@STRING{JPA = {J. Phys. A}} @STRING{PRE = {Phys. Rev. E}} @STRING{PRL = {Phys. Rev. Lett.}} @ARTICLE{BaeKetLoeRobVidHoeKuhSto2008, AUTHOR = {A. B\"acker and R. Ketzmerick and S. L\"ock and M. Robnik and G. Vidmar and R. H\"ohmann and U. Kuhl and H.-J. St\"ockmann}, TITLE = {Dynamical tunneling in mushroom billiards}, JOURNAL = PRL, VOLUME = {100}, PAGES = {174103}, YEAR = {2008}, NOTE = {}, CHECKED = {}, URL = {http://link.aps.org/abstract/PRL/v100/e174103}, ABSTRACT = {We study the fundamental question of dynamical tunneling in generic two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling rates. Experimentally, we use microwave spectra to investigate a mushroom billiard with adjustable foot height. Numerically, we obtain tunneling rates from high precision eigenvalues using the improved method of particular solutions. Analytically, a prediction is given by extending an approach using a fictitious integrable system to billiards. In contrast to previous approaches for billiards, we find agreement with experimental and numerical data without any free parameter.}, } @ARTICLE{TomUllBae2008, AUTHOR = {S. Tomsovic and D. Ullmo and A. B\"acker }, TITLE = {Residual Coulomb interaction fluctuations in chaotic systems: the boundary, random plane waves, and semiclassical theory}, URL = {http://link.aps.org/abstract/PRL/v100/e164101}, NOTE = {}, ANNOTE = {}, JOURNAL = PRL, PAGES = {164101} VOLUME = {100}, YEAR = {2008}, wasURL = {http://arxiv.org/abs/0712.0225}, wasJOURNAL = {arXiv:0712.0225v1 [cond-mat.mes-hall]}, wasYEAR = {2007}, ABSTRACT = {New fluctuation properties arise in problems where both spatial integration and energy summation are necessary ingredients. The quintessential example is given by the short-range approximation to the first order ground state contribution of the residual Coulomb interaction. The dominant features come from the region near the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates. Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as compared to Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly to significant errors, which can be corrected within a purely semiclassical approach. }, } @ARTICLE{BaeKetLoeSch2008, AUTHOR = {A. B\"acker and R. Ketzmerick and S. L\"ock and L. Schilling}, TITLE = {Regular-to-chaotic tunneling rates using a fictitious integrable system}, JOURNAL = PRL, VOLUME = {100}, PAGES = {104101}, YEAR = {2008}, NOTE = {}, ANNOTE = {}, CHECKED = {}, URL = {http://link.aps.org/abstract/PRL/v100/e104101}, ABSTRACT = {We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.}, } @ARTICLE{BaeKetMon2007, AUTHOR = {A. B\"acker and R. Ketzmerick and A. Monastra}, TITLE = {Universality in the flooding of regular islands by chaotic states}, JOURNAL = PRE, VOLUME = {75}, PAGES = {066204 (11 pages)}, YEAR = {2007}, NOTE = {}, ANNOTE = {}, URL = {http://link.aps.org/abstract/PRE/v75/e066204}, ABSTRACT = { We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.}, } @ARTICLE{Bae2007b, AUTHOR = {A. B\"acker}, TITLE = {Quantum Chaos in Billiards}, JOURNAL = {Computing in Science \& Engineering}, % VOLUME = {9}, % Annote = {May/June 2007 (Vol. 9, No. 3) }, NOTE = {vol. 9, no. 3, May/June 2007, pp. 60--64}, % PAGES = {60-64}, % YEAR = {2007}, URL = {}, ABSTRACT = {}, } @ARTICLE{Bae2007a, AUTHOR = {A. B\"acker}, TITLE = {Computational Physics Education with Python}, JOURNAL = {Computing in Science \& Engineering}, NOTE = {}, % VOLUME = {9}, % Annote = {May/June 2007 (Vol. 9, No. 3) }, % NOTE = {May/June}, % PAGES = {30-33}, % YEAR = {2007}, NOTE = {vol. 9, no. 3, May/June 2007, pp. 30--63}, URL = {}, ABSTRACT = {}, } @ARTICLE{FeiBaeKetRotHucBur2006, AUTHOR = {J. Feist and A. B\"acker and R. Ketzmerick and S. Rotter and B. Huckestein and J. Burgd\"orfer}, TITLE = {Nano-wires with surface disorder: Giant localization lengths and quantum-to-classical crossover}, JOURNAL = PRL, VOLUME = {97}, PAGES = {116804 (4 pages)}, YEAR = {2006}, NOTE = {}, ANNOTE = {}, CHECKED = {}, HTTP = {http://link.aps.org/abstract/PRL/v97/e116804}, ABSTRACT = {We investigate electronic quantum transport through nano--wires with one--sided surface roughness. A magnetic field perpendicular to the scattering region is shown to lead to exponentially diverging localization lengths in the quantum--to--classical crossover regime. This effect can be quantitatively accounted for by tunneling between the regular and the chaotic components of the underlying mixed classical phase space. }, } @ARTICLE{BaeKetMon2005, AUTHOR = {A. B\"acker and R. Ketzmerick and A. Monastra}, TITLE = {Flooding of regular islands by chaotic states }, JOURNAL = PRL, VOLUME = {94}, PAGES = {054102 (4 pages)}, YEAR = {2005}, NOTE = {}, ANNOTE = {}, URL = {http://link.aps.org/abstract/PRL/v94/e054102}, ABSTRACT = { We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect can be observed for island chains with just 10 islands.