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Waltner D, Richter K. Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales. Phys Rev E 2019; 100:042212. [PMID: 31770924 DOI: 10.1103/physreve.100.042212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2019] [Indexed: 06/10/2023]
Abstract
Despite considerable progress during the past decades in devising a semiclassical theory for classically chaotic quantum systems a quantitative semiclassical understanding of their dynamics at late times (beyond the so-called Heisenberg time T_{H}) is still missing. This challenge, corresponding to resolving spectral structures on energy scales below the mean level spacing, is intimately related to the quest for semiclassically restoring unitary quantum evolution. Guided through insights for quantum graphs we devise a periodic-orbit resummation procedure for spectra of quantum chaotic systems invoking periodic-orbit self-encounters as the structuring element of a hierarchical phase space dynamics. Quantum unitarity is reflected in real-valued spectral determinants with zeros giving discrete energy levels. We propose a way to purely semiclassically construct such real spectral determinants based on two major underlying mechanisms. (i) Complementary contributions to the spectral determinant from regrouped pseudo-orbits of duration T<T_{H} and T_{H}-T are complex conjugate to each other. (ii) Contributions from long periodic orbits involving multiple traversals along shorter orbits cancel out. We furthermore discuss implications for interacting N-particle quantum systems with a chaotic classical large-N limit that have recently attracted particular interest in the context of many-body quantum chaos.
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Affiliation(s)
- Daniel Waltner
- Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, D-47048 Duisburg, Germany
| | - Klaus Richter
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
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2
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Li J, Tomsovic S. Exact relations between homoclinic and periodic orbit actions in chaotic systems. Phys Rev E 2018; 97:022216. [PMID: 29548081 DOI: 10.1103/physreve.97.022216] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 11/07/2022]
Abstract
Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulas expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This enables an explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.
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Affiliation(s)
- Jizhou Li
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| | - Steven Tomsovic
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
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3
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Li J, Tomsovic S. Geometric determination of classical actions of heteroclinic and unstable periodic orbits. Phys Rev E 2017; 95:062224. [PMID: 28709367 DOI: 10.1103/physreve.95.062224] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2017] [Indexed: 06/07/2023]
Abstract
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase-space areas bounded by segments of stable and unstable manifolds and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following from the exact relations are given with error estimates. They arise from asymptotic scaling of certain bounded phase-space areas. The actions of infinite subsets of periodic orbits are determined by their periods and the locations of the limiting homoclinic points on which they accumulate.
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Affiliation(s)
- Jizhou Li
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
| | - Steven Tomsovic
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
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4
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King CG, Horsley SAR, Philbin TG. Perfect Transmission through Disordered Media. PHYSICAL REVIEW LETTERS 2017; 118:163201. [PMID: 28474922 DOI: 10.1103/physrevlett.118.163201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2016] [Indexed: 06/07/2023]
Abstract
The transmission of a wave through a randomly chosen "pile of plates" typically decreases exponentially with the number of plates, a phenomenon closely related to Anderson localization. In apparent contradiction, we construct disordered planar permittivity profiles which are complex valued (i.e., have reactive and dissipative properties) that appear to vary randomly with position, yet are one-way reflectionless for all angles of incidence and exhibit a transmission coefficient of unity. In addition to these complex-valued "random" planar permittivity profiles, we construct a family of real-valued, two-way reflectionless and perfectly transmitting disordered permittivity profiles that function only for a single angle of incidence and a narrow frequency range.
