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Maslov indices and singularities of Integrable systems

Maslov indices and singularities of Integrable systems. JM Robbins, University of Bristol J Foxman S Creagh, H Dullen, G Tanner, H Waalkens. Outline. Singularities of integrable systems Maslov index Monodromy and Maslov index Singularity formula for Maslov index

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Maslov indices and singularities of Integrable systems

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  1. Maslov indices and singularities of Integrable systems JM Robbins, University of Bristol J Foxman S Creagh, H Dullen, G Tanner, H Waalkens

  2. Outline • Singularities of integrable systems • Maslov index • Monodromy and Maslov index • Singularity formula for Maslov index • Examples, including Toda chain

  3. Integrable systems

  4. Regular values

  5. Singularities

  6. Global topology

  7. Maslov index

  8. Maslov index

  9. Maslov index as winding number

  10. n=1 x x

  11. n=1

  12. n=2 X X

  13. n=2

  14. Quantization of integrable systems

  15. EBK Quantization Rule

  16. EBK Quantization Rule

  17. V y x Monodromy - Example 

  18. Monodromy of integrable systems

  19. Monodromy and Maslov indices (Creagh, Dullen, JMR, Tanner Waalkens (2005))

  20. L Butler, C Viterbo (1990)

  21. Singularity formula for Maslov index

  22. Nondegenerate corank-1 singularities

  23. Transversality

  24. Formula

  25. Formula

  26. Formula

  27. n=1 example

  28. n=2 example

  29. n=2 example

  30. n=2 example

  31. Rotational symmetry in Rn

  32. r p pj rj p(j-1) r(j-1) r(j) p(j) Singularities

  33. Angular momentum Maslov indices

  34. 3 1 2 4 The Periodic Toda Chain Henon,Flaschka Kac & Moerbeke Gutzwiller, Sklyanin

  35. Lax pair

  36. rth Inequivalent Lax pair

  37. j ( ) * 0 * Integrability

  38. Singularities of Toda chain

  39. Nondegeneracy

  40. Nondegeneracy

  41. Maslov index and Holonomy of Lax bundles

  42. Questions • Other Lax pairs • Higher Maslov classes 2 H4n-3 (Trofimov, Suzuki) • Higher-order semiclassical quantization of Toda chain (Littlejohn, Colin de Verdiere)

  43. V y x Quantum Monodromy- Example from Child et al (1999)

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