The reduced effect of a single scattering with a low-mass particle via a point interaction

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in λ to the map , where G∈B(L2(Rn)) is a heavy-particle observable, ρ∈B1(Rn) is the density matrix corresponding to the state of the light particle, is the mass ratio of the light particle to the heavy particle, Sλ∈B(L2(Rn)⊗L2(Rn)) is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on G.