}, } @ARTICLE{BaeFueSch2004, AUTHOR = {A. B\"acker and S. F\"urstberger and R. Schubert}, TITLE = {Poincar\'e Husimi representation of eigenstates in quantum billiards}, JOURNAL = PRE, VOLUME = {70}, PAGES = {036204 (10 pages)}, YEAR = {2004}, URL = {http://link.aps.org/abstract/PRE/v70/e036204}, ABSTRACT = {For the representation of eigenstates on a Poincar\'e section at the boundary of a billiard different variants have been proposed. We compare these Poincar\'e Husimi functions, discuss their properties and based on this select one particularly suited definition. For the mean behaviour of these Poincar\'e Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincar\'e Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincar\'e Husimi functions in the case of ergodic systems is shown. }, } @ARTICLE{BaeFueSchSte2002, AUTHOR = {A. B\"acker and S. F\"urstberger and R. Schubert and F. Steiner}, TITLE = {Behaviour of boundary functions for quantum billiards}, JOURNAL = JPA, VOLUME = {35}, PAGES = {10293-10310}, YEAR = {2002}, URL = {http://stacks.iop.org/0305-4470/35/10293}, ABSTRACT = {We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore they form a reduced representation of the quantum system, analogous to the Poincar\'e section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different lima\c{c}on billiards and the circle billiard, and good agreement is found. Furthermore we derive an asymptotic completeness relation for the boundary functions. }, } @ARTICLE{BaeKeaPra2002, AUTHOR = {A. B\"acker and {J. P.} Keating and {S. D.} Prado}, TITLE = {Orbit bifurcations and wavefunction autocorrelations}, JOURNAL = {Nonlinearity}, VOLUME = {15}, PAGES = {1417-1433}, YEAR = {2002}, URL = {http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp02-3.ps.gz}, ABSTRACT = {It was recently shown (Keating \& Prado, {\it Proc. R. Soc. Lond. A} {\bf 457}, 1855-1872 (2001)) that, in the semiclassical limit, the scarring of quantum eigenfunctions by classical periodic orbits in chaotic systems may be dramatically enhanced when the orbits in question undergo bifurcation. Specifically, a bifurcating orbit gives rise to a scar with an amplitude that scales as $h{\alpha}$ and a width that scales as $\hbar^{\omega}$, where $\alpha$ and $\omega$ are bifurcation-dependent scar exponents whose values are typically smaller than those ($\alpha=\omega=1/2$) associated with isolated and unstable periodic orbits. We here analyze the influence of bifurcations on the autocorrelation function of quantum eigenstates, averaged with respect to energy. It is shown that the length-scale of the correlations around a bifurcating orbit scales semiclassically as $\hbar^{1-\alpha}$, where $\alpha$ is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps.}, } @ARTICLE{Bae2003, AUTHOR = {A. B\"acker}, TITLE = {Numerical aspects of eigenvalues and eigenfunctions of chaotic quantum systems}, JOURNAL = {in: {\it The Mathematical Aspects of Quantum Chaos I}, M. Degli Esposti and S. Graffi (Eds.), Springer Lecture Notes in Physics {\bf 618}}, VOLUME = {}, PAGES = {91--144}, YEAR = {2003}, URL = {http://www.springer.de/cgi-bin/search_book.pl?isbn=3-540-02623-1}, ABSTRACT = {We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two--dimensional area--preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area--preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigenvectors and compare them with the corresponding random matrix distributions. The Husimi representation allows for a direct comparison of the localization of the eigenstates in phase space with the corresponding classical structures. Examples for a perturbed cat map and the standard map with different parameters are