Abstract
We investigate spacing statistics for ensembles of various real random matrices where the matrix-elements have various Probability Distribution Function (PDF: <em>f(x)</em>) including Gaussian. For two modifications of 2 × 2 matrices with various PDFs, we derive the spacing distributions <em>p(s)</em> of adjacent energy eigenvalues. Nevertheless, they show the linear level repulsion near s = 0 as <em>αs</em> where <em>α</em> depends on the choice of the PDF. More interestingly when <em>f</em>(<em>x</em>) = <em>xe</em><sup>−x<sup>2</sup></sup> (<em>f</em>(0) = 0), we get cubic level repulsion near s = 0: <em>p(s)</em> ~ s<sup>3</sup>e<sup>−s<sup>2</sup></sup>.We also derive the distribution of eigenvalues <em>D</em>(ε) for these matrices.
Highlights
Due to matrix mechanics of Heisenberg and Method of Linear Combination of Atomic Orbitals (LCAO) one can visualize the eigenspectrum of various systems with time-reversal symmetry as result of diagonalization of a real symmetric matrix where the matrix element are calculated using the inter-particle interaction
Random Matrix Theory [1,2,3,4,5] originated by considering the level spacing statistics P (S) between eigenvalues of real symmetric matrices
All the spacing distributions display the linear level repulsion near s = 0 wherein α depends on the type of Probability Distribution Functions (PDFs) and the type of matrix (1) being used
Summary
Due to matrix mechanics of Heisenberg and Method of Linear Combination of Atomic Orbitals (LCAO) one can visualize the eigenspectrum of various systems with time-reversal symmetry as result of diagonalization of a real symmetric matrix where the matrix element are calculated using the inter-particle interaction. Random Matrix Theory [1,2,3,4,5] originated by considering the level spacing statistics P (S) between eigenvalues of real symmetric matrices Wigner surmised [1,2,3,4,5] that the spacing distribution of adjacent eigenvalues of N number of n × n Gaussian random real symmetric matrices will again be given by (3).
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