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Affiliation(s)
- C G King
- Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
| | - S A R Horsley
- Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
| | - T G Philbin
- Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
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5
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Engl T, Urbina JD, Richter K. Periodic mean-field solutions and the spectra of discrete bosonic fields: Trace formula for Bose-Hubbard models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062907. [PMID: 26764774 DOI: 10.1103/physreve.92.062907] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2015] [Indexed: 06/05/2023]
Abstract
We consider the many-body spectra of interacting bosonic quantum fields on a lattice in the semiclassical limit of large particle number N. We show that the many-body density of states can be expressed as a coherent sum over oscillating long-wavelength contributions given by periodic, nonperturbative solutions of the, typically nonlinear, wave equation of the classical (mean-field) limit. To this end, we construct the semiclassical approximation for both the smooth and oscillatory parts of the many-body density of states in terms of a trace formula starting from the exact path integral form of the propagator between many-body quadrature states. We therefore avoid the use of a complexified classical limit characteristic of the coherent state representation. While quantum effects such as vacuum fluctuations and gauge invariance are exactly accounted for, our semiclassical approach captures quantum interference and therefore is valid well beyond the Ehrenfest time where naive quantum-classical correspondence breaks down. Remarkably, due to a special feature of harmonic systems with incommensurable frequencies, our formulas are generically valid also in the free-field case of noninteracting bosons.
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Affiliation(s)
- Thomas Engl
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Juan Diego Urbina
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
| | - Klaus Richter
- Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
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6
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Fujii M, Yamashita K. Semiclassical quantization of nonadiabatic systems with hopping periodic orbits. J Chem Phys 2015; 142:074104. [PMID: 25701999 DOI: 10.1063/1.4907910] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems consisting of fast and slow degrees of freedom (DOFs) by extending Gutzwiller's trace formula to a nonadiabatic form. The quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow DOF, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels. In addition to the semiclassical quantization condition, we also discuss chaotic dynamics involved in the classical limit of nonadiabatic dynamics.
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Affiliation(s)
- Mikiya Fujii
- Department of Chemical System Engineering, School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
| | - Koichi Yamashita
- Department of Chemical System Engineering, School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
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7
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Barak H, Kay KG. Initial-value representation of the semiclassical zeta function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:012912. [PMID: 25679687 DOI: 10.1103/physreve.91.012912] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2014] [Indexed: 06/04/2023]
Abstract
This work casts the semiclassical zeta function in a form suitable for practical calculations of energy levels for rather general systems. To accomplish this, the zeta function is approximated by applying an initial-value representation (IVR) treatment to the traces of the transfer matrix that appear when the function is expanded in cumulants. Because this approach does not require searches for periodic orbits or special trajectories obeying double-ended boundary conditions, it is easily applicable to multidimensional systems with smooth potentials. Calculations are presented for the energy levels of three two-dimensional systems, including one that is classically integrable, one having mixed phase space, and one that is almost fully chaotic. The results show that the present treatment is far more numerically efficient than a previously proposed IVR method for the zeta function [Barak and Kay, Phys. Rev. E 88, 062926 (2013)]. The approach described here successfully resolves nearly all energy levels in the range investigated for the first two systems as well as energy levels in spectral regions that are not too highly congested for the highly chaotic system.
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Affiliation(s)
- Haim Barak
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Kenneth G Kay
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
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Barak H, Kay KG. Semiclassical initial-value representation of the transfer operator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062926. [PMID: 24483550 DOI: 10.1103/physreve.88.062926] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2013] [Indexed: 06/03/2023]
Abstract
The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.
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Affiliation(s)
- Haim Barak
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Kenneth G Kay
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
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9
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Waltner D, Gnutzmann S, Tanner G, Richter K. Subdeterminant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052919. [PMID: 23767610 DOI: 10.1103/physreve.87.052919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2013] [Indexed: 06/02/2023]
Abstract
We study the implications of unitarity for pseudo-orbit expansions of the spectral determinants of quantum maps and quantum graphs. In particular, we advocate to group pseudo-orbits into subdeterminants. We show explicitly that the cancellation of long orbits is elegantly described on this level and that unitarity can be built in using a simple subdeterminant identity which has a nontrivial interpretation in terms of pseudo-orbits. This identity yields much more detailed relations between pseudo-orbits of different lengths than was known previously. We reformulate Newton identities and the spectral density in terms of subdeterminant expansions and point out the implications of the subdeterminant identity for these expressions. We analyze furthermore the effect of the identity on spectral correlation functions such as the autocorrelation and parametric cross-correlation functions of the spectral determinant and the spectral form factor.
