In this letter, by an approach that employs Weyl symbols for operators, a semiclassical theory is developed for the off-diagonal function in the eigenstate thermalization hypothesis, which is for off-diagonal elements $\langle E_i\left|O\right|E_j\rangle$ of an observable $O$ on the energy basis. It is shown analytically that the matrix of $O$ has a banded structure, possessing a bandwidth $w_b$ that scales linearly with $\hslash$, a phase-space gradient of the classical Hamiltonian $\langle \left|\mathbf{\nabla }{H}_{\mathrm{cl}}\right|\rangle$ and an $O$-dependent property. This predicts that the thermalization timescale of a quantum system may be inversely proportional to the phase-space gradient of the Hamiltonian, aligning with intuitions in classical thermalization. This approach also elucidates the origin of a ${\rho }_{\mathrm{dos}}^{-1/2}$ scaling of the off-diagonal function. The analytical predictions are checked numerically in the Lipkin-Meshkov-Glick model.

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• Eigenstate thermalization
• Quantum chaos
• Quantum statistical mechanics
• Quantum thermodynamics