Discrete symmetries and the periodic-orbit expansions

Lauritzen, Bent

doi:10.1103/physreva.43.603

Cited by 49 publications

(54 citation statements)

References 12 publications

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“…In any event, it is trivial to see that the second line equals the first regardless of how it was derived. We now follow reference [23] and identify the first term as giving the contribution to the even states and the second term as giving the contribution to the odd states. The exponential factor in Eq.…”

Section: A2 Stability Factor Calculationmentioning

confidence: 99%

“…(3) comes from considering just the pure bounce orbit. In the presence of a reflection symmetry in y, this orbit lies on the symmetry axis and its amplitudes decomposes between the even and odd states in a specific way [23]. We start by defining the larger eigenvalue of W 0 appearing in Eq.…”

Section: A2 Stability Factor Calculationmentioning

confidence: 99%

“…(23). This requires that we evaluate the retarded Green's function connecting the initial position x ′ near Σ R to a point x ′′ under the barrier and…”

Section: The Semiclassical Green's Functionmentioning

confidence: 99%

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A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

Creagh

Whelan

1999

Annals of Physics

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It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.

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Section: A2 Stability Factor Calculationmentioning

confidence: 99%

Section: A2 Stability Factor Calculationmentioning

confidence: 99%

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A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

Creagh

Whelan

1999

Annals of Physics

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“…Thus, for the partial density ρ α of states belonging to the irreducible representation α of the symmetry group we write ρ α (E) ≈ρ α (E) + ρ α,osc (E) (2) where the average term is given by the truncated Weyl expansion. The oscillating term ρ α,osc has previously been treated for both discrete [3][4][5] and continuous [6,7] symmetries. To first order it yields a trace formula similar to the Gutzwiller trace formula but with the periodic orbits given in a reduced phase-space.…”

Section: Introductionmentioning

confidence: 99%

Weyl Expansion for Symmetric Potentials

Lauritzen¹,

Whelan²

1995

Annals of Physics

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We present a semiclassical expansion of the smooth part of the density of states in potentials with some form of symmetry. The density of states of each irreducible representation is separately evaluated using the Wigner transforms of the projection operators. For discrete symmetries the expansion yields a formally exact but asymptotic series inh, while for the rotational SO(n) symmetries the expansion requires averaging over angular momentum as well as energy. A numerical example is given in two dimensions, in which we calculate the leading terms of the Weyl expansion as well as the leading periodic orbit contributions to the symmetry reduced level density.

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“…This is particularly important for cycle expansions [19,20], for which a correct treatment of symmetry significantly accelerates convergence of the expansions. An uncoupled lattice has symmetry group S X × G X where X is the number of lattice sites (equal to two in this paper), S X is the symmetric group, of order X!, corresponding to permutations of the lattice points, and G is the symmetry group of the dynamics.…”

Section: Symmetrymentioning

confidence: 99%

Periodic orbit theory of two coupled Tchebyscheff maps

Dettmann

2004

Chaos, Solitons & Fractals

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Coupled map lattices have been widely used as models in several fields of physics, such as chaotic strings, turbulence, and phase transitions, as well as in other disciplines, such as biology (ecology, evolution) and information processing. This paper investigates properties of periodic orbits in two coupled Tchebyscheff maps. Then zeta function cycle expansions are used to compute dynamical averages appearing in Beck's theory of chaotic strings. The results show close agreement with direct simulation for most values of the coupling parameter, and yield information about the system complementary to that of direct simulation.

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Discrete symmetries and the periodic-orbit expansions

Cited by 49 publications

References 12 publications

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

Weyl Expansion for Symmetric Potentials

Periodic orbit theory of two coupled Tchebyscheff maps

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