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Affiliation(s)
- Daniel Waltner
- Weizmann Institute of Science, Physics Department, Rehovot, Israel
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10
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Vergini EG. Semiclassical approach to long time propagation in quantum chaos: predicting scars. PHYSICAL REVIEW LETTERS 2012; 108:264101. [PMID: 23004984 DOI: 10.1103/physrevlett.108.264101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2012] [Indexed: 06/01/2023]
Abstract
We present two powerful semiclassical formulas for quantum systems with classically chaotic dynamics, one of them being the Fourier transform of the other. The first formula evaluates the autocorrelation function of a state constructed in the neighborhood of a short periodic orbit, where the propagation for times greater than the Ehrenfest time is computed through the contribution of homoclinic orbits. The second formula evaluates the square of the overlap of the proposed state with the eigenstates of the system, providing valuable information about the scarring phenomenon. Both expressions are successfully verified in the Bunimovich stadium billiard.
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Affiliation(s)
- Eduardo G Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica., Avenida del Libertador 8250, (C1429BNP) Buenos Aires, Argentina
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11
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Keating JP, Müller S. Resummation and the semiclassical theory of spectral statistics. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.0178] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random-matrix theory. To do so, we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits. This allows us to obtain both the non-oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework, thus overcoming a significant problem in previous approaches. In addition, our results extend beyond the universal regime to describe the system-specific approach to the semiclassical limit.
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Affiliation(s)
| | - Sebastian Müller
- Cavendish Laboratory, University of CambridgeJJ Thomson Avenue, Cambridge CB3 0HE, UK
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12
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Takahashi S, Takatsuka K. Phase quantization of chaos in the semiclassical regime. J Chem Phys 2007; 127:084112. [PMID: 17764234 DOI: 10.1063/1.2772274] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, nonconvergence in the trace formula, and so on. These difficulties have been hampering the progress of quantum chaos, and it is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This forms a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems. We show here that energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature. In this argument there is no distinction between integrability and nonintegrability of classical dynamics. Then we present numerical evidence that chaos can be indeed quantized by means of amplitude-free quasicorrelation functions and Heller's frozen Gaussian method. This is called phase quantization. Finally, we revisit the work of Yamashita and Takatsuka [Prog. Theor. Phys. Suppl. 161, 56 (2007)] who have shown explicitly that the semiclassical spectrum is quite insensitive to smooth modification (rescaling) of the amplitude factor. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.
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Affiliation(s)
- Satoshi Takahashi
- Department of Basic Science, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan.
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Takatsuka K, Takahashi S, Koh YW, Yamashita T. Energy quantization of chaos with the semiclassical phases alone. J Chem Phys 2007; 126:021104. [PMID: 17228932 DOI: 10.1063/1.2431178] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
The mechanism of energy quantization is studied for classical dynamics on a highly anharmonic potential, ranging from integrable, mixed, and chaotic motions. The quantum eigenstates (standing waves) are created by the phase factors (the action integrals and the Maslov index) irrespective of the integrability, when the amplitude factors are relatively slowly varying. Indeed we show numerically that the time Fourier transform of an approximate semiclassical correlation function in which the amplitude factors are totally removed reproduces the spectral positions (energy eigenvalues) accurately in chaotic regime. Quantization with the phase information alone brings about dramatic simplification to molecular science, since the amplitude factors in the lowest order semiclassical approximation diverge exponentially in a chaotic domain.
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Affiliation(s)
- Kazuo Takatsuka
- Department of Basic Sciences, Graduate School of Arts and Sciences, The University of Tokyo, Komaba, 153-8902, Tokyo, Japan
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Teramoto H, Takatsuka K. A semiclassical theory for nonseparable rovibrational motions in curved space and its application to energy quantization of nonrigid molecules. J Chem Phys 2006; 125:194301. [PMID: 17129097 DOI: 10.1063/1.2372759] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The nonseparability of vibrational and rotational motions of a nonrigid molecule placed in the rotationally isotropic space induces several important effects on the dynamics of intramolecular energy flow and chemical reaction. However, most of these studies have been performed within the framework of classical mechanics. We present a semiclassical theory for the motions of such nonrigid molecules and apply to the energy quantization of three body atomic cluster. It is shown numerically that the semiclassical spectum given without the correct account of the rotational symmetry suffers from unnecessary broadening of the resultant spectral lines and moreover from spurious peaks.