shown. Billiard systems and the corresponding quantum billiards are another important class of systems (which are also relevant to applications, for example in mesoscopic physics). We provide a detailed exposition of the boundary integral method, which is one important method to determine the eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several methods to determine the eigenvalues from the Fredholm equation and illustrate them for the stadium billiard. The occurrence of spurious solutions is discussed in detail and illustrated for the circular billiard, the stadium billiard, and the annular sector billiard. We emphasize the role of the normal derivative function to compute the normalization of eigenfunctions, momentum representations or autocorrelation functions in a very efficient and direct way. Some examples for these quantities are given and discussed. }, } @ARTICLE{BaeManHucKet2002, AUTHOR = {A. B\"acker and A. Manze and B. Huckestein and R. Ketzmerick}, TITLE = {Isolated resonances in conductance fluctuations and hierarchical states}, JOURNAL = PRE, VOLUME = {66}, PAGES = {016211 (8 pages)}, YEAR = {2002}, URL = {http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp02-1.ps.gz}, ABSTRACT = {We study the isolated resonances occurring in conductance fluctuations of quantum systems with a classically mixed phase space. We demonstrate that the isolated resonances and the resonant scattering states can be associated to eigenstates of the closed system. They can all be categorized as hierarchical or regular, depending on where the corresponding eigenstates live in the classical phase space. }, } @ARTICLE{BaeSch2002b, AUTHOR = {A. B\"acker and R. Schubert}, TITLE = {Autocorrelation function of eigenstates in chaotic and mixed systems}, JOURNAL = JPA, VOLUME = {35}, PAGES = {539-564}, YEAR = {2002}, URL = {http://stacks.iop.org/0305-4470/35/539}, ABSTRACT = {We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation length. For localized states, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales. }, } @ARTICLE{BaeSch2002a, AUTHOR = {A. B\"acker and R. Schubert}, TITLE = {Amplitude distribution of eigenfunctions in mixed systems}, JOURNAL = JPA, VOLUME = {35}, PAGES = {527-538}, YEAR = {2002}, URL = {http://stacks.iop.org/0305-4470/35/527}, ABSTRACT = {We study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space. For an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and good agreement with numerical computations for the family of lima\c{c}on billiards is found. The natural extension of our result to more general systems, e.g. with a potential, is also discussed. }, } @ARTICLE{DulBae2001, AUTHOR = {H. R. Dullin and A. B\"acker}, TITLE = {About ergodicity in the family of lima\c{c}on billiards}, JOURNAL = {Nonlinearity}, VOLUME = {14}, PAGES = {1673-1687}, YEAR = {2001}, URL = {http://stacks.iop.org/0951-7715/14/1673}, ABSTRACT = {By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of lima\c{c}on billiards. The statistics of these bifurcations show that the sizes of the stable intervals decrease with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle node bifurcations. This shows explicitly that if in this one parameter family of maps ergodicity occurs for more than one parameter the set of these parameter values has a complicated structure. }, } @ARTICLE{BaeSte2001, AUTHOR = {A. B\"acker and F. Steiner}, TITLE = {Quantum chaos and quantum ergodicity}, JOURNAL = {in {\it Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems}, B. Fiedler (ed.), 717--752, Springer-Verlag Berlin/Heidelberg}, VOLUME = {}, PAGES = {}, YEAR = {2001}, URL = {http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-5.ps.gz}, ABSTRACT = {We report on some of our results which have been achieved within the {\it DFG Schwerpunktprogramm ``Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme'' (1994-2000)}. One main point of our research programme was the search for universal statistical properties of energy spectra and eigenfunctions of quantum mechanical systems whose classical dynamics is chaotic. The mode-fluctuation distribution $P(W)$ has been proposed as a universal signature of quantum chaos and a conjecture on its limit distribution has been put forward. The conjecture turns out to be mathematically