A classical particle whose energy is strictly larger than the potential energy at large distances from the origin moves off to infinity for suitable initial configurations. The corresponding statement for quantum mechanical systems is definitely wrong: A potential that vanishes at infinity can localize particles at all energies if the potential oscillates in a suitable way and resonance phenonema occur. This effect is the antithesis of tunneling. However, the potential must not tend to zero too rapidly; for so-called short range potentials, there is no localization at positive energies, independently of the shape of the potential. The Kato Conjecture concerns the transition from short range to long range. As far as I know, Kato never made any such conjecture; rather, the term refers to a result in this context proved by Kato.