BGS conjecture

The Bohigas–Giannoni–Schmit (BGS) conjecture also known as the random matrix conjecture) for simple quantum mechanical systems (ergodic with a classical limit) few degrees of freedom holds that spectra of time reversal-invariant systems whose classical analogues are K-systems show the same fluctuation properties as predicted by the GOE (Gaussian orthogonal ensembles).^{[1][2][further explanation needed]}

Alternatively, the spectral fluctuation measures of a classically chaotic quantum system coincide with those of the canonical random-matrix ensemble in the same symmetry class (unitary, orthogonal, or symplectic).^{[further explanation needed]}

That is, the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble as the distance of a few spacings between eigenvalues of a chaotic Hamiltonian operator generically statistically correlates with the spacing laws for eigenvalues of large random matrices.^{[further explanation needed]}

A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards:^{[further explanation needed]}

• Energy levels: ${\displaystyle -{\frac {\hbar ^{2}}{2{\mathit {m}}}}\bigtriangledown ^{2}\psi +{\mathit {V}}({\mathit {x}})\psi ={{\mathit {E}}_{\mathit {i}}}\psi }$ ^{[definition needed]}
• Spectral density: ${\displaystyle \rho ({\mathit {x}})=\sum _{\mathit {i}}\delta ({\mathit {x}}-{\mathit {E}}_{\mathit {i}})}$
• Average spectral density: ${\displaystyle \langle \rho ({\mathit {x}})\rangle }$
• Correlation: ${\displaystyle \langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle -\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }$
• Unfolding: ${\displaystyle \rho ({\mathit {x}})\rightarrow {\frac {\rho ({\mathit {x}})}{\langle \rho ({\mathit {x}})\rangle }}}$
• Unfolded correlation: ${\displaystyle {\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle }{\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }}-1}$
• BGS conjecture: ${\displaystyle {\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle }{\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }}-1={\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle _{\operatorname {RMT} }}{\langle \rho ({\mathit {x}})\rangle _{\operatorname {RMT} }\langle \rho ({\mathit {y}})\rangle _{\operatorname {RMT} }}}-1}$

The conjecture remains unproven despite supporting numerical evidence.^{[citation needed]}

References

1. Bohigas, O.; Giannoni, M. J.; Schmit, C. (2010), "Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws", Spectral Distributions in Nuclei and Statistical Spectroscopy, World Scientific Publishing Co. Pte. Ltd., pp. 420–423, doi:10.1142/9789814287395_0024 (inactive 1 July 2025), ISBN 978-981-4287-39-5, retrieved 2025-03-06
2. Bohigas, O.; Giannoni, M.J.; Schmit, C. (1984). "Spectral properties of the Laplacian and random matrix theories". Journal de Physique Lettres. 45 (21): 1015–1022. doi:10.1051/jphyslet:0198400450210101500. ISSN 0302-072X.

Links

• the BGS conjecture in Scholarpedia.