The Wave Front Set of the Wigner Distribution and Instantaneous Frequency

We prove a formula expressing the gradient of the phase function of a function f:ℝ ^d ↦ℂ as a normalized first frequency moment of the Wigner distribution for fixed time. The formula holds when f is the Fourier transform of a distribution of compact support, or when f belongs to a Sobolev space H ^d/2+1+ε (ℝ ^d ) where ε>0. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.     

Weyl transforms associated with the Hankel transform in Clifford analysis

In this paper, we define the Hankel–Wigner transform in Clifford analysis and therefore define the corresponding Weyl transform. We present some properties of this kind of Hankel–Wigner transform, and then give the criteria of the boundedness of the Weyl transform and compactness on the L^p space. Copyright © 2005 John Wiley & Sons, Ltd.     

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a “quasi-probability density” on ℝ² which may take negative values and must satisfy intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.      

Bulk universality for Wigner matrices

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e^?U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C⁶( \input amssym $\Bbb R$ ) with at most polynomially growing derivatives and ν(x) ≥ Ce^?C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-frequency propagation for the Schrödinger equation on the torus

The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schrödinger equation on the standard d-dimensional torus Td. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigner distributions of the initial data. We characterize them in terms of a new object, the resonant Wigner distribution, which describes high-frequency effects associated to the fraction of the energy of the sequence of initial data that concentrates around the set of resonant frequencies in phase-space T^*Td. This construction is related to that of the so-called two-microlocal semiclassical measures. We prove that any limit μ of the Wigner distributions corresponding to solutions to the Schrödinger equation on the torus is completely determined by the limits of both the Wigner distribution and the resonant Wigner distribution of the initial data; moreover, μ follows a propagation law described by a family of density-matrix Schrödinger equations on the periodic geodesics of Td. Finally, we present some connections with the study of the dispersive behavior of the Schrödinger flow (in particular, with Strichartz estimates). Among these, we show that the limits of sequences of position densities of solutions to the Schrödinger equation on T² are absolutely continuous with respect to the Lebesgue measure.     

Weyl Pseudo-Differential Operator and Wigner Transform on the Poincaré Disk

The purpose of this paper is to investigate some relations between the kernel of a Weyl pseudo-differential operator and the Wigner transform on Poincaré disk defined in our previous paper [11]. The composition formula for the class of the operators defined in [11] has not been proved yet. However, some properties and relations, which are analogous to the Euclidean case, between the Weyl pseudo-differential operator and the Wigner transform have been investigated in [11]. In the present paper, an asymptotic formula for the Wigner transform of the kernel of a Weyl pseudo-differential operator as 0 is given. We also introduce a space of functions on the cotangent bundle T ^* D whose definition is based on the notion of the Schwartz space on the Poincaré disk. For an S ¹-invariant symbol in that space, we obtain a formula to reproduce the symbol from the kernel of the Weyl pseudo-differential operator.     

On the Spectral Distribution of Gaussian Random Matrices

We consider the empirical spectral distribution （ESD） of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotaeh approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O（n^-1） to the Wigner distribution function uniformly on every compact intervals [u,v] within the limiting support （-1, 1）. Furthermore, the variance of the ESD for such an interval is proved to be （πn）^-2 logn asymptotically which surprisingly enough, does not depend on the details （e.g. length or location） of the interval, This property allows us to determine completely the covariance function between the values of the ESD on two intervals.     

Weyl Transforms,the Heat Kernel and Green Function of a Degenerate Elliptic Operator

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ² related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L²(ℝ²). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e^−tL, t > 0, are given. Communicated by B.-W. Schulze (Potsdam) Mathematics Subject Classifications (2000): 47G30, 47E05.     

