Semiclassical trace formulas in the presence of continuous symmetries

Creagh, Stephen C.; Littlejohn, Robert G.

doi:10.1103/physreva.44.836

Cited by 118 publications

(187 citation statements)

References 25 publications

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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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Section: Introductionmentioning

confidence: 99%

Weyl Expansion for Symmetric Potentials

Lauritzen¹,

Whelan²

1995

Annals of Physics

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“…Second, in a system with one integral of motion in addition to the energy, the periodic orbits are not isolated, but appear in one-parameter families instead. Hence, we should use the modified trace formula [10] that involves the summation over the periodic orbit families (p.o.f. ):…”

Section: B Adiabatic Limit In the Extended Phase Spacementioning

confidence: 99%

Semiclassical theory of spin orbit interaction in the extended phase space

Pletyukhov

Zaitsev

2003

J. Phys. A: Math. Gen.

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“…where M β represents symmetry-reduced monodromy matrix [3], which characterizes linear stability of the periodic orbit. Varying external parameters in the Hamiltonian, each periodic orbit continuously changes its properties, and one of the eigenvalues of M β might eventually approaches 1.…”

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confidence: 99%

Periodic-orbit bifurcations as the origin of nuclear deformations

Arita¹

2006

Phys. Scr.

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Semiclassical analysis of shell structures in realistic nuclear potentials are presented using periodicorbit theory. We adopted r α potential model and examined classical-quantum correspondence using Fourier transformation technique. Spin-orbit coupling is also taken into account in the model Hamiltonian. Gross shell structure for a certain combination of surface diffuseness and spin-orbit parameters are investigated and its relation to pseudospin symmetry is discussed. Analysis of superdeformed shell structure in realistic model is also presented.PACS numbers: 21.60.-n, 03.65.Sq, 31.15.Gy SHELL STRUCTURE AND PERIODIC ORBITSNuclear deformations are intimately related with shell structures in single-particle energy spectra of deformed Hamiltonian. Using the semiclassical theory, quantum level density g(E) can be represented in terms of classical periodic orbits, and one obtains the trace formula[1, 2](1) g 0 (E) is average part of the level density, and the oscillating part is expressed as the sum over all periodic orbits β in corresponding classical Hamiltonian system. S β = β p · dr is action integral along the orbit β, and ν β is Maslov phase determined by the number of conjugate points along the orbit. In the trace formula, each periodic orbit contribution is an oscillatory function of energy since action S β (E) is an increasing function of E. The energy scale of this oscillation is given by δE ≈ 2π /T β . From this relation, one can see that short periodic orbits (having small T β ) is associated with gross structure (having large δE).The amplitude factor A β in the trace formula (1) has significant dependence on the stability of the orbit. In standard stationary-phase approximation, it is proportional to the stability factor;where M β represents symmetry-reduced monodromy matrix [3], which characterizes linear stability of the periodic orbit. Varying external parameters in the Hamiltonian, each periodic orbit continuously changes its properties, and one of the eigenvalues of M β might eventually approaches 1. At this point, a continuous family of quasi-periodic orbits appear in neighborhood of the periodic orbit β in direction to eigenvector belonging to the above unit eigenvalue of M β . It usually accompany the appearance of new periodic orbit from that local family (or, inversely, disappearance of another periodic orbit into the family), namely bifurcation of periodic orbit occur at this point. One should also note that this periodic-orbit bifurcation is related with the restoration of local dynamical symmetry in neighborhood of the periodic orbit. Since these quasi-periodic orbits make coherent contribution in the periodic orbit sum, we can expect a significant enhancement of shell effect in quantum level density. Actually, our previous works [4,5] clearly show that the bifurcations of short periodic orbits play essential role in emergence of gross shell structures. The divergence of stability factor (2) at bifurcation point is due to the breakdown of standard stationary-phase method, and can be remedi...

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Semiclassical trace formulas in the presence of continuous symmetries

Cited by 118 publications

References 25 publications

Weyl Expansion for Symmetric Potentials

Weyl Expansion for Symmetric Potentials

Semiclassical theory of spin orbit interaction in the extended phase space

Periodic-orbit bifurcations as the origin of nuclear deformations

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