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Abstract.We find that characteristics of quantum tunneling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical evidences for the standard map together with a rigorous statement for the Hénon map are presented demonstrating that the complex classical paths which contribute to the semiclassical propagator are dense in the Julia set. Chaotic tunneling can thus be characterized by the transitivity of the dynamics and high density of the trajectories on the Julia set.

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Isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to iso-length spectrality of the corresponding domains. Although an explicit and exact trace formula such as Poisson's summation formula or Selberg's trace formula is not known to exist for such planar domains, equivalence between isospectrality and iso-length spectrality in a certain setting can be proved by employing the matrix representation of "transplantation of eigenfunctions". As an application of the equivalence, transplantable pairs of domains, which are all isospectral pair of planar domains and therefore counter examples of Kac's question "can one hear the shape of a drum?", are numerically enumerated and it is found at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to Sunada.

We have revealed that the barrier-tunneling process in non-integrable systems is strongly linked to chaos in complex phase space by investigating a simple scattering map model. The semiclassical wavefunction reproduces complicated features of tunneling perfectly and it enables us to solve all the reasons why those features appear in spite of absence of chaos on the real plane. Multi-generation structure of manifolds, which is the manifestation of complex-domain homoclinic entanglement created by complexified classical dynamics, allows a symbolic coding and it is used as a guiding principle to extract dominant complex trajectories from all the semiclassical candidates. 05.45.Mt, 03.65.Ge, 03.65.Sq, Tunneling phenomenon is peculiar to quantum mechanics and no counterparts exist in classical mechanics. Features of tunneling are nevertheless strongly influenced by the underlying classical dynamics [1][2][3][4][5][6]. In particular, chaotic features appearing in tunneling have been paid attention to in connection with real-domain chaos [2][3][4].A promising approach to see the connection of these two opposite concepts is to carry out the complex semiclassical analysis, which allows us to describe and interpret the tunneling phenomenon in terms of complex classical trajectories [7]. It has been shown that the complex semiclassical theory can successfully be applied even in classically chaotic systems and the origin of characteristic structures of the wavefunction inherent in chaotic systems is explained by the complex classical dynamics [5]. A significant role of almost real-domain homoclinic trajectories in the energy barrier tunneling has been pointed out based on the trace formula approach [8]. Recently, it is found that fringed pattern appears in the wavefunction of the two-dimensional barrier tunneling problem as a result of interference between oscillatory Lagrangian manifold [9]. They have shown a detailed scenario describing how such interference emerges in accordance with the divergent movement of singularities on the complex t-plane.In the present paper, we shall report the strong connection between the barrier-tunneling process in nonintegrable systems and the chaos in complex phase space by analyzing a simple scattering map model. In particular, it will be shown that even though the real-domain classical dynamics exhibits no chaos, i.e. null topological entropy, complex-domain chaos can make tunneling process complicated. Moreover our present analysis suggests that chaotic tunneling can be understood in a uni-fied manner from the viewpoint of complex-domain chaos not only in case of dynamical but also energy-barrier tunneling processes.We first introduce a scattering map model which is described by the following HamiltonianA set of classical equations of motion is given as (q j+1 , p j+1 ) = (q j , +T ′ (p j ), p j − V ′ (q j+1 )), where prime denotes a differentiation with respect to the corresponding argument. Note that the real-valued dynamics of our scattering map does not create chaos in contrast to th...