Spectral shift functions of all orders

Let n and let H₀,V be self-adjoint operators such that V is bounded and V(H₀-i)⁻ᵖn/p for p=1,…,n. We prove the existence, uniqueness up to polynomial summands, and regularity properties of all higher order spectral shift functions associated to the perturbation theory of t↦ H₀+tV (t∈[0,1]). Such perturbations arise in situations in noncommutative geometry and mathematical physics with dimension $n-1$. Our proof is simpler than that of {NuSkJST,NuSkJOT} and works at all orders regardless of n.