Making Almost Commuting Matrices Commute

Suppose two Hermitian matrices A, B almost commute (\({\Vert [A,B] \Vert \leq \delta}\)). Are they close to a commuting pair of Hermitian matrices, A′, B′, with \({\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon}\) ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every \({\epsilon > 0}\) there exists a δ > 0, independent of the size N of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how δ depends on \({\epsilon}\) . We give uniform bounds relating δ and \({\epsilon}\) . The proof is constructive, giving an explicit algorithm to construct A′ and B′. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.     

Ruijsenaars' Commuting Difference Operators as Commuting Transfer Matrices

For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL=LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space. Received: 27 December 1995 / Accepted: 11 November 1996     

From Alternating Sign Matrices to the Gaussian Unitary Ensemble

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the 6-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.     

Spectral Analysis of Unitary Band Matrices

 This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only. Received: 29 April 2002 / Accepted: 7 August 2002 Published online: 20 January 2003 Communicated by B. Simon     

Level Compressibility of Certain Random Unitary Matrices

The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of this quantity for different random unitary matrices describing models with intermediate spectral statistics. The computations are based on the approach developed by G. Tanner for chaotic systems. The main ingredient of the method is the determination of eigenvalues of a transition matrix whose matrix elements equal the squared moduli of matrix elements of the initial unitary matrix. The principal result of the paper is the proof that the level compressibility of random unitary matrices derived from the exact quantisation of barrier billiards and consequently of barrier billiards themselves is equal to 1/2 irrespective of the height and the position of the barrier.     

Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

We provide an elementary proof for a theorem due to Petz and Réffy which states that for a random n × n unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) k × k submatrix converges in distribution, after multiplying by a normalization factor and as , to a matrix of independent complex Gaussian random variables with mean 0 and variance 1.      

Recurrence for Discrete Time Unitary Evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector ${\phi}$ . In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to ${\phi}$ . We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.     

Clustering of Periodic Orbits and Ensembles of Truncated Unitary Matrices

In present article we consider a combinatorial problem of counting and classification of periodic orbits in dynamical systems on an example of the baker’s map. Periodic orbits of a chaotic system can be organized into a set of clusters, where orbits from a given cluster traverse approximately the same points of the phase space but in a different time-order. We show that counting of cluster sizes in the baker’s map can be turned into a spectral problem for matrices from truncated unitary ensemble (TrUE). We formulate a conjecture of universality of the spectral edge in the eigenvalues distribution of TrUE and utilize it to derive asymptotics of the second moment of cluster distribution in the regime when both the orbit lengths and the parameter controlling closeness of the orbit actions tend to infinity. The result obtained allows to estimate the size of average cluster for various numbers of encounters in periodic orbit.     

A Factorization of Determinant Related to Some Random Matrices

We consider the expectation of the determinant det(–X)^–1for Im >0 associated with some random N×Nmatrices and factorize it into NStieltjes transforms of probability measures. Moreover, using this factorization, we investigate the limiting behavior of the logarithm of the quantity as N.     

Asymptotic Infinitesimal Freeness with Amalgamation for Haar Quantum Unitary Random Matrices

We consider the limiting distribution of U_NA_NU_N^*{U_NA_NU_N^*} and B _N (and more general expressions), where A _N and B _N are N × N matrices with entries in a unital C*-algebra B{\mathcal B} which have limiting B{\mathcal B}-valued distributions as N → ∞, and U _N is a N × N Haar distributed quantum unitary random matrix with entries independent from B{\mathcal B}. Under a boundedness assumption, we show that U_NA_NU_N^*{U_NA_NU_N^*} and B _N are asymptotically free with amalgamation over B{\mathcal B}. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko.     

Random Dirac Operators with Time Reversal Symmetry

Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness, are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO^*(2L), and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely continuous distribution.     