equivalent to a hypothesis on the value distribution of dynamical zeta functions on the critical line and has been successfully tested for several chaotic systems. For certain systems this can be expressed in terms of the Selberg zeta function. For a large class of ergodic systems the quantum ergodicity theorem holds, which (roughly speaking) states that almost all eigenfunctions become equidistributed in the semiclassical limit. Particular attention has been paid to the question of subsequences of exceptional, non-quantum ergodic eigenfunctions, and their counting function. Such eigenfunctions are for example bouncing-ball modes occurring in billiards with two parallel walls (like the stadium or the Sinai billiard). Also ``scarred'' eigenfunctions showing localization along unstable periodic orbits could give rise to a non-quantum ergodic subsequence of eigenfunctions. Furthermore, the rate by which the classical limit is approached has been studied.
We conclude by giving a short summary of the other topics studied by our group within the Schwerpunktprogramm. }, } @ARTICLE{BaeHaa99, AUTHOR = {A. B\"acker and G. Haag}, TITLE = {Spectral statistics for quantized skew translations on the torus,}, JOURNAL = JPA, VOLUME = {32}, PAGES = {L393-L398}, YEAR = {1999}, URL = {http://stacks.iop.org/0305-4470/32/L393}, ABSTRACT = {We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level-spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.}, } @ARTICLE{BaeHaa99, AUTHOR = {A. B\"acker and G. Haag}, TITLE = {Spectral statistics for quantized skew translations on the torus,}, JOURNAL = JPA, VOLUME = {32}, PAGES = {L393-L398}, YEAR = {1999}, URL = {http://stacks.iop.org/0305-4470/32/L393}, ABSTRACT = {We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level-spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.}, } @ARTICLE{BaeSch99, AUTHOR = {A. B\"acker and R. Schubert}, TITLE = {Chaotic eigenfunctions in momentum space}, JOURNAL = JPA, VOLUME = {32}, PAGES = {4795-4815}, YEAR = {1999}, URL = {http://stacks.iop.org/0305-4470/32/4795}, ABSTRACT = {We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigenstates, such as scars, or bouncing ball modes. We also find examples, where the localization is more strongly pronounced in position space than in momentum space, which we discuss in detail. Finally applications and generalizations are discussed. }, } @ARTICLE{AurBaeSchTag2000, AUTHOR = {R. Aurich and A. B\"acker and R. Schubert and M. Taglieber}, TITLE = {Maximum norms of chaotic quantum eigenstates and random waves}, JOURNAL = {in: {\it Equadiff 99}, Proceedings of the International Conference on Differential Equations, Berlin, Germany 1--7 August 1999 B. Fiedler, K. Gr\"oger and J. Sprekels (eds.), World Scientific, Singapore}, PAGES = {1461--1463}, YEAR = {2000}, ABSTRACT = {The growth of the maximum norms of quantum eigenstates with increasing energy is studied for classically chaotic systems. The maximum norms provide a measure for localization effects in eigenfunctions. For the maxima of superpositions of random functions an upper bound can be derived and is found to be in good agreement with the numerical results for the eigenfunctions of chaotic quantum billiards.}, } @ARTICLE{AurBaeSchTag99, AUTHOR = {R. Aurich and A. B\"acker and R. Schubert and M. Taglieber}, TITLE = {Maximum norms of chaotic quantum eigenstates and random waves}, JOURNAL = {Physica D}, VOLUME = {129}, PAGES = {1--14}, YEAR = {1999}, ABSTRACT = {The growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated. The maximum norms provide a measure for localization effects in eigenfunctions. An upper bound for the maxima of random superpositions of random functions is derived. For the random-wave model this gives the bound $c \sqrt{\ln E}$ in the semiclassical limit $E\to \infty$. The growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic. The maximum norms of these systems are numerically shown to be conform with the random-wave model. Furthermore, the distribution of the locations of the maximum norms is discussed. }, } @ARTICLE{AltBaeDemGraHofRehRic98, AUTHOR = {H. Alt and A. B\"acker and C. Dembowski and H.