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).Template:Explain

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).Template:Explain

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.Template:Explain

A simple example of the unfolded quantum energy levels in a classically chaotic system correlating like that would be Sinai billiards:Template:Explain

• Energy levels: $-\frac{{\hslash }^2}{2𝑚}{▽}^2\psi +𝑉(𝑥)\psi ={𝐸}_{𝑖}\psi$Template:Definition needed
• Spectral density: $\rho (𝑥)=\sum _{𝑖}\delta (𝑥-{𝐸}_{𝑖})$
• Average spectral density: $⟨\rho (𝑥)⟩$
• Correlation: $⟨\rho (𝑥)\rho (𝑦)⟩-⟨\rho (𝑥)⟩⟨\rho (𝑦)⟩$
• Unfolding: $\rho (𝑥)\to \frac{\rho (𝑥)}{⟨\rho (𝑥)⟩}$
• Unfolded correlation: $\frac{⟨\rho (𝑥)\rho (𝑦)⟩}{⟨\rho (𝑥)⟩⟨\rho (𝑦)⟩}-1$
• BGS conjecture: $\frac{⟨\rho (𝑥)\rho (𝑦)⟩}{⟨\rho (𝑥)⟩⟨\rho (𝑦)⟩}-1=\frac{⟨\rho (𝑥)\rho (𝑦){⟩}_{\mathrm{RMT}}}{⟨\rho (𝑥){⟩}_{\mathrm{RMT}}⟨\rho (𝑦){⟩}_{\mathrm{RMT}}}-1$

The conjecture remains unproven despite supporting numerical evidence.Template:Citation needed

Links

• the BGS conjecture in Scholarpedia.