On the Witten index in terms of spectral shift functions

Abstract

We study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path {A(t)}t=−∞∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by Wr(DA)=limλ↑0(−λ)trL2(R;H)((H1−λI)−1−(H2−λI)−1),Ws(DA)=limt↑∞trL2(R;H)(e−tH1−e−tH2), Wr(DA)=limλ↑0⁡(−λ)trL2(R;H)((H1−λI)−1−(H2−λI)−1),Ws(DA)=limt↑∞⁡trL2(R;H)(e−tH1−e−tH2), whenever these limits exist. These regularized indices coincide with the Fredholm index of DA whenever the latter is Fredholm. In situations where DA ceases to be a Fredholm operator in L2(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A+, A-) associated with the pair (A+, A-) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A+, A-), denoted by ξL(0+; A+, A-) and ξL(0-; A+, A-), we prove that 0 is also a right Lebesgue point of ξ(•; H2, H1), denoted by ξL(0+; H2, H1), and that Wr(DA)=Ws(DA)=ξL(0+;H2,H1)=[ξL(0+;A+,A−)+ξL(0−;A+,A−)]/2, Wr(DA)=Ws(DA)=ξL(0+;H2,H1)=[ξL(0+;A+,A−)+ξL(0−;A+,A−)]/[ξL(0+;A+,A−)+ξL(0−;A+,A−)]22, the principal result of this paper. In the special case where dim(H) < ∞, we prove that the Witten indices of DA are either integer, or half-integer-valued. Show more