Toward pruning theory of the Stokes geometry for the quantum Hénon map

Abstract

The Stokes geometry for the propagator of the quantum Hénon map is studied in the light of recent developments of the exact WKB analysis. As the simplest possible situation the Hénon map satisfying the so-called horseshoe condition is closely analyzed, together with listing up local bifurcation patterns of the Stokes geometry. This is exactly in the same spirit as pruning theory for the classical horseshoe system, and the present paper is placed as the first step to establish pruning theory of the Stokes geometry for chaotic systems. Our analysis reveals that the birth and death of the WKB solutions caused by the Stokes phenomenon do not occur in a local but entirely global manner, reflecting topological nature encoded in the Stokes geometry. We derive an explicit general formula to enumerate the number of WKB solutions in the asymptotic region and obtain its growth rate, which is shown to be less than the topological entropy of the corresponding classical dynamics. The relations to the diffraction catastrophe integrals and one-step multiply folding maps are also discussed.

Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems

“…However, a naive treatment of the Stokes phenomenon was shown to break down. In particular, we demonstrated that there is a situation where even exponentially decaying contributions should be dropped, which is one piece of evidence suggesting that the Stokes phenomenon for the normal form Hamiltonian systems must be highly non-trivial [36].…”

“…As seen in the phase space portrait drawn in figure 11, the system has two symmetric wells located at the positions q = ±1 respectively, and nonlinear resonance like equi-energy contours appear around each well. Figure 11: Equi-energy contours for the Hamiltonian (36).…”

“…Figure 13: The Riemann surface of p(q) for the Hamiltonian (36). The Riemann surface forms a 9-fold torus.…”

“…In the following, we discuss the tunnelling splitting ∆E n for the normal form Hamiltonian (36) based on the semiclassical analysis. Note however that the semiclassical analysis performed here will not fully be based on the semiclassical formula (3), and could be done only with a heuristic recipe.…”

Communication: Systematic elimination of Stokes divergences emanating from complex phase space caustics

“…Significant progress has been made towards eliminating such divergences in discrete time systems based on Adachi's work on the Principle of Exponential Dominance (PED). 4,[8][9][10] However the PED requires comparison of trajectory pairs which can only be found numerically via a root search in the complex trajectory manifold. In continuous time dynamics, such a root search is quite challenging and has been successful so far only for relatively short times.…”