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

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

2010-04-09

arxiv.org/abs/1004.1582v2

Adv.Math.227:319-420,2011

Mathematics

Spectral Theory

85 pages, minor corrections incorporated

Scientific paper

We compute the Fredholm index, ind(D_A), of the operator D_A = (d/dt) + A on L^2(\bbR;\cH) associated with the operator path {A(t)}_{t=-\infty}^{\infty}, where (A f)(t) = A(t) f(t) for a.e. t\in\bbR, and appropriate f \in L^2(\bbR;\cH), via the spectral shift function xi(.;A_+,A_-) associated with the pair (A_+, A_-) of asymptotic operators A_{\pm}=A(\pm\infty) on the separable complex Hilbert space \cH 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)=(D_A {D_A}^*, {D_A}^* D_A) in this relative trace class context. This formula is then used to identify the Fredholm index of D_A with xi(0;A_+,A_-). In addition, we prove that ind(D_A) coincides with the spectral flow SpFlow ({A(t)}_{t=-\infty}^\infty) of the family {A(t)\}_{t\in\bbR} and also relate it to the (Fredholm) perturbation determinant for the pair (A_+, A_-). 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.

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