Gaussian pseudo-Orthogonal Ensemble of Real Random Matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866915386697449472 |
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| author | Kumar, Sachin Kumar, Amit Yusuf, S M |
| author_facet | Kumar, Sachin Kumar, Amit Yusuf, S M |
| contents | Here, using two real non-zero parameters $λ$ and $μ$, we construct Gaussian pseudo-orthogonal ensembles of a large number $N$ of $n \times n$ ($n$ even and large) real pseudo-symmetric matrices under the metric $η$ using $ \altmathcal {N}=n(n+1)/2$ elements independently drawn from a Gaussian random population and investigate the statistical properties of the eigenvalues. When $λμ>0$, we show that the pseudo-symmetric matrix is similar to a real symmetric matrix, consequently, all the eigenvalues are real and so the spectral distributions satisfy Wigner's statistics. But when $λμ<0$ the eigenvalues are either real or complex conjugate pairs. We find that these real eigenvalues exhibit intermediate statistics. We show that the diagonalizing matrices ${ \cal D}$ of these pseudo-symmetric matrices are pseudo-orthogonal under a constant metric $ζ$ as $ \altmathcal{D}^t ζ\altmathcal{D}= ζ$, and hence they belong to a pseudo-orthogonal group. These pseudo-symmetric matrices serve to represent the parity-time (PT)-symmetric quantum systems having exact (un-broken) or broken PT-symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_04588 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Gaussian pseudo-Orthogonal Ensemble of Real Random Matrices Kumar, Sachin Kumar, Amit Yusuf, S M Quantum Physics Statistical Mechanics Strongly Correlated Electrons Mathematical Physics Here, using two real non-zero parameters $λ$ and $μ$, we construct Gaussian pseudo-orthogonal ensembles of a large number $N$ of $n \times n$ ($n$ even and large) real pseudo-symmetric matrices under the metric $η$ using $ \altmathcal {N}=n(n+1)/2$ elements independently drawn from a Gaussian random population and investigate the statistical properties of the eigenvalues. When $λμ>0$, we show that the pseudo-symmetric matrix is similar to a real symmetric matrix, consequently, all the eigenvalues are real and so the spectral distributions satisfy Wigner's statistics. But when $λμ<0$ the eigenvalues are either real or complex conjugate pairs. We find that these real eigenvalues exhibit intermediate statistics. We show that the diagonalizing matrices ${ \cal D}$ of these pseudo-symmetric matrices are pseudo-orthogonal under a constant metric $ζ$ as $ \altmathcal{D}^t ζ\altmathcal{D}= ζ$, and hence they belong to a pseudo-orthogonal group. These pseudo-symmetric matrices serve to represent the parity-time (PT)-symmetric quantum systems having exact (un-broken) or broken PT-symmetry. |
| title | Gaussian pseudo-Orthogonal Ensemble of Real Random Matrices |
| topic | Quantum Physics Statistical Mechanics Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/1802.04588 |