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1.
A pseudospectral scheme with centered time‐differencing for solving the Wigner function (WF) equation is investigated. Stability, second‐order accuracy in time, and spectral accuracy in space are proved for the WF equation with a potential in a periodic setting. In addition, normalization and energy conservation properties, and Ehrenfest's theorem are discussed. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 62–87, 2017 相似文献
2.
We formulate and solve the problem of finding a distribution function F(r,p,t) such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems. 相似文献
3.
Jacek Chmieliński 《Journal of Mathematical Analysis and Applications》2004,289(2):571-583
Investigating the stability of the Wigner equation where the class of its approximate solutions is defined by the inequality
4.
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.
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5.
Zhonggen SU 《Frontiers of Mathematics in China》2013,8(3):609-641
Let X n be a standard real symmetric (complex Hermitian) Wigner matrix, y 1, y 2, . . . , y n a sequence of independent real random variables independent of X n . Consider the deformed Wigner matrix H n,α = n -1/2X n + n -α /2 diag (y 1, . . . , y n ), where 0<α <1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform m n,α (z ) converges in probability to the corresponding Stieltjes transform m (z ). In this paper, we shall give the asymptotic estimate for the expectation Em n ,α (z ) and varianceVar(m n ,α (z )), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation. 相似文献
6.
We present a rigorous derivation of classical molecular dynamics (MD) from quantum molecular dynamics (QMD) that applies to the standard Hamiltonians of molecular physics with Coulomb interactions. The derivation is valid away from possible electronic eigenvalue crossings. 相似文献
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Jos B. Almeida Vasudevan Lakshminarayanan 《Journal of Computational and Applied Mathematics》2003,160(1-2):17-26
By combining the definition of the Wigner distribution function (WDF) and the matrix method of optical system modeling, we can evaluate the transformation of the former in centered systems with great complexity. The effect of stops and lens diameter are also considered and are shown to be responsible for nonlinear clipping of the resulting WDF in the case of coherent illumination and nonlinear modulation of the WDF when the illumination is incoherent. As an example, the study of a single lens imaging system illustrates the applicability of the method. 相似文献
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Gitta Kutyniok 《Journal of Mathematical Analysis and Applications》2003,277(2):589-608
In this paper we construct a general class of time-frequency representations for LCA groups which parallel Cohen's class for the real line. For this, we generalize the notion of ambiguity function and Wigner distribution to the setting of general LCA groups in such a way that the Plancherel transform of the ambiguity function coincides with the Wigner distribution. Furthermore, properties of the general ambiguity function and Wigner distribution are studied. In detail we characterize those groups whose ambiguity functions and Wigner distributions vanish at infinity or are square-integrable. Finally, we explicitly construct Cohen's class for the group of p-adic numbers, p prime. 相似文献
11.
Mark Fannes Dé nes Petz 《Proceedings of the American Mathematical Society》2003,131(7):1981-1988
Let be an arbitrary self-adjoint matrix and be an (random) Wigner matrix. We show that is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that is positive definite whenever the noncommutative random variables and are in free relation, with semicircular.
12.
V. P. Maslov 《Mathematical Notes》1999,66(6):701-715
We study the asymptotics of the spectrum for certain averaged operators related to the Liouville operator. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 849–866, December, 1999. 相似文献
13.
David Borwein Jonathan M. Borwein Armin Straub 《Journal of Mathematical Analysis and Applications》2014
Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective. 相似文献
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M. Nedjalkov I. Dimov P. Bordone R. Brunetti C. Jacoboni 《Mathematical and Computer Modelling》1997,25(12):33-53
The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions. 相似文献
16.
S.K. Giannopoulou 《Applicable analysis》2013,92(3):624-645
We develop a technique for uniformizing WKB functions which fail to correctly represent wave fields on caustics due to geometric singularities of ray fields. The uniformization technique is based on appropriate asymptotic surgery of the Wigner transform of the WKB functions, in different regions of the phase space. We present the details of the computation for the model example of the semiclassical Airy equation and we explain how the method can be applied to higher dimensional WKB functions for fold caustics 相似文献
17.
We consider ensembles of N×N Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as N tends to infinity. 相似文献
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19.
For a wide class of nonlinearities satisfying
we show that any nonnegative solution of the quasilinear equation over the entire must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.
0$\space in $(0,a)$\space and $f(u)<0$\space in $(a,\infty)$ ,}\end{displaymath}">
we show that any nonnegative solution of the quasilinear equation over the entire must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.
20.
曾岳生 《高校应用数学学报(A辑)》1991,6(3):365-369
本文求得了四维Euclid空间中liouville方程的依赖于两个任意复Fueter解析函数的一般解,并讨论了一般情形. 相似文献