On the Global Limiting Absorption Principle for Massless Dirac Operators

Carey, Alan L.; Gesztesy, Fritz; Kaad, Jens; Levitina, Galina; Nichols, Roger; Potapov, Denis; Sukochev, Fedor

doi:10.1007/s00023-018-0675-5

Cited by 17 publications

(19 citation statements)

References 38 publications

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Section: Comments and Open Problemsmentioning

confidence: 72%

“…3. In addition to the classical results of Chernoff [13,14] and the results in [35] for the smooth case, criteria for essential self-adjointness of general first order matrixvalued differential operators with rough coefficients on R d were recently obtained by completely different methods in [11], and in [5] in the elliptic case. In [11], the authors consider abstract operators which, in our case, have the form…”

Section: Comments and Open Problemsmentioning

confidence: 99%

“…Hypothesis 2.1 of [11] specialized to this case assumes that the A j 's and all their partial derivatives are Lipschitz and bounded as |x| → ∞. So if the arguments from our Comments 1 and 2 above are correct, then the extension of our results to the unbounded case covers this case of Theorem 2.4 in [11]. Turning to [5], it was surprising to read the authors' claim that, in the elliptic case, one can dispense with the Chernoff-type condition (7.1) (see [5,Remark 3.11]).…”

Section: Comments and Open Problemsmentioning

confidence: 99%

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Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

Nenciu

Obermeyer

2021

Commun. Math. Phys.

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We prove essential self-adjointness of Dirac operators with Lorentz scalar potentials which grow sufficiently fast near the boundary ∂ of the spatial domain ⊂ R d . On the way, we first consider general symmetric first order differential systems, for which we identify a new, large class of potentials, called scalar potentials, ensuring essential self-adjointness. Furthermore, using the supersymmetric structure of the Dirac operator in the two dimensional case, we prove confinement of Dirac particles, i.e. essential self-adjointness of the operator, solely by magnetic fields B assumed to grow, near ∂ , faster than 1/ 2dist(x, ∂ ) 2 . Contents

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Section: Comments and Open Problemsmentioning

confidence: 72%

Section: Comments and Open Problemsmentioning

confidence: 99%

Section: Comments and Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

Nenciu

Obermeyer

2021

Commun. Math. Phys.

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“…2 [11, p. 190], [12, p. 44]). Recently, some of us needed the extension of (1.5) to general k ∈ N, k 3, in [1], but were not able to find it in the literature; hence, this note aims at closing this gap.…”

Section: Det H ((I H − A)(i H − B)) = Det H (I H − A)det H (I H − B)mentioning

confidence: 99%

The product formula for regularized Fredholm determinants

Britz

Carey

Gesztesy

et al. 2021

Proc. Amer. Math. Soc. Ser. B

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For trace class operators A, B ∈ B 1 (H) (H a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form det H ((I H − A)(I H − B)) = det H (I H − A)det H (I H − B). When trace class operators are replaced by Hilbert-Schmidt operators A, B ∈ B 2 (H) and the Fredholm determinant det H (I H − A), A ∈ B 1 (H), by the 2nd regularized Fredholm determinant det H,2 (I H − A) = det H ((I H − A) exp(A)), A ∈ B 2 (H), the product formula must be replaced by det H,2 ((I H − A)(I H − B)) = det H,2 (I H − A)det H,2 (I H − B) × exp(− tr H (AB)). The product formula for the case of higher regularized Fredholm determinants det H,k (I H −A), A ∈ B k (H), k ∈ N, k 2, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.

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“…In the context of massless Dirac operators in dimension n = 2 we also refer to [60] (see also [59]). We also note that a global limiting absorption principle for H 0 on R for all n ∈ N, n 2, was proved in [40], [104], [22].…”

mentioning

confidence: 99%

The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

Carey¹,

Gesztesy²,

Levitina³

et al. 2021

Preprint

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Interacting, Massless Dirac Operators 27 5. The Green's Functions of H0(m) and H0 29 6. Trace Ideal Properties of• −δ 34 7. Powers of Resolvents and Trace Ideals 44 8. The Spectral Shift Function: Abstract Facts 48 9. Representing ξ( • ; S, S0) in Terms of Regularized Fredholm Determinants 56 10. Analysis of Im(FH,H 0 (λ + i0)), λ ∈ R 63 11. Analysis of GH,H 0 85 12. Analysis of ξ( • ; H, H0) and an Application to the Witten Index for a Class of Non-Fredholm operators 106 Appendix A. Some Remarks on Block Matrix Operators 115 Appendix B. Asymptotic Results for Hankel Functions 118 B.1. Asymptotics of H (1) ν (ζ) as |ζ| → ∞ 119 B.2. Asymptotics of H (1) ν (ζ) as |ζ| → 0 125 Appendix C. Expansions and estimates of the Free Dirac Green's Matrix 128 Appendix D. A Product Formula for Modified Fredholm Determinants 141 Appendix E. Notational Conventions 147 References 150

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On the Global Limiting Absorption Principle for Massless Dirac Operators

Cited by 17 publications

References 38 publications

Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

The product formula for regularized Fredholm determinants

The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

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