For sub-quadratic confining potentials, the quantum smoothing effect and the classical escape rate of Hamiltonian trajectories are equivalent up to an O(1/R) factor.
Spectral asymptotics in the semi-classical limit , volume 268 of London Mathematical Society Lecture Note Series
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Smoothing effect and quantum-classical correspondence for the Schr{\"o}dinger equation with confining potential
For sub-quadratic confining potentials, the quantum smoothing effect and the classical escape rate of Hamiltonian trajectories are equivalent up to an O(1/R) factor.