DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES

@article{Marenko1967DISTRIBUTIONOE,
  title={DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES},
  author={V A Mar{\vc}enko and Leonid A. Pastur},
  journal={Mathematics of The Ussr-sbornik},
  year={1967},
  volume={1},
  pages={457-483}
}
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… 

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References

SHOWING 1-5 OF 5 REFERENCES
On the Distribution of the Roots of Certain Symmetric Matrices
TLDR
The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
Reviews of Topical Problems: Energy Spectrum Structure and Quantum States of Disordered Condensed Systems
CONTENTS 1. Introduction. Choice of Model 549 2. Formulation of the Problem. General Considerations 552 3. The Behavior of the Spectral Density Near the True Boundary of the Spectrum 553 4. The
The uniqueness of the solution of Cauchy's problem, Uspehi Mat
  • Russian) MR
  • 1948
On degenerate regular perturbations. II: Quasicontinuous and continuous spectrum
  • Z. Eksper. Teoret. Fiz
  • 1947