Vibrating quantum billiards on Riemannian manifolds

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 9, 2001, Pages 2305-2315

  Porter, Mason A ^a   Liboff, Richard L ^a  

^a Cornell University ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035438228     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127401003449     Document Type: Review

References (32)

1. Abraham, R., Marsden, J. E. & Ratiu, T. [1988] Manifolds, Tensor Analysis and Applications, Applied Mathematical Sciences, Vol. 75, 2nd edition (Springer-Verlag, NY).
2. Allen, L. & Eberly, J. H. [1987] Optical Resonance and Two-Level Atoms (Dover Publications, NY).
3. Arnold, V. I. [1988] Geometrical Methods in the Theory of Ordinary Differential Equations, Series of Comprehensive Studies in Mathematics, Vol. 250, 2nd edition (Springer-Verlag, NY).
4. Badrinarayanan, R. & José, J. V. [1995] "Spectral properties of a Fermi accelerating disk," Physica D83, 1-29.
5. Benenti, S. [1980] "Separability structures on Riemannian manifolds," in Differential Geometrical Methods in Mathematical Physics: Proc., Aix-en-Provence and Salamanca 1979, Lecture Notes in Mathematics, Vol. 836 (Springer-Verlag, NY), pp. 512-538.
6. Blümel, R. & Esser, B. [1994] "Quantum chaos in the Born-Oppenheimer approximation," Phys. Rev. Lett. 72(23), 3658-3661.
7. Blümel, R. & Reinhardt, W. P. [1997] Chaos in Atomic Physics (Cambridge University Press, Cambridge, England)
8. Butkov, E. [1968] Mathematical Physics (Addison-Wesley Reading, MA).
9. Casati, G. (ed.) [1985] Chaotic Behavior in Quantum Systems (Plenum, NY).
10. Casati, G. & Chirikov, B. [1995] Quantum Chaos (Cambridge, NY).
11. do Carmo, M. P. [1992] Riemannian Geometry (Birkhäuser, Boston, MA).
12. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 42 (Springer-Verlag, NY).
13. Gutzwiller, M. C. [1990] Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, Vol. 1 (Springer-Verlag, NY).
14. Johnson, C. [1987] Numerical Solution of Partial Differential Equations by the Finite Element Method (Cambridge University Press, Cambridge, England).
15. Katok, A. & Hasselblatt, B. [1995] Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, NY).
16. Liboff, R. L. [1998] Introductory Quantum Mechanics, 3rd edition (Addison-Wesley, San Francisco, CA).
17. Liboff, R. L. [1999] "Many faces of the Helmholtz equation," Phys. Essays 12, 1-8.
18. Liboff, R. L. [2000] "Quantum billiard chaos," Phys. Lett. A269, 230-233.
19. Liboff, R. L. & Porter, M. A. [2000] "Quantum chaos for the radially vibrating spherical billiard," Chaos 10(2), 366-370.
20. Lichtenberg, A. J. & Lieberman, M. A. [1992] Regular and Chaotic Dynamics, Applied Mathematical Sciences, Vol. 38, 2nd edition (Springer-Verlag, NY).
21. Lucan, J. [1998] Quantum Dots (Springer, NY).
22. MacDonald, S. W. & Kaufman, A. N. [1988] "Wave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equation," Phys. Rev. A37, p. 3067.
23. Moon, P. & Spencer, D. E. [1988] Field Theory Handbook, 2nd edition (Springer-Verlag, Berlin, Germany).
24. Sakurai, J. J. [1994] Modern Quantum Mechanics, revised edition (Addison-Wesley, Reading, MA).
25. Simmons, G. F. [1991] Differential Equations with Applications and Historical Notes, 2nd edition (McGraw-Hill, NY).
26. Strang, G. [1988] Linear Algebra and its Applications, 3rd edition (Harcourt Brace Jovanovich College Publishers, Orlando, FL).
27. Temam, R. [1997] Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, 2nd edition (Springer-Verlag, NY).
28. Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2 (Springer-Verlag, NY).
29. Wong, S. [1990] Introductory Nuclear Physics (Prentice Hall, Englewood Cliffs, NJ).
30. Zaren, H., Vahala, K. & Yariv, A. [1989] "Gain spectra of quantum wires with inhomogeneous broadening," IEEE J. Quant. Electron. 25, p. 705.
31. Zwanziger, J. W., Koenig, M. & Pines, A. [1990] "Berry's phase," Ann. Rev. Phys. Chem. 41, 601-646.
32. Zwillinger, D. (ed.), [1996] Standard Mathematical Tables and Formulae, 30th edition (CRC Press, Boca Raton, FL).