The spectral shift function and the witten index

Abstract

We survey the notion of the spectral shift function of two operators and recent progress on its connection with the Witten index. We begin with classical definitions of the spectral shift function ξ(・;H2,H1) under various assumptions on the pair of operators (H2,H1) in a fixed Hilbert space and then discuss some of its properties. We then present a new approach to defining the spectral shift function and discuss Krein’s Trace Theorem. In particular, we describe a proof that does not use complex analysis [53] and develop its extension to general σ-finite von Neumann algebras M of type II and unbounded perturbations from the predual of M. We also discuss the connection between the theory of the spectral shift function and index theory for certain model operators. We start by introducing various definitions of the Witten index, (an extension of the notion of Fredholm index to non-Fredholm operators). Then 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 being a separable, complex Hilbert space). The setup permits the operator family A(t) on H to be an unbounded relatively trace class perturbation of the unbounded self-adjoint operator A−, and no discrete spectrum assumptions are made on the asymptotes A±. When there is a spectral gap for the operators A± at zero, it is shown that the operator DA is Fredholm and the Fredholm index can be computed as ind(DA) = ξ(0+; |D∗A |2, |DA |2) = ξ(0;A+,A−). When 0 ∈ σ(A+) (or 0 ∈ σ(A−)), the operator DA ceases to be Fredholm. However, under the additional assumption that 0 is a right and a left Lebesgue point of ξ(・;A+, A−), it is proved that 0 is also a right Lebesgue point of ξ(・; |D∗A |2, |DA |2). For the resolvent (resp., semigroup) regularized Witten index Wr(DA) (resp., Ws(DA)) the following equality holds, Wr(DA) = Ws(DA) = ξ(0+; |D∗A |2, |DA |2) = [ξ(0+;A+,A−) + ξ(0−;A+,A−)]/2. We also study a special example, when the perturbation of the unbounded self-adjoint operator A− is not assumed to be relatively trace class. In this example [Formula presented] is the differentiation operator on L2(ℝ) and the perturbation is given by the multiplication operator by a (bounded) realvalued function f on R. Under certain assumptions on f it is proved that [Formula presented].