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

We compute the Fredholm index, index(D-A), of the operator D-A = (d/dt) + A on L-2(R; H) associated with the operator path {A(t)}(t)(infinity) = -infinity, where (Af)(t) = A(t) f (t) for ac. t is an element of R, and appropriate f is an element of L-2(R; H), via the spectral shift function xi(A(+), A(-)) associated with the pair (A(+), A(-)) of asymptotic operators A(+/-) = A(+/-infinity) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A(-). We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function xi(A(+), A(-)) for the pair (A(+), A(-)), and the corresponding spectral shift function xi(H-2, H-1) for the pair of operators (H-2, H-1) = (DAD*(A), D*D-A(A)) in this relative trace class context, xi(lambda; H-2, H-1) = 1/pi integral(lambda 1/2)(-lambda 1/2) xi(v; (A(+), A_)dv/(lambda-v(2))(1/2) for a.e. lambda > 0. This formula is then used to identify the Fredholm index of D-A with xi(0; A(+), A(-)). In addition, we prove that index(D-A) coincides with the spectral flow SpFlow({A(t)}(t)(infinity)=-infinity) of the family {A(t)}(t is an element of R) and also relate it to the (Fredholm) perturbation determinant for the pair (A(+), A(-)): index(D-A) = SpFlow({A(t)}(t)(infinity)=-infinity) = xi(0; A(+), A_) = pi(-1) lim(epsilon down arrow 0) Im(In(det(H)((A(+) - i epsilon I)(A(-) - i epsilon I)(-1)))) = xi(0(+); H-2,H-1), with the choice of the branch of In(det(H)(.)) on C+ such that lim(Im(z)->+infinity) In(det(H)((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. (C) 2011 Elsevier Inc. All rights reserved.