Descripción



The correspondence between quantum and classical mechanics has received much attention in the last 30 years. This topic is interesting per se, and also for the de velopment of semiclassical theories [1, 2] which are very often the only computational method applicable to the study of multidimensional realistic problems. The semiclassical quantization of chaotic systems is based, according to Gutzwiller [3], on properties of the periodic orbits (POs) of the system which are of unstable nature. Unstable POs have also been shown to inuence the structure of individual eigenfunctions by inducing an anomalous accumulation of probability density in their neighborhoods [4]. This eect was systematically stud ied by Heller [5], who coined the term scar to refer to it. He explained this enhancement as the result of a co herent interference caused by recurrences along the PO circuit. Scars have been observed experimentally in microwave cavities [6], microcavity lasers [7], and optical bers [8]. They have also been shown to be relevant in technical applications in nanotechnology where their inuence in the tunnelling current in quantum wells was observed [9]. In a series of papers [10, 11], it was shown how non stationary wave functions highly localized along a given PO, the so called tube functions, can be constructed in a systematic way. More sophisticated constructions taking into account the dynamics up to the Ehrenfest time have been called scar functions [12]. They are localized (in phase space) not only on the PO, but also along their associated unstable and stable invariant manifolds [12, 13]. The stable and unstable manifolds of POs cross in a hi erarchical way at homoclinic and heteroclinic orbits [14], as pointed out in the pioneering work of Poincaré. These orbits establish bridges between POs and it is reasonable to use them in order to describe dynamical processes re lated to pares of POs. In fact, it was shown very recently how they control the uctuations on the scar function widths in the eigenspectrum [15] and on the structure of the corresponding wave functions [16]. Similarly, areas associated to heteroclinic circuits also leave their imprint quantum mechanically [17, 18]. The upper panel of Fig. 1 shows the probability den sity of a tube wave function on a quantum Poincaré sur face of section (the function of probability density being the Husimi function). As it can be seen, the picture mimic the transverse section of a tube; actually, in this case there are two dierent tubes corresponding to the two POs used to construct a wave with the appropriate quantum symmetry. On the other hand, the lower g ure shows the Husimi of a scar wave function. In this case, the probability density spreads along the manifolds (specied by curves). In this gure we notice a curi ous phenomenon; the intersection point at x = 0 and px ' 1.1 acts as a repeller of probability. We associate this phenomenon with the destructive interference of a wave moving along the tube and a wave evolving along the corresponding heterocinic orbit. In this work, we study the eect of phase space circuits along homoclinic and heteroclinic orbits by using scar functions. In particular, we analyze the required condi tions for constructive interference in terms of canonical invariants. In order to clearly show the interference pro cess, we plot the Husimi of scar states at a xed point in the circuit as we go towards the semiclassical limit, verifying a strong correlation between constructive inter ference and the energy localization of the scar function in the spectrum.  
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Universidades Politécnica y Autónoma de Madrid 
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