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The tunneling effect in multidimensional systems may be greatly influenced by the underlying chaotic dynamics which is generic in more than one-dimensional systems. Aiming at a classical dynamical description of chaotic tunneling, we have developed in the present paper a basic formulation for the complex-domain semiclassical wave-matrix of systems with 1.5 degrees of freedom, namely, periodically time-dependent 1D-scattering systems, which can be regarded as a minimal model of multidimensional systems. Using an autonomous 2D system equivalent to the periodically time-dependent system, the semiclassical expression of the wave-matrix described in the 1D time-dependent picture is derived. In the latter half of the paper, our semiclassical formulation is examined with two simple examples, i.e., an oscillating wall and an oscillating barrier. The semiclassical wave-matrix is constructed, paying particular attention to some problems which emerge by extending classical dynamics to complex domain, and the results are compared with the fully quantum counterparts. The semiclassical results agree quite well with the corresponding quantal results if complex classical trajectories are fully taken into account. The relationship between the semiclassical wave-matrix and Miller's classical S-matrix is also discussed.

Icosahedral quasicrystals and their approximants are generally described as packing of icosahedral clusters. Experimental studies show that clusters in various approximants are orderly arranged, such that their centers are located at the nodes (or vertices) of a periodic tiling composed of four basic polyhedra called the canonical cells. This so called canonical-cell geometry is likely to serve as a common framework for modeling how clusters are arranged in approximants, while its applicability seems to extend naturally to icosahedral quasicrystals. To date, however, it has not been proved yet if the canonical cells can tile the space quasiperiodically, though we usually believe that clusters in icosahedral quasicrystals are arranged such that quasiperiodic long-range order as well as icosahedral point symmetry is maintained. In this paper, we report for the first time an iterative geometrical transformation of the canonical cells defining a so-called substitution rule, which we can use to generate a class of quasiperiodic canonical-cell tilings. Every single step of the transformation proceeds as follows: each cell is first enlarged by a magnification ratio of τ 3 ( τ = golden mean ) and then subdivided into cells of the original size. Here, cells with an identical shape can be subdivided in several distinct manners depending on how their adjacent neighbors are arranged, and sixteen types of cells are identified in terms of unique subdivision. This class of quasiperiodic canonical-cell tilings presents the first realization of three-dimensional quasiperiodic tilings with fractal atomic surfaces. There are four distinct atomic surfaces associated with four sub-modules of the primitive icosahedral module, where a representative of the four submodules corresponds to the Σ = 4 coincidence site module of the icosahedral module. It follows that the present quasiperiodic tilings involve a kind of superlattice ordering that manifests itself in satellite peaks in the diffraction patterns. Moreover, we argue that the superlattice ordering causes the overall point symmetry of the tilings to reduce from icosahedral to tetrahedral, although the symmetry reduction is weak and mainly observed in the satellite peaks. The present discovery suggests that the canonical-cell geometry may play an important role in uncovering the mystery of structure and formation of icosahedral quasicrystals.