Spherical Functions on a Semisimple Lie Group, I

Harish-Chandra,

doi:10.2307/2372786

Cited by 371 publications

(156 citation statements)

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“…The angular average over the unitary supergroup can be performed by using the supersymmetric generalization [18,19,20] of the Harish Chandra Itzykson Zuber integral [22,23]. We define a second Hermitian supermatrix v &1 rv of the type (3.7) where v is superunitary and r diagonal.…”

Section: Graded Eigenvalue Methodsmentioning

confidence: 99%

Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Guhr

1996

Annals of Physics

122

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The transition from arbitrary to chaotic fluctuation properties in quantum systems is studied in a random matrix model. It is assumed that the Hamiltonian can be written as the sum of an arbitrary part and a chaos-producing part. The Gaussian ensembles are used to model the chaotic part. A closed integral representation for all the correlations in the case of broken time reversal invariance is derived by employing supersymmetry and the graded eigenvalue method. In particular, the two level correlation function is expressed as a double integral. For a fixed correlation index, exact diffusion equations are derived for all three universality classes which describe the transition to the chaotic regime from arbitrary initial conditions. As an application, the transition from Poisson regularity to chaos is discussed. The two level correlation function becomes a double integral for all values of the transition parameter.

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Section: Graded Eigenvalue Methodsmentioning

confidence: 99%

Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Guhr

1996

Annals of Physics

122

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“…There is no analytic, closed-form solution to this integral, although an exact determinantal expression was derived when averaging over the unitary group U(N) [35,36]. An asymptotic representation can be obtained when β ≫ 1.…”

Section: Joint Distribution Of the Eigenvaluesmentioning

confidence: 99%

Statistical properties of the linear tidal shear

2008

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Large-scale structures originate from coherent motions induced by inhomogeneities in the primeval gravitational potential. Here, we investigate the two-point statistics of the second derivative of the potential, the tidal shear, under the assumption of Gaussianity. We derive an exact closed form expression for the angular averaged, two-point distribution of the shear components which is valid for an arbitrary Lagrangian separation. This result is used to write down the two-point statistics of the shear eigenvalues in compact form. Next, we examine the large-scale asymptotics of the correlation of the shear eigenvalues and the alignment of the principal axes. The analytic results are in good agreement with measurements obtained from random realizations of the gravitational potential. Finally, we show that a number of two-point distributions of the shear eigenvalues are well approximated by Gaussian bivariates over a wide range of separation and smoothing scales. We speculate that the Gaussian approximation also holds for multiple point distributions of the shear eigenvalues. It is hoped that these results will be relevant for studies aimed at describing the properties of the (evolved) matter distribution in terms of the statistics of the primordial shear field.

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“…Ce résultat est à comparer avec l'expression classique ( [7], p. 291; [24], vol. II, p. 326) de la fonction d'Harish-Chandra pour la paire symétrique non compacte (Go, K') :…”

Section: Jn-unclassified

Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact

Lassalle¹

1978

Ann. Sci. École Norm. Sup.

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Spherical Functions on a Semisimple Lie Group, I

Cited by 371 publications

References 0 publications

Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Transitions toward Quantum Chaos: With Supersymmetry from Poisson to Gauss

Statistical properties of the linear tidal shear

Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact

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