-D. Gr\"af and R. Hofferbert and H. Rehfeld and A. Richter}, TITLE = {Mode fluctuation distribution for spectra of superconducting microwave billiards}, JOURNAL = PRE, VOLUME = {58}, PAGES = {1737-1742}, YEAR = {1998}, URL = {http://publish.aps.org/abstract/PRE/v58/p1737}, ABSTRACT = {High resolution eigenvalue spectra of several two- and three-dimensional superconducting microwave cavities have been measured in the frequency range below 20 GHz and analyzed using a statistical measure which is given by the distribution of the normalized mode fluctuations. For chaotic systems the limit distribution is conjectured to show a universal Gaussian, whereas integrable systems should exhibit a non-Gaussian limit distribution. For the investigated Bunimovich stadium and the 3D-Sinai billiard we find that the distribution is in good agreement with this prediction. We also study members of the family of lima\c{c}on billiards, having mixed dynamics. It turns out that in this case the number of approximately 1000 eigenvalues for each billiard does not allow to observe significant deviations from a Gaussian. }, } @ARTICLE{BaeSchSti98, AUTHOR = {A. B\"acker and R. Schubert and P. Stifter}, TITLE = {Rate of quantum ergodicity in {E}uclidean billiards}, JOURNAL = PRE, VOLUME = {57}, PAGES = {5425-5447}, YEAR = {1998}, NOTE = {erratum ibid. {\bf 58} (1998) 5192}, URL = {http://publish.aps.org/abstract/PRE/v57/p5425}, ABSTRACT = {For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdière and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.}, } @ARTICLE{BaeSchSti97, AUTHOR = {A. B\"acker and R. Schubert and P. Stifter}, TITLE = {On the number of bouncing-ball modes in billiards}, JOURNAL = JPA, VOLUME = {30}, PAGES = {6783--6795}, YEAR = {1997}, URL = {http://stacks.iop.org/0305-4470/30/6783}, ABSTRACT = {We study the number of bouncing ball modes $N_{\text{bb}}(E)$ in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically $N_{\text{bb}}(E) \sim \alpha E^\delta$ for $E\to\infty$, where $\delta\in ]\frac{1}{2},1[$ depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to $\delta=1$, which corresponds to the leading term in Weyl's law for the mean behaviour of the counting function of eigenstates. This result is discussed in the context of quantum ergodicity. We compare the theoretical results with the numerically determined counting function $N_{\text{bb}}(E)$ for the stadium billiard and the cosine billiard and find good agreement. }, } @ARTICLE{BaeChe98, AUTHOR = {A. B\"acker and N. Chernov}, TITLE = {Generating partitions for two-dimensional hyperbolic maps}, JOURNAL = {Nonlinearity}, VOLUME = {11}, PAGES = {79--87}, YEAR = {1998}, URL = {http://stacks.iop.org/0951-7715/11/79}, ABSTRACT = {For a class of two-dimensional hyperbolic maps (which includes certain billiard systems) we construct finite generating partitions. Thus, trajectories of the map can be labelled uniquely by doubly infinite symbol sequences, where the symbols correspond to the atoms of the partition. It is shown that the corresponding conditions are fulfilled in the case of the cardioid billiard, the stadium billiard (and other Bunimovich billiards), planar dispersing and semidispersing billiards. }, } @ARTICLE{AurBaeSte97, AUTHOR = {R. Aurich and A. B\"acker and F. Steiner}, TITLE = {Mode fluctuations as fingerprints of chaotic and non-chaotic systems}, JOURNAL = {Int. J. Mod. Phys. B}, VOLUME = {11}, PAGES = {805--849}, YEAR = {1997}, URL = {http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp96-2.html}, ABSTRACT = {The mode-fluctuation distribution $P(W)$ is studied for chaotic as well as for non-chaotic quantum billiards. This statistic is discussed in the broader framework of the $E(k,L)$ functions being the probability of finding $k$ energy levels in a randomly chosen interval of length $L$, and the distribution of $n(L)$, where $n(L)$ is the number of levels in such an interval, and their cumulants $c_k(L)$. It is demonstrated that the cumulants provide a possible measure for the distinction between chaotic and non-chaotic systems. The vanishing of the normalized cumulants $C_k$, $k\geq 3$, implies a Gaussian behaviour of $P(W)$, which is realized in the case of chaotic systems, whereas non-chaotic systems display non-vanishing values for these cumulants leading to a non-Gaussian behaviour of $P(W)$. For some integrable systems there exist rigorous proofs of the non-Gaussian