Weyl Expansion for Symmetric Potentials

Lauritzen, Bent; Whelan, Niall D.

doi:10.1006/aphy.1995.1109

Cited by 14 publications

(26 citation statements)

References 18 publications

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“…The first thing which is apparent is that there is a great distinction between the A 1 and B 1 states compared to the It is also interesting to numerically isolate the contributions from the various classes and compare them to (2.15) directly as done in Ref. [12]. This is a simple exercise since the entries in the character table are components of a unitary matrix which is readily inverted.…”

Section: Numerical Comparison Of Two Dimensional Resultsmentioning

confidence: 99%

Symmetry decomposition of potentials with channels

Whelan¹

1997

J. Phys. A: Math. Gen.

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We discuss the symmetry decomposition of the average density of states for the two dimensional potential V = x 2 y 2 and its three dimensional generalisation V = x 2 y 2 + y 2 z 2 + z 2 x 2 . In both problems, the energetically accessible phase space is non-compact due to the existence of infinite channels along the axes. It is known that in two dimensions the phase space volume is infinite in these channels thus yielding non-standard forms for the average density of states. Here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels, leading to terms which are essentially leading order. We verify these results numerically and also observe a peculiar numerical effect which we associate with the channels. In three dimensions, the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials -however there are still non-generic effects related to some of the group elements.

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Section: Numerical Comparison Of Two Dimensional Resultsmentioning

confidence: 99%

Symmetry decomposition of potentials with channels

Whelan¹

1997

J. Phys. A: Math. Gen.

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“…Our aim is to calculate the behaviour of the form factor, a defined for a given such desymmetrized spectrum by (1) where (...)E denotes an energy average, dace) = (da(E»)E is the corresponding mean density (Lauritzen & Whelan 1995), n a is the symmetry-related level degeneracy, and the normalization is chosen so that Ka(T) --71 as T --700. We do this by substituting in the semiclassical trace formula for d a (Robbins 1989, Lauritzen 1991, Cvitanovic & Eckhardt 1993 (2)…”

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confidence: 99%

Discrete symmetries and spectral statistics

Keating

Robbins

1997

J. Phys. A: Math. Gen.

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q~antum chaos; discrete s~etries; spectral statistics; random matrix theory We calculate the 2-point spectral statistics associated with a given irreduciole representation (i.e. symmetry c~ass) for time-reversal. invari~t systems possessmg discrete symmetries usmg semiclassical perioilic orbit theory. When the representation m question is real or pseudoreal, our results conform to those of the Gaussian Orthogonal Ensemble (GOE) of random matrices. When it is complex, we find instead Gaussian Unitary Ensemble (GUE) behaviour. This provides a direct semiclassical explanation for the recent observation by Leyvraz et aI (1996) of GUE correlations in the desYJllffietrized spectra of certain symmetric billiards in the absence of any time-reversal invariance breaking (e.g. magnetic) fields. Internal Accession Date Only

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“…The wrapped fundamental domain in this example is the restriction of the billiard to a sector of central angle φ with identification of points along its two bordering half-lines. As it is equal to a cone, this produces non-trivial wave propagation at the origin which gives rise to additional corrections in the level densitȳ ̺ sym (E) of symmetric states [28]. Analogue to that, mapping a bosonic system to its wrapped fundamental domain implies non-trivial wave propagation in the vicinity of the symmetry planes.…”

Section: Non-interacting Undistinguishable Particlesmentioning

confidence: 99%

The Weyl expansion for systems of independent identical particles

Hummel¹,

Urbina²,

Richter³

2013

J. Phys. A: Math. Theor.

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Abstract. We present a novel analytical approach for the calculation of the mean density of states in many-body systems made of confined indistinguishable and noninteracting particles. Our method makes explicit the intrinsic geometry inherent in the symmetrization postulate and, in the spirit of the usual Weyl expansion for the smooth part of the density of states in single-particle confined systems, our results take the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under elements of the group of permutations. Being asymptotic, our approximation gives increasingly better results for large excitation energies and we formally confirm that it coincides with the celebrated Bethe estimate in the appropriate region. Moreover, our construction gives the correct high energy asymptotics expected from general considerations, and shows that the emergence of the fermionic ground state is actually a consequence of an extremely delicate large cancellation effect. Remarkably, our expansion in cluster zones is naturally incorporated for systems of interacting particles, opening the road to address the fundamental problem about the interplay between confinement and interactions in many-body systems of identical particles.PACS numbers: 74.20.Fg, 75.10.Jm, 71.10.Li, 73.21.La A geometrical approach to the mean density of states in many-body quantum systems 2

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Weyl Expansion for Symmetric Potentials

Cited by 14 publications

References 18 publications

Symmetry decomposition of potentials with channels

Symmetry decomposition of potentials with channels

Discrete symmetries and spectral statistics

The Weyl expansion for systems of independent identical particles

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