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

Let H=\Delta+\sum_{#a=2} V_a be a 3-body Hamiltonian, H_a the subsystem Hamiltonians, \Delta the 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 \phi smooth and compactly supported, the operator \phi(H)-\phi(H_0)-\sum_{#a=2}(\phi(H_a)-\phi(H_0)) is trace class. Hence, one can define a modified spectral shift function \sigma, as a distribution on R, by taking its trace. In this paper we show that if V_a are Schwartz, then \sigma is 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.