The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA)index(DA), of the operator DA=(d/dt)+ADA=(d/dt)+A on L2(R;H)L2(R;H) associated with the operator path {A(t)}t=−∞∞, where (Af)(t)=A(t)f(t)(Af)(t)=A(t)f(t) for a.e. t∈Rt∈R, and appropriate f∈L2(R;H)f∈L2(R;H), via the spectral shift function ξ(⋅;A+,A−)ξ(⋅;A+,A−) associated with the pair (A+,A−)(A+,A−) of asymptotic operators A±=A(±∞)A±=A(±∞) on the separable complex Hilbert space HH in the case when A(t)A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A−A−.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A+,A−)ξ(⋅;A+,A−) for the pair (A+,A−)(A+,A−), and the corresponding spectral shift function ξ(⋅;H2,H1)ξ(⋅;H2,H1) for the pair of operators (H2,H1)=(DADA⁎,DA⁎DA) in this relative trace class context,ξ(λ;H2,H1)=1π∫−λ1/2λ1/2ξ(ν;A+,A−)dν(λ−ν2)1/2for a.e. λ>0.This formula is then used to identify the Fredholm index of DADA with ξ(0;A+,A−)ξ(0;A+,A−). In addition, we prove that index(DA)index(DA) coincides with the spectral flow SpFlow({A(t)}t=−∞∞) of the family {A(t)}t∈R{A(t)}t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A+,A−)(A+,A−):index(DA)=SpFlow({A(t)}t=−∞∞)=ξ(0;A+,A−)=π−1limε↓0Im(ln(detH((A+−iεI)(A−−iεI)−1)))=ξ(0+;H2,H1), with the choice of the branch of ln(detH(⋅))ln(detH(⋅)) on C+C+ such thatlimIm(z)→+∞ln(detH((A+−zI)(A−−zI)−1))=0.We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.