On the 1/n Expansion for Some Unitary Invariant Ensembles of Random Matrices

We present a version of the 1/n-expansion for random matrix ensembles known as matrix models. The case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated. In these cases we construct the expansion scheme for the Jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices, prove the asymptotic character of the scheme and give an explicit form of the first two terms. This allows us, in particular, to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices. We also find the asymptotic form of several related objects, such as smoothed squares of certain orthogonal polynomials, the normalized trace and the matrix elements of the resolvent of the Jacobi matrices, etc. Received: 9 November 2000 / Accepted: 26 July 2001     

Symmetry Properties of the k-Body Embedded Unitary Gaussian Ensemble of Random Matrices

We extend the recent study of the k-body embedded Gaussian ensembles by L. Benet, T. Rupp, and H. A. Weidenmüller (2001, Benet, Phys. Rev. Lett.87, 101601-1 and 2001, Ann. Phys. (N.Y.)292, 67) and by T. Asaga, L. Benet, T. Rupp, and H. A. Weidenmüller (cond-mat/0107363 and cond-mat/0107364). We show that central results of these papers can be derived directly from the symmetry properties of both the many-particle states and the random k-body interaction. We offer new insight into the structure of the matrix of second moments of the embedded ensemble and of the supersymmetry approach. We extend the concept of the embedded ensemble and define it purely group-theoretically.     

跷跷板模型中三阶中微子混合矩阵的幺正性破坏

对标准模型的一种简单扩充就是引入n个重的右手中微子且保持其SU(2)_L×U(1)_Y规范对称性. 通过对角化(3+n)×(3+n)阶中微子质量矩阵, 得到关于ν_e, ν_μ和ν_τ的有效 质量矩阵的精确的解析表达式. 结果表明, 在轻子带电弱流中出现的3×3中微子混合矩阵V必须不是严格幺正的. 如果通过跷跷板机制产生正确的轻的中微子的质量标度, 那么V的幺正性破坏的程度非常小, 几乎可以忽略. 类似的结论同样可以在第二类跷跷板模型中得到.     

时间反演过程中可逆性佯谬的消除

按目前量子力学中的定义，微观过程在时间反演下保持不变的条件是系统的哈密顿量满足关系 .证明了目前粒子物理学中电磁相互作用理论实际上不满足这个条件，由于至今的粒子物理实验都表明电磁相互作用在时间反演下是保持不变的，因此目前粒子物理学的时间反演变换理论必重建.作者提出一种更为合理的时间反演方案，按此方案，单一过程的电磁相互作用的跃迁几率对时间反演保持不变.但对非单一过程的宏观系统，由于各不同过程跃迁几率振幅的干涉效应，使得宏观过程时间反演的对称性被破坏，因此可以从根本上解决物理学上长期悬而未决的，孤立宏观系统演化过程不可逆但微观过程可逆的所谓可逆性佯谬问题.     

Geoacoustic Inversion Using Time Reversal of Ocean Noise

We present a passive geoacoustic inversion method using two hydrophones, which combines noise interferometry and time reversal mirror(TRIM) techniques. Numerical simulations are firstly performed, in which strong focusing occurs in the vicinity of one hydrophone when Green's function(GF) is back-propagated from the other hydrophone, with the position and strength of the focus being sensitive to sound speed and density in the bottom.We next extract the GF from the noise cross-correlation function measured by two hydrophones with 8025-m distance in the Shallow Water '06 experiment. After realizing the TRM process, sound speed and density in the bottom are inverted by optimizing focusing of the back-propagated GF. The passive inversion method is inherently environmentally friendly and low-cost.     

时变信道下的被动时间反转扩频水声通信

本文研究了时变信道条件下采用被动时间反转的直接序列扩频水声通信方案。多通道被动时间反转可通过对信道多径进行时间、空间聚焦实现信道匹配,但低信噪比、时变特性造成的信道特性失配对被动时间反转处理的性能造成严重影响。在垂直阵接收的基础上,本文采用码片级信道估计获取水声信道特性并进行周期性更新,并采用已判决码元产生的扩频码片作为信道估计训练序列,结合应用稀疏信道估计算法抑制零值抽头上的估计噪声,从而可有效改善时变、低信噪比条件下的被动时间反转处理的时、空多径聚焦效果,提高扩频通信性能。通过湖试实验比较了采用稀疏信道估计、传统信道估计算法的时反扩频接收机,以及经典直扩接收机的通信性能,实验结果表明：本文方案可在低信噪比获得较好的性能,并有效抑制时变信道对时反扩频通信性能的影响。