The off-diagonal Eigenstate Thermalization Hypothesis

Consider energy eigenstates $|E_i\rangle$ and $|E_j\rangle$ of a finite isolated system, and define the matrix elements of an operator $\hat{O}$ by

$$O_{ij}=\langle E_i|\hat{O}|E_j\rangle.$$

For $i\ne j$, write $O_{ij}=\Delta_{ij}$, and let $\langle\cdots\rangle_n$ denote averages over nearby energy levels of $i$ and $j$. Let $\Delta_i$ be the diagonal fluctuation described by the diagonal ETH statement.

Eigenstate Thermalization Hypothesis. The local statistics of the off-diagonal matrix elements satisfy

$$\langle|O_{ij}|^2\rangle_n=\langle\Delta_i^2\rangle_n F(E_i,E_j),$$

where $F$ is of order unity for $E_i=E_j$ and tends to zero as $|E_i-E_j|$ becomes large.

This extension of ETH controls off-diagonal matrix elements and is relevant to dynamical correlation functions and the approach to equilibrium. The source does not state a resolution status for this formulation.