BGS conjecture

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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).

• Energy levels: $-\frac{{\hslash }^2}{2{m}}{▽}^2\psi +{V}({x})\psi ={{E}}_{{i}}\psi$
• 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: $\frac{⟨\rho ({x})\rho ({y})⟩}{⟨\rho ({x})⟩⟨\rho ({y})⟩}-1=\frac{⟨\rho ({x})\rho ({y}){⟩}_{\mathrm{RMT}}}{⟨\rho ({x}){⟩}_{\mathrm{RMT}}⟨\rho ({y}){⟩}_{\mathrm{RMT}}}-1$

Links

• the BGS conjecture in Scholarpedia.