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