Smoothness and high energy asymptotics of the spectral shift function in many-body scattering

Abstract

Let H=Δ+\sum_{#a=2} V_a be a 3-body Hamiltonian, H_a the subsystem Hamiltonians, Δthe positive Laplacian of the Euclidean metric on X_0=R^n, V_a real-valued. Buslaev and Merkurev have shown that, if the pair potentials decay sufficiently fast, for ϕsmooth and compactly supported, the operator ϕ(H)-ϕ(H_0)-\sum_{#a=2}(ϕ(H_a)-ϕ(H_0)) is trace class. Hence, one can define a modified spectral shift function σ, as a distribution on R, by taking its trace. In this paper we show that if V_a are Schwartz, then σis in fact smooth away from the thresholds, and obtain its high energy asymptotics. In addition, we generalize this result to N-body scattering, N arbitrary.