behaviour which are also discussed. Our numerical results and the rigorous results for integrable systems suggest that a clear fingerprint of chaotic systems is provided by a Gaussian distribution of the mode-fluctuation distribution $P(W)$. }, } @ARTICLE{BaeDul97, AUTHOR = {A. B\"acker and H. R. Dullin}, TITLE = {Symbolic dynamics and periodic orbits for the cardioid billiard}, JOURNAL = JPA, VOLUME = {30}, PAGES = {1991--2020}, YEAR = {1997}, URL = {http://stacks.iop.org/0305-4470/30/1991}, ABSTRACT = {The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topologically well ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov - Sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. Furthermore, the statistical properties of periodic orbits are investigated. }, } @ARTICLE{BaeSteSti95, AUTHOR = {A. B\"acker and F. Steiner and P. Stifter}, TITLE = {Spectral statistics in the quantized cardioid billiard}, JOURNAL = PRE, VOLUME = {52}, PAGES = {2463--2472}, YEAR = {1995}, URL = {http://prola.aps.org/abstract/PRE/v52/i3/p2463_1}, ABSTRACT = {The spectral statistics of the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the level-spacing distribution is in good agreement with the GOE distribution of random-matrix theory. In case of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosine-modulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits instead of exclusively Gaussian peaks. }, } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @Misc{Bae95:Dip, AUTHOR = {A. B\"acker}, NOTE = {Diploma thesis, II. Institut f\"ur Theoretische Physik, Universit\"at Hamburg}, TITLE = {Spektrale {S}tatistiken des quantisierten {K}ardioid-{B}illards}, YEAR = {1995}, ABSTRACT = {In dieser Arbeit werden spektrale Statistiken des quantisierten Kardioid-Billards studiert. F\"ur das Kardioid-Billard ist mathematisch bewiesen, da{\ss} es ein stark chaotisches System ist, so da"s es sich ideal f"ur die Untersuchung einer m\"oglichen Manifestation des klassischen Chaos in der Quantenmechanik eignet. Die Untersuchung basiert auf den jeweils ersten 11000 quantenmechanischen Energieeigenwerten f\"ur gerade und ungerade Symmetrie. Betrachtet werden unter anderem die Level-Spacing-Verteilung, die Number-Variance, die Rigidity und die $E(k,L)$-Statistiken. Die Ergebnisse werden mit den Erwartungen des Modells der Zufallsmatrizen verglichen. Des weiteren wird die Wahrscheinlichkeitsverteilung des (geeignet normierten) fluktuierenden Anteils der spektralen Stufenfunktion untersucht. F\"ur beide Symmetrieklassen ergibt sich sehr gute \"Ubereinstimmung mit einer Gau{\ss}chen Normalverteilung, die im Fall klassisch (stark) chaotischer Systeme erwartet wird. Es wird eine Methode zur Bestimmung periodischer Orbits des Kardioid-Billards angegeben. Die periodischen Orbits sind die Grundlage zur Anwendung der Gutzwillerschen Spurformel, die den Zusammenhang zwischen klassischen und quantenmechanischen Gr\"o\"sen herstellt. Insbesondere wird mit Hilfe der Spur des Kosinus-modulierten Heat-Kernels die Bestimmung von geometrischen Eigenschaften der periodischen Orbits aus der Kenntnis der quantenmechanischen Eigenwerte untersucht. }, } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @PhdThesis{Bae98:PhD, AUTHOR = {A. B\"acker}, SCHOOL = {Abteilung Theo\-re\-ti\-sche Physik, Universit\"at Ulm}, TITLE = {Classical and Quantum Chaos in Billiards}, YEAR = {1998}, ABSTRACT = {In this work we study so--called billiard systems as model systems for classical and quantum chaos. They are given by the free motion of a point particle inside some (compact) domain with elastic reflections at the boundary. The types of motion occurring in billiards range from integrable (i.e.\ regular) to strongly chaotic. Of particular importance for both the classical and the quantized billiards are the periodic orbits which contain much information about the system, as they cover the phase space densely (for certain systems) and can be used to approximate orbits of the system. A systematic enumeration of the periodic orbits for a given system is only possible if a symbolic dynamics is available. One important example for a strongly chaotic (i.e.