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). 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). 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. A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards:
Energy levels: − ℏ 2 2 m ▽ 2 ψ + V ( x ) ψ = E i ψ {\displaystyle -{\frac {\hbar ^{2}}{2{\mathit {m}}}}\bigtriangledown ^{2}\psi +{\mathit {V}}({\mathit {x}})\psi ={{\mathit {E}}_{\mathit {i}}}\psi }
Spectral density: ρ ( x ) = ∑ i δ ( x − E i ) {\displaystyle \rho ({\mathit {x}})=\sum _{\mathit {i}}\delta ({\mathit {x}}-{\mathit {E}}_{\mathit {i}})}
Average spectral density: ⟨ ρ ( x ) ⟩ {\displaystyle \langle \rho ({\mathit {x}})\rangle }
Correlation: ⟨ ρ ( x ) ρ ( y ) ⟩ − ⟨ ρ ( x ) ⟩ ⟨ ρ ( y ) ⟩ {\displaystyle \langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle -\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }
Unfolding: ρ ( x ) → ρ ( x ) ⟨ ρ ( x ) ⟩ {\displaystyle \rho ({\mathit {x}})\rightarrow {\frac {\rho ({\mathit {x}})}{\langle \rho ({\mathit {x}})\rangle }}}
Unfolded correlation: ⟨ ρ ( x ) ρ ( y ) ⟩ ⟨ ρ ( x ) ⟩ ⟨ ρ ( y ) ⟩ − 1 {\displaystyle {\frac {\langle \rho ({\mathit {x}})\rho ({\mathit {y}})\rangle }{\langle \rho ({\mathit {x}})\rangle \langle \rho ({\mathit {y}})\rangle }}-1}
BGS conjecture: ⟨ ρ ( x ) ρ ( y ) ⟩ ⟨ ρ ( x ) ⟩ ⟨ ρ ( y ) ⟩ − 1 = ⟨ ρ ( x ) ρ ( y ) ⟩ RMT ⟨ ρ ( x ) ⟩ RMT ⟨ ρ ( y ) ⟩ RMT − 1 {\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.
References
Links the BGS conjecture in Scholarpedia.