. Appl. Math. 62 (2004), 323-335
MSC: Primary 37D50; Secondary 37J10, 81Q50
DOI: https://doi.org/10.1090/qam/2054602
MathSciNet review: MR2054602
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Three results are reported in this work. The first addresses the four 'elemental-polygon' billiards with sides replaced by circular non-overlapping concave elements. Any orbit of the resulting concave billiard is mapped onto a trajectory in the plane that is shown to diverge from the trajectory of the related polygon billiard. This mapping permits application of Lyapunov exponents relevant to an unbounded system to be applied to the bounded concave elemental polygon-billiards. It is shown that Lyapunov exponents for concave elemental polygon-billiards go to zero as the curvature of the concave billiard segments go to zero. The second topic considers the quantum analogue of this problem. A conjecture is introduced which implies that a characteristic quantum number exists below which the adiabatic theorem applies and above which quantum chaos ensues. This parameter grows large as side curvature of the given billiard grows small. Lastly, a correspondence property between classical and quantum chaos for the concave elemental-polygon billiards is described.


References [Enhancements On Off] (What's this?)

  • [1] S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of Billiard Flows and Quadratic-Differentials, J. Ann. Math., 124 293-311 (1986) MR 855297
  • [2] Y. G. Sinai, Introduction to Ergodic Theory, Springer, New York (1976) MR 0584788
  • [3] I. P. Cornfield, S. V. Fomin, and Y. G. Sinai, Ergodic Theory, Springer, New York (1982) MR 832433
  • [4] A. B. Katok and J. Strelcyn, Invariant Manifolds, Entropy of Billiards, Springer, New York (1986)
  • [5] V. Kozlov, Billiards: A Generic Introduction to the Dynamics of Systems with Impacts, AMS, Providence, RI (1991) MR 1118378
  • [6] R. L. Liboff, (a) The Polygon Quantum Billiard, J. Math. Phys. 35, 596-607 (1994), (b) Circular Sector Quantum Billiard and Allied Configurations, J. Math. Phys. 35, 2218-2228, (1994), (c) Nodal-surface conjectures for the convex quantum billiard, J. Math Phys. 35, 3881-3886 (1994) MR 1271917
  • [7] V. Amar, M. Paupri, and A. Scotti, Schrödinger Equation for Convex Plane Polygons: A Tiling Method for the Derivation of Eigenvalues and Eigenfunctions, J. Math. Phys. 32, 2442-2449 (1991) MR 1122533
  • [8] G. M. Zaslavsky, Chaos in Dynamic Systems, Harwood, New York (1985). In this work, the concave-segmented regular N-gon is labeled a 'star.' A rectangular billiard with a uniformly concave-segmented edge is labeled a 'caterpillar' billiard. MR 780371
  • [9] M. C. Gurtzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, (1990) MR 1077246
  • [10] L. E. Reichl, The Transition to Chaos, Springer, New York (1992) MR 2090889
  • [11] E. Ott, Chaos in Dynamical Systems, Cambridge, New York (1993)
  • [12] M. Wojtkowski, Principles for the Design of Billiards with Nonvanishing Lyapunov Exponents Comm. Math. Phys. 105, 391-414 (1986) MR 848647
  • [13] G. Benettin, L. Galgani, and J. M. Strelcyn, Phys. Rev. A14, Kolgomorov Entropy and Numerical experiments. 2338-2345, (1976)
  • [14] M. Casartelli, E. Diana, I. Galgani, and A. Scotti, Numerical Computations on a Stochastic Parameter Related to Kolgomorov Entropy, Phys. Rev. A13, 1921-1295, (1976)
  • [15] A. Barnett, D. Cohen, and E. J. Heller, Deformations and Dilations of Chaotic Billiards: Dissipation Rate and Quasiorthogonality of the Boundary Wave Functions, Phys. Rev. Lett. 85, 1412-1415 (2000)
