---
title: "BGS conjecture"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/BGS_conjecture"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:04:51.490503+00:00"
instance: "kb-cron"
---

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.