Maslov indices and singularities of Integrable systems

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
n1
x
x
11
n1
12
n2
X
X
13
n2
14
Quantization of integrable systems
15
EBK Quantization Rule
16
EBK Quantization Rule
17
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
n1 example
28
n2 example
29
n2 example
30
n2 example
31
Rotational symmetry in Rn
32
Singularities
33
Angular momentum Maslov indices
34
(No Transcript)
35
The Periodic Toda Chain
Henon,Flaschka Kac Moerbeke Gutzwiller, Sklyanin
36
Lax pair
37
Inequivalent Lax pair
38
Integrability
39
Singularities of Toda chain
40
Nondegeneracy
41
Nondegeneracy
42
Maslov index and Holonomy of Lax bundles
43
Questions

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

44
Quantum Monodromy- Example
from Child et al (1999)