On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Abstract

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, and the family of self-adjoint operators A(t) in L 2 (R; dx) studied here is explicitly given byHere φ : R → R has to be integrable on R and θ : R → R tends to zero as t → −∞ and to 1 as t → +∞ (both functions are subject to additional hypotheses). In particular, A(t), t ∈ R, has asymptotes (in the norm resolvent sense)The interesting feature is that D A violates the relative trace class condition introduced in [9, Hypothesis 2.1 (iv)]. 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. Introducingwhenever this limit exists. In the concrete example at hand, we prove2πˆR dx φ(x). Here ξ( · ; S 2 , S 1 ) denotes the spectral shift operator for the pair of self-adjoint operators (S 2 , S 1 ), and we employ the normalization, ξ(λ; H 2 , H 1 ) = 0, λ < 0. Contents

Trace formulas for a class of non-Fredholm operators: A review

“…First, rewriting Next, one notes that 10) implies the required convergence. Consequently, 11) completes the proof of item (i).…”

“…We will focus on examples of the non-Fredholm case motivated by recent progress in [10]- [14]. Remark 1.1.…”

Double operator integral methods applied to continuity of spectral shift functions

“…To keep the following sufficiently short, we only describe one of the consequences of our continuity results. We note, however, that it was precisely this consequence that was employed in recent applications to Witten index computations for certain classes of non-Fredholm Dirac-type operators without a mass gap in [7]- [9] (see also [10], [14]). To set this up, assume that A 0 and B 0 are fixed self-adjoint operators in the Hilbert space H, and there exists m ∈ N, m odd, such that,…”

“…The Witten index for these types of non-Fredholm Dirac-type operators (a concept extending the Fredholm index) is computed in terms of spectral shift functions and the latter are approximated by the spectral shift functions corresponding to the pseudo-differential approximants. Due to limitations of space we will not go into further details at this point but refer to [7]- [9] (see also [14]).…”

On the Global Limiting Absorption Principle for Massless Dirac Operators

“…This paper was motivated by recent investigations of the Witten index (a possible substitute for the Fredholm index) for classes of non-Fredholm operators, a prime example of which being the massless Dirac operator, see, for instance, [13]- [15].…”