Spectral Averaging and the Krein Spectral Shift

We provide a new proof of a theorem of Birman and Solomyak that if A(s) = A0+ sB with B ≥ 0 trace class and dμs(·) = Tr(BEA(s)(·)B), then ∫ 1 0 [dμs(λ)] ds = ξ(λ)dλ where ξ is the Krein spectral shift from A(0) to A(1). Our main point is that this is a simple consequence of the formula: d ds Tr(f(A(s)) = Tr(Bf ′(A(s))). Let A and C = A+B be bounded self-adjoint operators and suppose that B ≥ 0 and B is trace class. Then it is a fundamental result of Krein [5, 6, 7] that there is an L function ξA,C(λ) so that for any C function f (and, in particular, for all f ∈ C∞ 0 (R )), Tr(f(C)− f(A)) = ∫ f ′(λ)ξA,C(λ)dλ. (1) ξ ≥ 0 and is uniquely determined by (1) and the condition that ξ is L. This ξ has compact support (if A,C are bounded). Moreover, if B is finite rank, then 0 ≤ ξA,C(λ) ≤ rank(B) (2) and lim s→∞ [ ξA,A+sB(λ)− ξA,A−sB(λ) ] = rank(B). (3) In 1971, Javrjan [3] proved a remarkable formula for rank(B) = 1: Let B = (φ, · )φ and define dμs(λ) by ∫ edμs(λ) = (φ, eφ) and dη(λ) = ∫ s1 s0 [dμs(·)] ds, the average of the spectral measures for A+ sB. Then, Javrjan’s formula is dη(λ) = ξA+s0B,A+s1B(λ)dλ. (4) ∗ This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material. To be submitted to Proc. Amer. Math. Soc.