Journal Article10.1070/SM1967V001N04ABEH001994
Distribution of eigenvalues for some sets of random matrices
V A Marčenko,Leonid Pastur +1 more
TL;DR: In this article, the authors studied the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices in the energy spectra of disordered systems.
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Abstract: In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space ///v, a selfadjoint operator BN (re) of the form
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Citations
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Central Limit Theorem for Linear Spectral Statistics of Large Dimensional Kendall's Rank Correlation Matrices and its Applications.
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TL;DR: In this paper, the central limit theorem was established for the linear spectral statistics of the Kendall's rank correlation matrices under the Marchenko-Pastur asymptotic regime, in which the dimension diverges to infinity proportionally with the sample size.
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TL;DR: In this paper, the distribution function of the frequencies of normal modes of vibration of a disordered chain of one-dimensional harmonic oscillators is calculated analytically, in the limit when the chain becomes infinitely long.
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Energy spectrum structure and quantum states of disordered condensed systems
TL;DR: In this paper, the Spectral Density Near the True Boundary of the Spectrum (SBL) model is proposed to describe the behavior of the spectrum near impurity levels.
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