A role of new Stokes curves in the quantized Héon map
Akira Shudo
Department of Physics, Tokyo Metropolitan University,
1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

shudo@phys.metro-u.ac.jp

A discrete analog of the Feynman-type path integral,Z ∞Z ∞ihin< qn |U |q0 >=···
dq1 dq2 · · · dqn−1 exp S(q0 , · · · , qn )
~−∞−∞

(1)

gives quantum dynamics of the symplectic map. Here, S(q0 , · · · , qn ) represents thediscretized Lagrangian given as
S(q0 , · · · , qn ) =nX1
j=12(qj − qj−1 )2 −n−1X

V (qj ).

(2)

j=1

The form of S(q0 , · · · , qn )is derived so that applying the variational principle generates
the symplectic map. In fact, we can easily see that the condition ∂S(q0 , · · · , qn )/∂qj =0 (1 ≤ j ≤ n − 1) yields the classical map in the Lagrangian form. The simplest
possible choice of the potential function would be,V (q) = −q3− cq,3(3)where c denotes a real parameter controlling qualitative features of the underlying
area-preserving map, and possibly the corresponding quantum dynamics. This system
is related to the so-called Hénon family via an appropriate affine transformation with achange of parameter. There is a classification theorm claiming that non-trivial polynomial diffeomorphisms from C2 to C2 are written as composition of the Hénon map, andthe others, which are either elementary or affine map, are the mappings easily analyzed[1]. For this reason, the Hénon map can be regarded as a simplest possible polynomial
diffeomorphism creating non-trivial dynamics.
Here, we fix the initial coordinate q0 and regard the quantum propagator < qn |U n |q0 >as a function of the final coordinate qn . We then discuss Stokes geometry of I(qn ) ≡<qn |U n |q0 > in applying the saddle point method. In order to analyze Stokes geometry where more than 2 solutions appear as saddle point contribution, we refer to thework by Aoki, Kawai and Takei, in which a prescription to analyze Stokes phenomena inhigher-order differential equations has been provided within the exact WKB framework[2]. The work contains not only mathematical justification of the preceding work [3]in which necessity of introducing new Stokes curves around crossing points of ordinary
Stokes curves was pointed out in an ad-hoc way, but also claims that virtual turning
points should first be taken into account to construct complete Stokes geometry. They
also clarified that new Stokes curves play essentially the same part in Stokes geometry.
1Here, we discuss Stokes geometry of quantum propagator I(qn ) on the same line,
especially focusing on Stokes geomety in the horseshoe limit. Since a simple symbolic
dynamics (binary full shift) completely controlling the underlying classical dynamics
in the horseshoe limit, it can be viewed as the simplest possible situation in the senseof dynamical systems and thus would be the first object to be investigated. We shallemphasize that even in such simple cases virtual turning points and new Stokes curves
are crucial ingredients to obtain the transition matrices that describes the propagationof WKB solutions from qn = −∞ to qn = +∞.
In the horseshoe limit, all the turning points are located on the real plane, but as thenonlinear paremeter c decreases, some of them fall into purely imaginary plane. Suchevents occur as a result of coalescence of turning points. If we know how Stokes graphschange when such bifurcations occur, Stokes geometry in generic parameter values canbe traced from the horseshoe limit in principle. This is exactly the same strategy tostudy the pruning of the horseshoe structure [4].

References[1] S. Friedland and J. Milnor, Ergod. Theor. Dyn. Sys. 9(1989) 67-99.
[2] T. Aoki, T. Kawai and Y. Takei, in Analyse algébrique des perturbations singulières. I. (ed. by L. Boutet de Monvel) Hermann, (1994) 69-84.
[3] H.L. Berk, W.M. Nevins and K.V. Roberts, J. Math. Phys. 23(1982) 988-1002.
[4] P. Cvitanović, G. Gunaratne and I. Procaccia, Phys. Rev. A 38 (1988) 1503-1520.

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