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Affiliation(s)
- Hiroshi Teramoto
- Department of Basic Science, Graduate School of Arts and Science, University of Tokyo, Komaba, 153-8902 Tokyo, Japan
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Ushiyama H, Takatsuka K. Extended quantization condition for constructive and destructive interferences and trajectories dominating molecular vibrational eigenstates. J Chem Phys 2005; 122:224112. [PMID: 15974656 DOI: 10.1063/1.1924388] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The role of destructive quantum interference in semiclassical quantization of molecular vibrational states is studied. This aspect is crucial for correct quantization, since failure in the appropriate treatment of destructive interference quite often results in many spurious peaks and broad background to hide the true peaks. We first study the time-Fourier transform of the autocorrelation function without performing summation over the trajectories. The resultant quantity, the prespectrum which is a function of individual classical trajectories, provides a clear view about how destructive interference among the trajectories should function. It turns out that the prespectrum is oscillatory but never a random noise. On the contrary, it bears a systematic and regular structure, which is sometimes characterized in terms of very sharp and high peaks in the energy space of the sampled classical trajectories. We have found an extended quantization condition that is responsible for generating these peaks in the prespectrum, which we call the prior quantization condition. Integration of the prespectrum over the trajectory space is supposed to give "zero" (practically a small value of the order of the Planck constant) at a noneigenvalue energy, which is actually a materialization of the destructive interference. Besides, certain finite peaks in the prespectrum survive after the integration to form the true spikes (eigenvalues) in the final spectrum, if they satisfy an additional resonance condition. For these resonance components, the prior quantization condition is reduced to the Einstein-Brillouin-Keller quantization condition. Based on these analyses, we propose a rather conventional filtering technique to efficiently handle tedious computation for destructive interference, and numerically verify that it works well even for multidimensional chaotic systems. This filtering technique is further utilized to extract a few trajectories that dominate an eigenstate of molecular vibration.
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Affiliation(s)
- Hiroshi Ushiyama
- Department of Basic Science, Graduate School of Arts and Sciences, University of Tokyo, Komaba, 153-8902, Tokyo, Japan.
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Hotta K, Takatsuka K. Semiclassical quantization of chaos in terms of an amplitude-free quasi-correlation function. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/17/306] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Prange RE. Resurgence in quasiclassical scattering. PHYSICAL REVIEW LETTERS 2003; 90:070401. [PMID: 12633221 DOI: 10.1103/physrevlett.90.070401] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2002] [Indexed: 05/24/2023]
Abstract
In quasiclassical spectral theory, "resurgence" means that long periodic orbits can be expressed by short ones in such a way that the spectral determinant is real. The question has thus long been posed whether long scattering orbits can be expressed by short orbits in such a way as to make the quasiclassical scattering matrix unitary. We here find a resurgent and manifestly Hermitean expression for Wigner's R matrix, implying a unitary scattering matrix. The result is particularly important if the average resonance width is comparable with the average resonance spacing.
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Affiliation(s)
- R E Prange
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Street 38, D-01187 Dresden, Germany
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Keppeler S. Level spacings and periodic orbits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:027201. [PMID: 11497747 DOI: 10.1103/physreve.64.027201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2001] [Indexed: 05/23/2023]
Abstract
Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this representation for the nearest-neighbor level spacing (k=0). In a second part, we present an asymptotic evaluation for large spacings, where consistency with random matrix theory is achieved for large k. We also discuss the relation with the method of Bogomolny and Keating [Phys. Rev. Lett. 77, 1472 (1996)] for two-point correlations.
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Affiliation(s)
- S Keppeler
- Abteilung Theoretische Physik, Universität Ulm, Albert-Einstein-Allee 11, Germany.
