## Abstract

We study the model operator D_{A} = (d/dt) + A in L^{2}(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 ∈ L^{2}(R;H) (with H a separable, complex Hilbert space). Denoting by A_{±} the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A_{-}, and no discrete spectrum assumptions are made on A_{±}. Introducing H_{1} = D_{A}*D_{A}, H_{2} = D_{A}D_{A}*, the resolvent and semigroup regularized Witten indices of D_{A}, denoted by W_{r}(D_{A}) and W_{s}(D_{A}), are defined by (Formula Presented.), whenever these limits exist. These regularized indices coincide with the Fredholm index of D_{A} whenever the latter is Fredholm. In situations where D_{A} ceases to be a Fredholm operator in L^{2}(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A_{+}, A_{-}) associated with the pair (A_{+}, A_{-}) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A_{+}, A_{-}), denoted by ξ_{L}(0_{+}; A_{+}, A_{-}) and ξ_{L}(0_{-}; A_{+}, A_{-}), we prove that 0 is also a right Lebesgue point of ξ(•; H_{2}, H_{1}), denoted by ξ_{L}(0_{+}; H_{2}, H_{1}), and that (Formula Presented.), the principal result of this paper. In the special case where dim(H) < ∞, we prove that the Witten indices of D_{A} are either integer, or half-integer-valued.

Original language | English |
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Pages (from-to) | 1-61 |

Number of pages | 61 |

Journal | Journal d'Analyse Mathematique |

Volume | 132 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jun 2017 |