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Sub-quadratic repulsive potentials accelerate quantum particles and can relaxthe decay rate in the $x$ of the external potentials $V$ that guarantee theexistence of the quantum wave operators. In the case where the sub-quadraticpotential is $- |x|^{\alpha} $ with $0< \alpha < 2$ and the external potentialsatisfies $|V(x) | \leq C (1+|x|) ^{-(1- \alpha /2) - \varepsilon} $ with$\varepsilon>0$, Bony et. al [3] determined the existence and completeness ofthe wave operators, and Itakura [12, 13, 14] then obtained their results usingstationary scattering theory for more generalized external potentials. Based ontheir results, we naturally expect the following. If the decay power of theexternal potential $V$ is less than ${ -(1- \alpha /2) } $, V is included inthe short-range class. If the decay power is greater than or equal to ${ -(1-\alpha /2) } $, $V$ is included in the long-range class. In this study, wefirst prove the new propagation estimates for the time propagator that can beapplied to scattering theory. Second, we prove that the wave operators do notexist if the power is greater than or equal to $-(1- \alpha /2)$ and that thethreshold expectation of ${ -(1- \alpha /2) } $ is true using the newpropagation estimates.

Nonexistence of wave operators via strong propagation estimates for Schr\"{o}dinger operators with sub-quadratic repulsive potentials