Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics

Bollé, Désiré; Gesztesy, Fritz; Grosse, Harald; Schweiger, Wolfgang; Simon, Barry

doi:10.1063/1.527508

Cited by 75 publications

(84 citation statements)

References 45 publications

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“…Such expansions can be obtained under more general conditions; in particular the unperturbed operator H 0 may have variable coefficients (periodic, for example) and no assumption on the classical flow is needed. This kind of expansion is similar to the one used to prove index theorems for elliptic operators on compact manifolds; actually (11) is a tool in the proof of relative index theorems see [3,4,8].…”

Section: Differential Operatorsmentioning

confidence: 99%

Asymptotic behavior of regularized scattering phases for long range perturbations

Bouclet¹

2002

Journées Équations Aux Dérivées Partielles

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We define scattering phases for Schrödinger operators on R d as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the Laplacian and therefore they can replace the usual spectral shift function (SSF) of Birman-Krein's theory, which can be defined for only special short range perturbations (relatively trace class perturbations). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF.

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Section: Differential Operatorsmentioning

confidence: 99%

Asymptotic behavior of regularized scattering phases for long range perturbations

Bouclet¹

2002

Journées Équations Aux Dérivées Partielles

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“…This is sometimes dubbed topological invariance of the Witten index in the pertinent literature (see, e.g., [4], [5], [12], [15], and the references therein).…”

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

confidence: 99%

“…A theory for non-Fredholm operators was initiated in [4] and [15], however, it was technically quite a formidable problem at that time to construct a wide range of examples. This paper produces examples for which the Witten index may be calculated explicitly in the non-Fredholm case.…”

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

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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“…This latter assumption is violated in general for differential operators (although, not necessarily for certain classes of pseudo-differential operators). Indeed, as the bulk of the available literature focuses on systems with purely discrete spectra, there is relatively little work available in the way of index formulas for operators with essential spectrum except for [7], [8] and previous work by the present authors. Motivation for this study stems, for example, from the fact that the spectral flow is a useful tool in condensed matter theory where the operators that arise do have some essential spectrum [41].…”

Section: Introduction and Reviewmentioning

confidence: 99%

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

et al. 2016

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Abstract. Take a one-parameter family of self-adjoint Fredholm operators {A(t)} t∈R on a Hilbert space H, joining endpoints A ± . There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator D A = (d/dt) + A acting in L 2 (R; H), where A denotes the multiplication operator (Af )(t) = A(t)f (t) for f ∈ dom(A). Most results are about the case where the operators A(·) have compact resolvent. In this article we review what is known when these operators have some essential spectrum and describe some new results.Using the operators

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Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics

Cited by 75 publications

References 45 publications

Asymptotic behavior of regularized scattering phases for long range perturbations

Asymptotic behavior of regularized scattering phases for long range perturbations

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

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

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