Complex paths for regular-to-chaotic tunneling rates

In generic Hamiltonian systems tori of regular motion are dynamically separated from regions of chaotic motion in phase space. Quantum mechanically these phase-space regions are coupled by dynamical tunneling. We introduce a semiclassical approach based on complex paths for the prediction of dynamical tunneling rates from regular tori to the chaotic region. This approach is demonstrated for the standard map giving excellent agreement with numerically determined tunneling rates.Comment: 5 pages, 4 figure

Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation.Comment: 7 pages, 4 figure

Nambu-Hamiltonian flows associated with discrete maps

For a differentiable map $(x_1,x_2,..., x_n)\to (X_1,X_2,..., X_n)$ that has an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of the initial value, say $x_n$, of the map plays the role of time variable while the others remain fixed. We present various examples which exhibit the map-flow correspondence.Comment: 19 page

Quantum Dynamics of Atom-molecule BECs in a Double-Well Potential

We investigate the dynamics of two-component Bose-Josephson junction composed of atom-molecule BECs. Within the semiclassical approximation, the multi-degree of freedom of this system permits chaotic dynamics, which does not occur in single-component Bose-Josephson junctions. By investigating the level statistics of the energy spectra using the exact diagonalization method, we evaluate whether the dynamics of the system is periodic or non-periodic within the semiclassical approximation. Additionally, we compare the semiclassical and full-quantum dynamics.Comment: to appear in JLTP - QFS 200

Semiclassical Description of Tunneling in Mixed Systems: The Case of the Annular Billiard

We study quantum-mechanical tunneling between symmetry-related pairs of regular phase space regions that are separated by a chaotic layer. We consider the annular billiard, and use scattering theory to relate the splitting of quasi-degenerate states quantized on the two regular regions to specific paths connecting them. The tunneling amplitudes involved are given a semiclassical interpretation by extending the billiard boundaries to complex space and generalizing specular reflection to complex rays. We give analytical expressions for the splittings, and show that the dominant contributions come from {\em chaos-assisted}\/ paths that tunnel into and out of the chaotic layer.Comment: 4 pages, uuencoded postscript file, replaces a corrupted versio

Semiclassical transmission across transition states

It is shown that the probability of quantum-mechanical transmission across a phase space bottleneck can be compactly approximated using an operator derived from a complex Poincar\'e return map. This result uniformly incorporates tunnelling effects with classically-allowed transmission and generalises a result previously derived for a classically small region of phase space.Comment: To appear in Nonlinearit

Evanescent wave approach to diffractive phenomena in convex billiards with corners

What we are going to call in this paper "diffractive phenomena" in billiards is far from being deeply understood. These are sorts of singularities that, for example, some kind of corners introduce in the energy eigenfunctions. In this paper we use the well-known scaling quantization procedure to study them. We show how the scaling method can be applied to convex billiards with corners, taking into account the strong diffraction at them and the techniques needed to solve their Helmholtz equation. As an example we study a classically pseudointegrable billiard, the truncated triangle. Then we focus our attention on the spectral behavior. A numerical study of the statistical properties of high-lying energy levels is carried out. It is found that all computed statistical quantities are roughly described by the so-called semi-Poisson statistics, but it is not clear whether the semi-Poisson statistics is the correct one in the semiclassical limit.Comment: 7 pages, 8 figure

Invariant varieties of periodic points for some higher dimensional integrable maps

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}Comment: 24 page

Slow relaxation in weakly open vertex-splitting rational polygons

The problem of splitting effects by vertex angles is discussed for nonintegrable rational polygonal billiards. A statistical analysis of the decay dynamics in weakly open polygons is given through the orbit survival probability. Two distinct channels for the late-time relaxation of type 1/t^delta are established. The primary channel, associated with the universal relaxation of ''regular'' orbits, with delta = 1, is common for both the closed and open, chaotic and nonchaotic billiards. The secondary relaxation channel, with delta > 1, is originated from ''irregular'' orbits and is due to the rationality of vertices.Comment: Key words: Dynamics of systems of particles, control of chaos, channels of relaxation. 21 pages, 4 figure

Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a microwave billiard with threefold symmetry

Recently it has been shown that time-reversal invariant systems with discrete symmetries may display in certain irreducible subspaces spectral statistics corresponding to the Gaussian unitary ensemble (GUE) rather than to the expected orthogonal one (GOE). A Kramers type degeneracy is predicted in such situations. We present results for a microwave billiard with a threefold rotational symmetry and with the option to display or break a reflection symmetry. This allows us to observe the change from GOE to GUE statistics for one subset of levels. Since it was not possible to separate the three subspectra reliably, the number variances for the superimposed spectra were studied. The experimental results are compared with a theoretical and numerical study considering the effects of level splitting and level loss