Formal perturbative solution for the generalized crossing state

Abstract

We reconsider Baute et.al.’s crossing states |u_α⟩ as applied to the the time of arrival problem (A. Baute, R. Sala, J.P. Palao, J.G. Muga, I.L. Egusquiza Phys. Rev. A 61 (2000) 022118) and show that if we require a proper generalization of the notion of crossing states in the interacting case, then this crossing state cannot be arbitrary but is rather determined by the null space of the quantum particle’s time of arrival operator. We develop the construction of such a generalized crossing state |H,T,α⟩ and demonstrate that the crossing states |u_α⟩ in the presence of an interaction potential appear as the leading term in the asymptotic expansion of |H,T,α⟩ in the case of arbitrarily large momentum.