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21
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Main J, Dando PA, Belkic D, Taylor HS. Decimation and harmonic inversion of periodic orbit signals. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0305-4470/33/6/311] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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22
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Dahlqvist P. Error of semiclassical eigenvalues in the semiclassical limit - an asymptotic analysis of the Sinai billiard. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/42/307] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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23
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Harayama T, Shudo A. Periodic orbits and semiclassical quantization of dispersing billiards. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/25/17/019] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Prosen T, Robnik M. Failure of semiclassical methods to predict individual energy levels. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/2/003] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Carvalho TOD, Aguiar MAMD. Eigenfunctions of the Liouville operator, periodic orbits and the principle of uniformity. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/29/13/026] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Agam O, Fishman S. Quantum eigenfunctions in terms of periodic orbits of chaotic systems. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/9/010] [Citation(s) in RCA: 52] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Abstract
Integrals involving exp { –
k
f
(
z
)}, where |
k
| is a large parameter and the contour passes through a saddle of
f
(
z
), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ∊
2.386
, where ∊ (proportional to exp (—
A
│
k
│) where
A
is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.
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32
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Abstract
By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line
s
= ½ + i
t
(
t
real), a family of exact representations, parametrized by a real variable
K
, is found for the real function
Z
(
t
) = ζ(½ + i
t
) exp {iθ(
t
)}, where θ is real. The dominant contribution
Z
0
(
t,K
) is a convergent sum over the integers
n
of the Dirichlet series, resembling the finite ‘main sum ’ of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections
Z
3
(
t,K
),
Z
4
(
t,K
)... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large
K
,
Z
0
contains not only the main sum of RS but also its first correction. An estimate of high orders
m
≫ 1 when
K
<
t
1/6
shows that the corrections
Z
k
have the ‘factorial/power ’ form familiar in divergent asymptotic expansions, the least term being of order exp { ─½
K
2
t
}. Graphical and numerical exploration of the new representation shows that
Z
0
is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections
Z
3
and
Z
4
give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate
Z
0
). The accuracy increases with
K
, as do the numbers of terms in the sums for each of the
Z
m
. By regarding Planck’s constant
h
as a complex variable, the method for
Z
(
t
) can be applied directly to semiclassical approximations for spectral determinants ∆(
E, h
) whose zeros
E
=
E
j
(
h
) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.
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33
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Abstract
A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calculate the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic properties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.
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34
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Weiper FJ, Ankerhold J, Grabert H, Pollak E. Semiclassical Quantization of Nonseparable Systems Without Periodic Orbits. PHYSICAL REVIEW LETTERS 1996; 77:2662-2665. [PMID: 10062014 DOI: 10.1103/physrevlett.77.2662] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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35
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Qiu Y, Müller J, Burgdörfer J. Periodic-orbit spectra of hydrogen and helium. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1996; 54:1922-1935. [PMID: 9913680 DOI: 10.1103/physreva.54.1922] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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36
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Biechele P, Goodings DA, Lefebvre JH. Energy spectra of the hydrogen atom and the harmonic oscillator in two dimensions from Bogomolny's semiclassical transfer operator. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:3198-3208. [PMID: 9964627 DOI: 10.1103/physreve.53.3198] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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37
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Argaman N. Semiclassical analysis of the quantum interference corrections to the conductance of mesoscopic systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:7035-7054. [PMID: 9982149 DOI: 10.1103/physrevb.53.7035] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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38
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Kaplan L, Heller EJ. Overcoming the wall in the semiclassical Baker's map. PHYSICAL REVIEW LETTERS 1996; 76:1453-1456. [PMID: 10061727 DOI: 10.1103/physrevlett.76.1453] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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39
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Vattay G, Rosenqvist PE. Periodic orbit quantization beyond the semiclassical theory. PHYSICAL REVIEW LETTERS 1996; 76:335-339. [PMID: 10061431 DOI: 10.1103/physrevlett.76.335] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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40
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Reimann SM, Brack M, Magner AG, Blaschke J, Murthy MV. Circular quantum billiard with a singular magnetic flux line. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1996; 53:39-48. [PMID: 9912858 DOI: 10.1103/physreva.53.39] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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41
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Takami T. Semiclassical study of avoided crossings. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:2434-2442. [PMID: 9963686 DOI: 10.1103/physreve.52.2434] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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42
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Creagh SC. Quantum zeta function for perturbed cat maps. CHAOS (WOODBURY, N.Y.) 1995; 5:477-493. [PMID: 12780203 DOI: 10.1063/1.166119] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The behavior of semiclassical approximations to the spectra of perturbed quantum cat maps is examined as the perturbation parameter brings the corresponding classical system into the nonhyperbolic regime. The approximations are initially accurate but large errors are found to appear in the traces and in the coefficients of the characteristic polynomial after nonhyperbolic structures appear. Nevertheless, the eigenvalues obtained from them remain accurate up to large perturbations. (c) 1995 American Institute of Physics.
