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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| Affiliation: | 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 ![urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0007]( "urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0007") 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) ![urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0015]( "urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0015") 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 ![urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0023]( "urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0023") whenever this limit exists. In the concrete example at hand, we prove ![urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0024]( "urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0024") 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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