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The index formula and the spectral shift function for relatively trace class perturbations | | Fritz Gesztesy
; Yuri Latushkin
; Konstantin A. Makarov
; Fedor Sukochev
; Yuri Tomilov
; | | Date: |
9 Apr 2010 | | Abstract: | We compute the Fredholm index, ind(D_A), of the operator D_A = (d/dt) + A on
L^2(bR;cH) associated with the operator path {A(t)}_{t=-infty}^{infty},
where (A f)(t) = A(t) f(t) for a.e. tinbR, and appropriate f in
L^2(bR;cH), via the spectral shift function xi(.;A_+,A_-) associated with
the pair (A_+, A_-) of asymptotic operators A_{pm}=A(pminfty) 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)}_{tinbR} 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. | | Source: | arXiv, 1004.1582 | | Services: | Forum | Review | PDF | Favorites | |
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