  • [16] G. Cassati and T. Prosen, Mixing Property of Triangular Billiards, Phys. Rev. Lett. 83, 4729-4732 (1999)
  • [17] R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions, 2nd ed., Wiley, New York (1998)
  • [18] C. Jaffe and W. P. Reinhardt, Uniform Semiclassical Quantization of Regular and Chaotic Classical Dynamics on the Henon-Heiles Surface, J. Chem. Phys. 77, 5191-5203, (1982) MR 681223
  • [19] R. B. Shirts and W. P. Reinhardt, Approximate Constants of Motion for Classically Chaotic Vibrational Dynamics, J. Chem. Phys. 77, 5204-5217 (1982) MR 681224
  • [20] K. Sohlberg and R. B. Shirts, Semiclassical Quantization of a Nonintegrable System, J. Chem. Phys. 101, 7763-7778, (1994)
  • [21] N. Hungerbuehler, SIAM Review 42, 657 (2000)
  • [22] R. Markarian, S-O. Kamphorst, and S. Carvallo, A Lower Bound for Chaos on the Elliptical Stadium, Physica D115, 189-202 (1998) MR 1626592
  • [23] H. Makino, T. Harayama, and Y. Aizawa, Quantum-Classical Correspondences of the Berry-Robnik Parameter Through Bifurcations in Lemon Billiard Systems, Phys. Rev. E63, Art. No. 056203 (2001)
  • [24] U. Krause, Concave Perron-Frobenius Theory and Applications, Nonlinear Analysis 47, 1457-1466 (2001) MR 1977031
  • [25] J. Ding, Absolutely Continuous Invariant Measure on a Piecewise Concave Mapping, Nonlinear Analysis 28, 1133-1140 (1997) MR 1422805
  • [26] Y. Zheng and J. F. Greenleaf, The Effect of Concave and Convex Weight Adjustments on Self-Organizing Maps. IEEE Transactions on Neural Networks 7, 87-96 (1996)
  • [27] G. Benetin, Power-Law Behavior of Lyapunov Exponents in Some Conservative Dynamical Systems, Physica D13, 211-220 (1984) MR 775286
  • [28] M. Wojtkowski, Invariant Families of Cones and Lyapunov Exponents, Ergodic Th. and Dynam. Sys. 8, 145-161 (1985) MR 782793
  • [29] R. Blümel and W. P. Reinhardt, Chaos in Atomic Physics, Cambridge, New York (1997)
  • [30] P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics, Physica D2, 495-412 (1981) MR 625449
  • [31] B. Chirikov, F. Izrailev, and D. Shepelyansky, Quantum chaos: Localization vs. Ergodicity, Physica D33,77-88 (1988) MR 984612
  • [32] R. L. Liboff, The Many Faces of the Helmholtz Equation, Phys. Essays 12, 492 (1999) MR 1783862
  • [33] M. A. Pinsky, The eigenvalues of an equilateral triangle, Siam J. Math. Anal 11, 819-949 (1980); 16, 848 (1985) MR 586910
  • [34] S. W. McDonald and A. N. Kaufman, Wave chaos in the stadium, Phys. Rev. A37, 3067-3078 (1988) MR 937657
  • [35] G. Alessandrini, Nodal lines of the fixed membrane problem in general convex domains, Comm. Math. Helv. 69, 142-154 (1994) MR 1259610
  • [36] R. L. Liboff, Function-Mixing Hypothesis and Quantum Chaos, Physica D93, 137 (1996) MR 1391462
  • [37] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol 1, Interscience, New York (1966)
  • [38] R. L. Liboff, Introductory Quantum Mechanics, 3rd. ed., Addison Wesley, San Francisco (1998)
  • [39] L. Kaplan and E. J. Heller, Weak Quantum Ergodicity, Physica D121, 1-18, (1998) MR 1644378
  • [40] G. Casati, ed., Chaotic Behavior in Quantum Systems, Plenum, New York (1985)
  • [41] R. L. Liboff, Bohr Correspondence Principle for Large Quantum Numbers, Found. Phys. 5, 271-293 (1975)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 37D50, 37J10, 81Q50

Retrieve articles in all journals with MSC: 37D50, 37J10, 81Q50


Additional Information

DOI: https://doi.org/10.1090/qam/2054602
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society