The spectral flow, the Fredholm index, and the spectral shift function

Pushnitski, Alexander

doi:10.1090/trans2/225/10

Cited by 20 publications

(60 citation statements)

References 8 publications

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“…In fact, the method we employ here to make this extension is of interest in any dimension. Moreover we consider A ± which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions found in all of the previous papers [9] , [14], and [22], can be proved, even in the non-Fredholm case. The significance of our new methods is that, besides being simpler, they also allow a wide class of examples such as pseudodifferential operators in higher dimensions.…”

Section: W R (D D D Dsupporting

confidence: 54%

On the index of a non-Fredholm model operator

Carey¹,

Gesztesy²,

Levitina³

et al. 2016

Operators and Matrices

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, and its relation to spectral flow along this path, has a long history, but it is mostly focussed on the case where the operators A(t) all have purely discrete spectrum. [14], that these values of the two spectral functions are equal, resolves the index = spectral flow question in this case. This relationship between spectral shift functions was generalized to non-Fredholm operators in [9] again under the relatively trace class perturbation hypothesis. Introducing the operators H H H H In this situation it asserts that the Witten index of D D D D A A A A , denoted by W r (D D D D A A A A ) , a substitute for the Fredholm index in the absence of the Fredholm property of D D D D A A A A , is given byHere one assumes that ξ ( ·;A − ,A + ) possesses a right and left Lebesgue point at 0 denoted by ξ L (0 ± ;A + ,A − ) (and similarly for ξ L (0 + ;H H H H 2 ,H H H H 1 ) ).When the path {A(t)} t∈R consists of differential operators, the relatively trace class perturbation assumption is violated. The simplest assumption that applies (to differential operators in 1+1 dimensions) is to admit relatively Hilbert-Schmidt perturbations. This is not just an incremental improvement. In fact, the method we employ here to make this extension is of interest in any dimension. Moreover we consider A ± which are not necessarily Fredholm and we establish that the relationships between the two spectral shift functions found in all of the previous papers [9] , [14], and [22], can be proved, even in the non-Fredholm case. The significance of our new methods is that, besides being simpler, they also allow a wide class of examples such as pseudodifferential operators in higher dimensions. Most importantly, we prove the above formula for the Witten index in the most general circumstances to date.Mathematics subject classification (2010): Primary 47A53, 58J30; Secondary 47A10, 47A40.

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Section: W R (D D D Dsupporting

confidence: 54%

On the index of a non-Fredholm model operator

Carey¹,

Gesztesy²,

Levitina³

et al. 2016

Operators and Matrices

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“…Then lim n→∞ R n S n T * n − RST * Bp(H) = 0. To set up approximations for A + , we now deviate from the usual approximation procedure originally employed in [12] and [27]: We introduce …”

Section: The Strategy Employed and Statement Of Resultsmentioning

confidence: 99%

“…[12]) that D A is a Fredholm operator in L 2 (R; H). Moreover, as shown in [12] (and earlier in [27] under a simpler set of hypotheses on the family A(·)), the Fredholm index of D A may then be computed as follows, index(D A ) = ξ(0 + ; H 2 , H 1 ) = ξ(0; A + , A − ).…”

Section: Of Course a =´⊕ R Dt A(t)mentioning

confidence: 98%

“…The latter paper was particularly motivated by [27] which, in turn, was motivated by [30], which relates the Fredholm index and spectral flow for operators with compact resolvent. The model operators considered there provide prototypes for more complex situations.…”

Section: Introductionmentioning

confidence: 99%

“…They arise in connection with investigations of the Maslov index, Morse theory, Floer homology, Sturm oscillation theory, etc. The principle aim in [27] and [12] was to extend the compact resolvent results in [30] to a relatively trace class perturbation approach, permitting essential spectra.…”

Section: Introductionmentioning

confidence: 99%

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

Carey

Gesztesy

Levitina

et al. 2015

Mathematische Nachrichten

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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, 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

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