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).¹²

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: ${\displaystyle -{\frac {\hbar ^{2}}{2{\mathit {m}}}}\bigtriangledown ^{2}\psi +{\mathit {V}}({\mathit {x}})\psi ={{\mathit {E}}_{\mathit {i}}}\psi }$
• 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.

• the BGS conjecture in Scholarpedia.