A Fredholm determinant for semiclassical quantization

Cvitanović, Predrag; Rosenqvist, Per E.; Vattay, Gábor; Rugh, Hans Henrik

doi:10.1063/1.165992

Cited by 43 publications

(85 citation statements)

References 29 publications

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“…Unlike scattering systems quantised with geometric orbits [3,5,13,19], here there are no subleading resonances, a fact also observed in Refs. [5,7].…”

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confidence: 51%

Semiclassical Quantization Using Diffractive Orbits

Whelan¹

1996

Phys. Rev. Lett.

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Diffraction, in the context of semiclassical mechanics, describes the manner in which quantum mechanics smooths over discontinuities in the classical mechanics. An important example is a billiard with sharp corners; its semiclassical quantisation requires the inclusion of diffractive periodic orbits in addition to classical periodic orbits. In this paper we construct the corresponding zeta function and apply it to a scattering problem which has only diffractive periodic orbits. We find that the resonances are accurately given by the zeros of the diffractive zeta function.

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“…Unlike scattering systems quantised with geometric orbits [3,5,13,19], here there are no subleading resonances, a fact also observed in Refs. [5,7].…”

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confidence: 51%

Semiclassical Quantization Using Diffractive Orbits

Whelan¹

1996

Phys. Rev. Lett.

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“…"=g"""n"e'"~( )t"/( det(M" -1))', where P p labels the primitive periodic orbits with n"bounces from the boundary, r is the number of repetitions, M" is the rth power of the monodromy matrix for the primitive orbit, and the action Sp = pL p is the momentum times the length of the orbit. The Dg is [8,15] Z(E) = P"op (1 -A 't~e't~t "). (Such a simple expression for Z is only possible if the eigenvalues of Mp are A", A ', i.e.…”

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confidence: 99%

Exact and Quasiclassical Fredholm Solutions of Quantum Billiards

Georgeot

Prange

1995

Phys. Rev. Lett.

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Quantum billiards are much studied as perhaps the simplest case which presents the central difficulty that the quasiclassical approximation is expressed as a divergent series. We find here, using the Fredholm method, an exact Green's function for billiards expressed as a ratio of absolutely convergent series. We make the quasiclassical approximation to this ratio. The method provides a convergence argument for previous results and an extension of results obtained for the spectrum to the full Green's function.It has been more than twenty years [1] since physicists realized that there is much we have failed to understand in the simplest quantum problems which are not exactly solvable. Among the questions studied are correlations between energy levels and the degree of pseudorandomness of the levels. Similar questions can be asked of wave functions, matrix elements, and scattering amplitudes. Attempts are made to classify the quantum problems, in particular, by the nature of the classical limit, e.g. , whether the classical limit is chaotic [2].Of these problems, doubtless the most thoroughly studied is that of the wave mechanics of a real time independent nonseparable two-dimensional potential. Two further specializations are also commonly made: First (in spite of much evidence that the wave functions are very interesting [3]), the study is restricted to the spectrum, and second, attention is confined to billiards, that is, systems for which the motion is free in the interior, and for which the essential problem is posed by the imposition of conditions on the boundary. Some celebrated billiards are the Sinai billiard, the Bunimovich stadium billiard, and billiards on spaces of negative curvature. Billiards gain

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“…However, if either the requirement of at University of Michigan on March 13, 2015 http://ptps.oxfordjournals.org/ Downloaded from hyperbolicity, or of the finite subshift, or both are not fulfilled, the spectral determinant is no longer an entire function, with delicate consequences for the spectra of evolution operators. 17), 19) We are far from having a global theory for non-hyperbolic dynamics; partial answers can, however, be given relating isolated features in the dynamics to spectral properties of the Perron-Frobenius operator. We will focus on a special case of non-hyperbolicity in what follows, that is, intermittent maps with complete symbolic dynamics and a single marginal fixed point.…”

Section: §2 Perron-frobenius Operators and Cycle Expansionsmentioning

confidence: 97%

Cycle Expansions for Intermittent Maps

Artuso

Cvitanović

Tanner

2003

Prog. Theor. Phys. Suppl.

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In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays. §1. IntroductionIn fluid dynamics one often observes long periods of regular dynamics (laminar phases) interrupted by irregular chaotic bursts, with the distribution of laminar phase intervals well described by a power law. This phenomenon is called intermittency, 1), 2) and it is a very general aspect of dynamics, a shadow cast by nonhyperbolic and marginally stable phase space regions. Complete hyperbolicity is indeed the exception rather than the rule: almost any dynamical system of interest exhibits a mixed phase space where islands of stability or regular regions coexist with hyperbolic regions with dynamics mixing exponentially fast. The trajectories on the border between chaos and regular dynamics are marginally stable, and trajectories from chaotic regions which come close to marginally stable regions can stay 'glued' there for arbitrarily long times. These intervals of regular motion are interrupted by irregular bursts as the trajectory is re-injected into the chaotic part of the phase space. How the trajectories are precisely 'glued' to the marginally stable region is often hard to describe. What coarsely looks like a border of an island in a Hamiltonian system with a mixed phase space, will under magnification dissolve into infinities of island chains of decreasing sizes and cantori.3)The existence of marginal or nearly marginal orbits is due to incomplete intersections of stable and unstable manifolds in a Smale horseshoe type dynamics. Following * )

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A Fredholm determinant for semiclassical quantization

Cited by 43 publications

References 29 publications

Semiclassical Quantization Using Diffractive Orbits

Semiclassical Quantization Using Diffractive Orbits

Exact and Quasiclassical Fredholm Solutions of Quantum Billiards

Cycle Expansions for Intermittent Maps

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