An integer-valued version of the Birman—Krein formula

Pushnitski, Alexander

doi:10.1007/s10688-010-0041-y

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…The Krein spectral shift functions are widely used in scattering theory, see [21][22][23][27][28][29]. For a pair of selfadjoint operators H 0 , H on a Hilbert space H, the spectral shift function ξ(λ) is defined by the Lifshits trace formula…”

Section: Spectral Shift Functionmentioning

confidence: 99%

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Spectral Action Beyond the Weak-Field Approximation

Iochum

Lévy

Vassilevich

2012

Commun. Math. Phys.

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The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p → ∞, the action decays as 1/p 4 in any even dimension.Recent advances [12,14] in explaining some key features of gravity and Standard Model through the spectral action of noncommutative geometry brought this subject to a focus of interest in theoretical physics. In noncommutative geometry, all information is encoded in a spectral triple (A, H, D), where A is an algebra acting on a Hilbert space H and D is a selfadjoint operator on H which plays the role of a Dirac operator [13,14,19]. In this approach, the action is the so-called spectral action introduced by Chamseddine and Connes [9-11]and defined for Λ ∈ R + which plays the role of a cut-off (and needed to make D/Λ dimensionless) and for a function f such that, of course, f (D 2 /Λ 2 ) is trace-class. In general, one chooses f ≥ 0 since the action Tr f (D 2 /Λ 2 ) ≥ 0 will have the correct sign for an Euclidean action. This action is the appropriate one in the framework of noncommutative geometry to reproduce several physical situations like the Einstein-Hilbert action in gravitation or the Yang-Mills-Higgs action in the standard model of particle physics [14], and the positivity of the function f implies positivity of actions for gravity, Yang-Mills or Higgs couplings, and the Higgs mass term is negative. Till the end of this Section we shall present a non-technical summary of our results to give a more physics-oriented reader a chance to appreciate them without going through the mathematics of the rest of this paper.Let M = R 2m be an even dimensional real plane, d = 2m ≥ 2, endowed with a spin structure given by the spinor bundle S = C 2 m . We denote by D the free Dirac operator and by D A the standard Dirac operator with a gauge connection A acting on the Hilbert space H := L 2 (M, S). We will use the notations and conventions from [25, eq. (3.26)], namely in local coordinates

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Section: Spectral Shift Functionmentioning

confidence: 99%

“…Since we want to make a connection to (22) and (23), we are obliged to suppose that f is a Laplace transform but eventually of a distribution since, typically, we want to allow f (x) to be e −x .…”

Section: Calculation Of the Spectral Actionmentioning

confidence: 99%

Spectral Action Beyond the Weak-Field Approximation

Iochum

Lévy

Vassilevich

2012

Commun. Math. Phys.

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“…8.7] (see also [3,), but also this definition fixes ξ( · ; A + , A − ) only up to an integer as will be discussed in some detail in Appendix A. For additional discussions addressing the open integer in ξ( · ; A + , A − ) we refer to [28]. ✸ Next, we apply Theorem A.1 to the pair (A + , A − ), identifying A 0 with A − , A with A + , and B with the operator of multiplication by φ, assuming again Hypothesis 3.1.…”

Section: The Strategy Employed and Statement Of Resultsmentioning

confidence: 99%

“…8.7] (see also [3,), but also this definition fixes ξ( • ; A + , A − ) only up to an integer as will be discussed in some detail in Appendix A. For additional discussions addressing the open integer in ξ( • ; A + , A − ) we refer to [28]. ✸…”

Section: The Strategy Employed and Statement Of Resultsmentioning

confidence: 99%

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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The Birman–Schwinger principle on the essential spectrum

Pushnitski

2011

Journal of Functional Analysis

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Let H 0 and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H 0 and H corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair H 0 , H .

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An integer-valued version of the Birman—Krein formula

Abstract: Abstract. We discuss an identity in abstract scattering theory which can be interpreted as an integer-valued version of the Birman-Krein formula.

Cited by 3 publications

References 22 publications

Spectral Action Beyond the Weak-Field Approximation

Spectral Action Beyond the Weak-Field Approximation

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

The Birman–Schwinger principle on the essential spectrum

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