On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example |
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Authors: | Alan Carey Fritz Gesztesy Galina Levitina Denis Potapov Fedor Sukochev Dima Zanin |
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Institution: | 1. Mathematical Sciences Institute, Australian National University, Kingsley St., Canberra, ACT 0200, Australia and School of Mathematics and Applied Statistics, University of Wollongong, NSW, Australia;2. Department of Mathematics, University of Missouri, Columbia, MO, USA;3. School of Mathematics and Statistics, UNSW, Kensington, NSW, Australia |
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Abstract: | Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , . |
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Keywords: | Fredholm and Witten index spectral shift function Primary: 47A53 58J30 Secondary: 47A10 47A40 |
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