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43
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Georgeot B, Prange RE. Exact and quasiclassical Fredholm solutions of quantum billiards. PHYSICAL REVIEW LETTERS 1995; 74:2851-2854. [PMID: 10058042 DOI: 10.1103/physrevlett.74.2851] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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44
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Shudo A, Ikeda KS. Complex Classical Trajectories and Chaotic Tunneling. PHYSICAL REVIEW LETTERS 1995; 74:682-685. [PMID: 10058821 DOI: 10.1103/physrevlett.74.682] [Citation(s) in RCA: 53] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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45
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Tomsovic S, Ullmo D. Chaos-assisted tunneling. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:145-162. [PMID: 9961952 DOI: 10.1103/physreve.50.145] [Citation(s) in RCA: 107] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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46
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Shudo A, Shimizu Y, Seba P, Stein J, Stöckmann HJ, Zyczkowski K. Statistical properties of spectra of pseudointegrable systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3748-3756. [PMID: 9961661 DOI: 10.1103/physreve.49.3748] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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47
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Szeredi T, Goodings DA. Classical and quantum chaos of the wedge billiard. II. Quantum mechanics and quantization rules. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:3529-3544. [PMID: 9961011 DOI: 10.1103/physreve.48.3529] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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48
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Szeredi T, Goodings DA. Classical and quantum chaos of the wedge billiard. I. Classical mechanics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:3518-3528. [PMID: 9961010 DOI: 10.1103/physreve.48.3518] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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49
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Dietz B, Smilansky U. A scattering approach to the quantization of billiards- The inside-outside duality. CHAOS (WOODBURY, N.Y.) 1993; 3:581-589. [PMID: 12780063 DOI: 10.1063/1.165962] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We present some recent results on the semiclassical quantization of billiards using an approach which is based on the strong link between the billiard interior and exterior problems. That is, the spectrum of the interior problem is extracted from the scattering matrix of the exterior problem. Once this is put on a rigorous basis, the semiclassical approximation is used to derive the semiclassical zeta function and the spectral density. The duality between the inside and outside problems prevails also in the classical description and offers new insight into this quantization procedure. The relation between the present approach and the more standard quantization methods is also discussed and illustrated with some numerical results.
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Affiliation(s)
- Barbara Dietz
- Department of Nuclear Physics, The Weizmann Institute of Science, 76100 Rehovot, Israel
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50
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Cvitanovic P, Rosenqvist PE, Vattay G, Rugh HH. A Fredholm determinant for semiclassical quantization. CHAOS (WOODBURY, N.Y.) 1993; 3:619-636. [PMID: 12780067 DOI: 10.1063/1.165992] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We investigate a new type of approximation to quantum determinants, the "quantum Fredholm determinant," and test numerically the conjecture that for Axiom A hyperbolic flows such determinants have a larger domain of analyticity and better convergence than the Gutzwiller-Voros zeta functions derived from the Gutzwiller trace formula. The conjecture is supported by numerical investigations of the 3-disk repeller, a normal-form model of a flow, and a model 2-D map.
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Affiliation(s)
- Predrag Cvitanovic
- Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, DenmarkUnite de Mathematiques Pures et Appliquees, Ecole Normale Superieure de Lyon 46, Allee d'Italie, F-69364 Lyon, France
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