We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the non-unitary quantum propagator, and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as → 0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates {ψ( )} →0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker map, for which the probability density in position space is observed to have self-similarity properties. In open systems the lack of unitarity of the quantum propagator gives rise to non-orthogonal decaying eigenstates with complex energies (resonances), the imaginary parts of which are interpreted as decay rates. In the case of open chaotic systems the classical mechanics is structured in phase space around fractal sets associated with trajectories that remain trapped for infinite times, either in the future (forward-trapped set K + ) or in the past (backward-trapped set K − ). The mean density of resonances is believed (but not in general proved) to be determined by the fractal dimension of the invariant set K 0 = K + ∩ K − , the classical repeller. This is the fractal Weyl law [4,5,6]. (Note that this is different to the resonance statistics in weakly open systems, for which the size of the opening vanishes in the semiclassical limit [7]).Much less is known about the resonance (or Gamow) eigenstates. These are important in many areas of physics [8] and chemistry [9], because they have marked influence on observable quantities such as scattering cross sections and reaction rates (they are a component of the Siegert pseudostates basis in terms of which the scattering wavefunctions and S matrix, for example, can conveniently be expanded [8]). Following the well established idea that in the semiclassical limit timeindependent quantum properties should be related to time-independent classical sets, it is natural to expect that long-lived eigenstates of open systems should be determined by the structure of K + and K − . This was tested numerically for some right eigenstates of the open kicked rotator in [10], where the term 'quantum fractal eigenstates' was coined.We here significantly extend the notion of quantum fractal eigenstates in several new directions. First, we draw the important distinctions between left and right eigenstates of the non-unitary propagator, and between states that are 'short-lived' and 'long-lived' with respect to the Ehrenfest time. Second, we show that in the semiclassical limit the long-lived left eigenstates concentrate on K + , while the long-lived right eigenstates concentrate on K − . In chaotic systems the eigenstates thus inherit the intricate fractal structure of the underlying classical trapped sets (this property has also been observed by Nonnenmacher and Rubin [11]). Third, we find that in the s...

We extend the semiclassical theory of scarring of quantum eigenfunctions Á n (q) by classical periodic orbits to include situations where these orbits undergo generic bifurcations. It is shown that jÁ n (q)j 2 , averaged locally with respect to position q and the energy spectrum fE n g, has structure around bifurcating periodic orbits with an amplitude and length-scale whose~dependence is determined by the bifurcation in question. Speci cally, the amplitude scales as~¬ and the length-scale as~!, and values of the scar exponents, ¬ and !, are computed for a variety of generic bifurcations. In each case, the scars are semiclassically wider than those associated with isolated and unstable periodic orbits; moreover, their amplitude is at least as large, and in most cases larger. In this sense, bifurcations may be said to give rise to superscars. The competition between the contributions from di¬erent bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed. We argue that there is a resulting universal~scaling in the semiclassical asymptotics of these moments for irregular states in systems with mixed phase-space dynamics. Finally, a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps.

Systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are, in the semiclassical limit, a combination of Poisson and randommatrix, with relative weightings determined by the corresponding measures of regular and chaotic orbits. We here identify an additional component in long-range spectral statistics, associated with periodic orbit bifurcations, which can be semiclassically large. This is illustrated for a family of perturbed cat maps.

We numerically show fractal Weyl law behavior in an open Hamiltonian system that is described by a smooth potential and which supports numerous above-barrier resonances. This behavior holds even relatively far away from the classical limit. The complex resonance wave functions are found to be localized on the fractal classical repeller.

We study temporal networks of characters in literature focusing on Alice's Adventures in Wonderland (1865) by Lewis Carroll and the anonymous La Chanson de Roland (around 1100). The former, one of the most influential pieces of nonsense literature ever written, describes the adventures of Alice in a fantasy world with logic plays interspersed along the narrative. The latter, a song of heroic deeds, depicts the Battle of Roncevaux in 778 A.D. during Charlemagne's campaign on the Iberian Peninsula. We apply methods recently developed by Taylor et al. [26] to find time-averaged eigenvector centralities, Freeman indices and vitalities of characters. We show that temporal networks are more appropriate than static ones for studying stories, as they capture features that the time-independent approaches fail to yield.

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Assuming the selectivity filter of KcsA potassium ion channel may exhibit quantum coherence, we extend a previous model by Vaziri and Plenio (2010 New J. Phys. 12 085001) to take into account Coulomb repulsion between potassium ions. We show that typical ion transit timescales are determined by this interaction, which imposes optimal input/output parameter ranges. Also, as observed in other examples of quantum tunneling in biological systems, the addition of moderate noise helps coherent ion transport.

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