\ ergodic, mixing, $K$-- and Bernoulli system) billiard is the cardioid billiard, for which a binary symbolic dynamics is introduced. The corresponding partition of the Poincar\'e section is mapped to a topologically well-ordered symbol plane. In the symbol plane the pruning front, separating allowed and forbidden symbol sequences, is obtained from orbits running either into or through the cusp of the cardioid. With the symbolic dynamics the periodic orbits can be enumerated systematically, and we give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. Furthermore, we investigate the statistical properties of periodic orbits. More generally, a theorem for the construction of finite generating partitions (i.e.\ symbolic dynamics) for a class of two-dimensional hyperbolic maps (which includes certain billiard systems) is proven. Applied to the cardioid billiard the theorem gives a proof of the binary symbolic dynamics and for the stadium billiard we obtain (with some additional considerations) a proof of a previously conjectured symbolic dynamics. The quantum analogue of the classical billiards are given by the stationary Schr\"odinger equation with appropriate boundary conditions. An important topic in quantum chaos is the investigation of the statistics of eigenvalues. Here we study the mode--fluctuation distribution $P(W)$ which is the distribution of the suitably normalized fluctuating part of the spectral staircase function, for the cardioid billiard, the stadium billiard and some integrable systems. For this statistics it has been conjectured that strongly chaotic systems show a Gaussian limit distribution, whereas integrable systems exhibit a non-Gaussian limit distribution. Our computations give clear numerical evidence in favour of the Gaussian limit distribution for chaotic systems. The connection between the classical and the quantized systems is provided by the Gutzwiller trace formula in terms of the periodic orbits of the classical system. For the cardioid billiard we study the inverse problem of quantum chaos with the periodic orbits determined using the symbolic dynamics, i.e.\ to conclude from the quantum mechanical eigenvalue spectrum on the properties of the classical length spectrum. The periodic orbits are also used for obtaining semiclassical approximations to the eigenvalues for the cardioid billiard using the Gutzwiller trace formula. Here we use the cosine quantization to obtain semiclassical approximations to the eigenvalues, whose accuracy is compared with the exact eigenvalues. One important result on the properties of eigenfunctions in ergodic systems is the quantum ergodicity theorem, which (roughly speaking) states that almost all eigenfunctions become equidistributed in the semiclassical limit. An example for a subsequence of eigenfunctions which do not become equidistributed is given by the so--called bouncing--ball modes, which occur for example in the stadium billiard. For this system we study the counting function numerically and find good agreement with analytical results. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied for three billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or so--called scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution. A measure for the strength of the localization of eigenfunctions is given by their $L^\infty$--norm, which is the maximal value of the modulus of the normalized eigenfunction. For the cardioid billiard and two stadium billiards we study the $L^\infty$--norms using up to $6\,000$ eigenfunctions and compare these results with an analytic upper bound obtained from the random wave model. We also study the distribution of the locations of the maximal values. Furthermore we give a discussion of the relation between the random wave model, the amplitude distribution of eigenfunctions and the quantum ergodicity theorem. In systems with mixed phase space a typical phenomenon are bifurcations of periodic orbits, if a parameter characterizing the systems is varied. As a prototypical system we investigate the family of lima\c{c}on billiards given by a deformation of the circle billiard, where the cardioid billiard is the limiting case. For this family we study the bifurcations, the spectral statistics and the influence of the bifurcations on the quantum mechanical energy spectrum. We also investigate some aspects of eigenfunctions in these systems. }, }