Cantori for the stadium billiard

Although almost all orbits in the stadium billiard are "chaotic," there are many regular orbits as well-the ordered periodic orbits and cantori. The symmetries of the stadium are exploited to find maximizing and saddle periodic orbits. Cantori are maximizing quasiperiodic orbits; they have caustics. Transport in the stadium should be impeded by cantori, just as in any twist map; this is particularly important for those orbits that are nearly glancing.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard

We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.     

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.Partially supported by NSF Grant DMS 88-02643     

Spectra and Wavefunctions in a Ray-Splitting Sinai Microwave Billiard and their Semiclassical Interpretation

Experimental results on spectra and wave functions of a ray-splitting microwave billiard are presented. The billiard is formed by a flat rectangular microwave cavity with a quarter-circle insert made of teflon in one of the corners. Using the Gutzwiller trace formula, the contribution of the periodic orbits of the billiard to the density of states are determined. The wave functions, many of them showing scars associated with periodic orbits, are interpreted in terms of the semiclassical Green function.     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

Semiclassical limit for a truncated Hamiltonian

In the numerical calculation of the eigenenergies of a polynomial Hamiltonian, the majority of the levels depend on the cutoff of the basis used. By analyzing the finite Hamiltonian matrix as corresponding to a classical "Action Billiard" we are able to explain several features of the full spectrum using semiclassical periodic orbit theory. There are a large number of low-period orbits which interfere at the higher energies contained in the billiard. In this range the billiard becomes more regular than the untruncated Hamiltonian, as reflected by the Berry-Robnik level spacing distribution. (c) 1996 American Institute of Physics.     

Experimental test of a trace formula for a chaotic three-dimensional microwave cavity

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.     

Test of Trace Formulas for Spectra of Superconducting Microwave Billiards

Experimental tests of various trace formulas, which in general relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart, for spectra of superconducting microwave billiards of varying chaoticity are reviewed by way of examples. For a two-dimensional Bunimovich stadium billiard the application of Gutzwiller's trace formula is shown to yield correctly locations and strengths of the peaks in the Fourier transformed quantum spectrum in terms of the shortest unstable classical periodic orbits. Furthermore, in two-dimensional billiards of the Limaçon family the transition from regular to chaotic dynamics is studied in terms of a recently derived general trace formula by Ullmo, Grinberg and Tomsovic. Finally, some salient features of wave dynamical chaos in a fully chaotic three-dimensional Sinai microwave billiard are discussed. Here the reconstruction of the spectrum is not as straightforward as in the two-dimensional cases and a modified trace formula as suggested by Balian and Duplantier will have eventually to be applied.     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

Semiclassical Poincare map for integrable systems

The semiclassical Poincare map is applied to integrable systems and in particular to the rectangular billiard. The zeroes of the functional determinant are shown to give EBK quantization. The transfer operator is explicitly unitary and finite, resulting in a finite expansion of the Euler product over periodic orbits.     

Linearly stable orbits in 3 dimensional billiards

We construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart. It shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing Lyapunov exponents cannot be easily extended to 3 dimensions.Supported in part by the NSF Grant DMS-8807077 and the Sloan Foundation     

Phase space scars and quantum billiards

A semiclassical expression is derived for the spectral Wigner function of ergodic billiards in terms of a sum over contributions from classical periodic orbits. It represents a generalization of a similar formula by Berry, which does not immediately apply to billiard systems. These results are a natural generalization of Gutzwiller's trace formula for the density of states. Our theory clarifies the origin of scars in the eigenfunctions of billiard systems. However, in its present form, it is unable to predict what states will be dominated by individual periodic orbits. Finally, we compare some of the predictions of our theory with numerical results from the stadium. Within the limitations of numerical resolution, we find agreement between the two.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

Dynamical zeta functions for Artin''s billiard and the Venkov-Zograf factorization formula

Dynamical zeta functions are expected to relate the Schrödinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationship is exact in the case of surfaces of constant negative curvature. The recently found factorization of the Selberg zeta function for the modular surface is known to correspond to a decomposition of the Schrödinger operator's eigenfunctions into two sets obeying different boundary condition on Artin's billiard. Here we express zeta functions for Artin's billiard in terms of generalized transfer operators, providing thereby a new dynamical proof of the above interpretation of the factorization formula. This dynamical proof is then extended to the Artin-Venkov-Zograf formula for finite coverings of the modular surface.     

Soft Billiards with Corners

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are acceptable. The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a scattering function. We define such an asymptotic scattering function and prove the existence of it, explain how can it be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Hénon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.     

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Quantifying intermittency in the open drivebelt billiard

A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient lim(t→∞)tP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.     

Big islands in dispersing billiard-like potentials

We derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in smooth Hamiltonian approximations of scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, universality classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter ε→0, islands of polynomial size in ε appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect.     

Aharonov-Bohm effect in spherical billiard

Using Gutzwiller's periodic orbit theory, we study the quantum level density of a spherical billiard in the presence of a magnetic flux line added at its center, especially discuss the influence of the magnetic flux strength on the quantum level density. The Fourier transformed quantum level density of this system has allowed direct comparison between peaks in the level density and the length of the periodic orbits. For particular magnetic flux strength, the amplitude of the peaks in the level density decreased and some of the peaks disappeared. This result suggests that Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits. This phenomenon will provide a new experimental testing ground for exploring Aharonov-Bohm effect.