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) is a quadratic transform that takes an oscillatory function u (x): ℝⁿ ↦ ℂ^d to a phase-space density W (x, k) = W [u ](x, k): ℝ²ⁿ ↦ ℂ^{d ×d} , resolving it over an additional set of 'wavenumber' variables. The WT and its variations have been heavily used in quantum mechanics, semiconductors, homogenization of wave equations, timefrequency analysis, signal processing, pseudodifferential operators etc. The WT however has a fundamental difficulty: WTs exhibit artifacts, collectively known as ‘interference terms’, and can be arbitrarily more complicated than the original wavefunction. A very successful, well established way to go around this is using the Wigner measures (WMs), a semiclassical approximation to the WT. We propose a different approach, namely smoothing the WT with an appropriate kernel. Such smoothed WTs (SWTs) have been used with great success in signal processing. They have not been used in the treatment of PDEs, a fundamental obstacle being the lack of exact equations governing their evolution. We present the machinery which allows the coarse-scale reformulation of a broad class of wave problems in terms of the SWT, along with numerical experiments which clearly show the validity and applicability of the method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ?^d ? Sp (d, ?), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d² + 2d, whereas if H ? Sp (d, R), then dim H ≤ d² + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universality of random matrices and local relaxation flow

Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N ^−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

Representations of SO(3) and angular polyspectra

We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere S²={(x,y,z):x²+y²+z²=1}. Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch–Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the Cosmic Microwave Background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and “simulation” of Clebsch–Gordan coefficients.     

Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

Let I ^* and I ^⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici–Pata bijection between I ^* and I ^⊞ . The class type W of symmetric distributions in I ^⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ^⊞ and the free counterpart under Λ of the positive distributions in I ^* is established. It is shown that the class type W does not include all symmetric distributions in I ^⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I ^*. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.     

Rate of convergence to the semi-circular law

A stochastic bound of order O _P (n ^–1/2 ) for the Kolmogorov distance between the spectral distribution function of an n×n matrix from Wigner ensemble and the distribution function of the semi-circular law is obtained. The result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded.Research supported by the DFG-Forschergruppe FOR 399/1-1 ``Spektrale Analyse, Asymptotische Verteilungen und Stochastische Dynamiken'.Research supported by the DFG-Forschergruppe FOR 399/1-1 ``Spektrale Analyse, Asymptotische Verteilungen und Stochastische Dynamiken'.Partially supported by Russian Foundation for Fundamental Research Grants NN02-01-00233, 00-15-96019. Partially supported by INTAS N99-01317, DFG-RFBR N99-01-04027. Mathematics Subject Classification (2000): 60F05     

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

We give a formula for the one-parameter strongly continuous semigroups ${e^{-tL^{\lambda}}}We give a formula for the one-parameter strongly continuous semigroups e^-tLl{e^{-tL^{\lambda}}} and e^{-t [(A)\tilde]}{e^{-t \tilde{A}}}, t > 0 generated by the generalized Hermite operator L^l, l ? R\{0}{L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}} respectively by the generalized Landau operator ?. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L ²(R ²ⁿ ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L ² estimate for the solutions of initial value problems for the heat equations governed by L ^λ respectively ?, in terms of L ^p norm, 1 ≤ p ≤ ∞ of the initial data are given.     

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j Symbols, and Character Localization

In this paper, we employ a technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index J) of generic Wigner matrix elements D^J_MM￠(g){D^{J}_{MM'}(g)} . We use this result to derive asymptotic formulae for the character χ ^J (g) of an SU(2) group element and for Wigner’s 3j symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for χ ^J (g) is in fact exact. The result hints at a “Duistermaat-Heckman like” localization property for discrete sums.     

Asymptotics of the density matrix for a system of a large number of identical particles

The Wigner equation is considered for a system of a large numberN of identical particles with interaction factor of the order of 1/N. In both the Bose and the Fermi cases, we construct the asymptotics of the solution of the Cauchy problem for this equation with regard to the exchange effect for the case in which the Planck constant is of the order ofN ^−1/d , whered is the space dimension. This asymptotics is interpreted in terms of the theory of the complex germ on a curved phase space equivalent to the space of functions with values on the Riemann sphere in the Fermi case and on the Lobachevskii plane in the Bose case. The classical equations of motion in both cases are reduced to the Vlasov equation; since the phase space is infinite-dimensional, the complex germ is subjected to additional conditions depending on the type of statistics. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 84–106, January, 1999.     

The Favoured Solutions of the Moment Problem

The sequence of Catalan numbers is known to be the moments of the Wigner law. A characterization of this sequence is given in terms of some determinants and the moment sequence of the q-Gaussian law is characterized in a similar fashion.