Counting Function of Characteristic Values and Magnetic Resonances

Bony, Jean-François; Bruneau, Vincent; Raikov, Georgi

doi:10.1080/03605302.2013.777453

Cited by 21 publications

(41 citation statements)

References 34 publications

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“…The introduction of randomness should be a beginning of a new development in the subject. We also do not discuss the important case of resonances for magnetic Schrödinger operators and refer to Alexandrova-Tamura [4], Bony-Bruneau-Raikov, [26] and Tamura [258,259] for 6 Small prizes are offered by the author for the first proofs of the conjectures within five years of the publication of this survey: a dinner in a restaurant for an even numbered conjecture and in a restaurant, for an odd numbered one.…”

Section: Some Recent Developmentsmentioning

confidence: 99%

Mathematical study of scattering resonances

Zworski

2017

Bull. Math. Sci.

118

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Section: Some Recent Developmentsmentioning

confidence: 99%

Mathematical study of scattering resonances

Zworski

2017

Bull. Math. Sci.

118

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“…For interesting applications of this circle of ideas see also [1], [3], [7], [38]. Next we strengthen Hypothesis 4.1 as follows:…”

Section: On the Notion Of An Index Of Meromorphic Operator-valued Funmentioning

confidence: 63%

On the index of meromorphic operator-valued functions and some applications

Behrndt¹,

Gesztesy

Holden

et al. 2017

Functional Analysis and Operator Theory for Quantum Physics

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Dedicated with great pleasure to Pavel Exner at the occasion of his 70th birthday.Abstract. We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided. Contents 1. Introduction 1 2. On Factorizations of Analytic Operator-Valued Functions 3 3. Algebraic Multiplicities of Zeros of Analytic Fredholm Operators 7 4. On the Notion of an Index of Meromorphic Operator-Valued Functions 10 5. Abstract Perturbation Theory and Applications to Birman-Schwinger-Type Operators 12 6. An Index Formula for the Weyl-Titchmarsh Function Associated to Closed Extensions of Dual Pairs 17 References 22where (with the symbol denoting contour integrals)2) for 0 < ε < ε 0 and D(z 0 ; ε 0 )\{z 0 } ⊂ ρ(T ); here D(z 0 ; r 0 ) ⊂ C is the open disk with center z 0 and radius r 0 > 0, and C(z 0 ; r 0 ) = ∂D(z 0 ; r 0 ) the corresponding circle. The geometric multiplicity m g (z 0 ; T ) of an eigenvalue z 0 ∈ σ p (T ) is defined by m g (z 0 ; T ) = dim(ker((T − z 0

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“…Formally, our Theorem 3.1 resembles the results of [14] on compactly supported V , which however are less precise than (3.3) and (3.4): the right-hand side of the analogue of (3.3) (resp., of (3.4)) in [14] is − 1 2 Φ 0 (λ)(1 + o(1)) (resp., ± 1 4 Φ 0 (λ)(1 + o(1))). A problem closely related to the analysis of the SSF ξ(·; H 0 + V, H 0 ) as E → Λ q for a given q ∈ Z + , is the investigation of accumulation of resonances of H 0 + V at Λ q performed in [8,9,10]. The asymptotic distribution of resonances near the Landau levels for the operators H ± considered in this article, is studied in [12].…”

Section: Resultsmentioning

confidence: 99%

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

Bruneau¹,

Raikov²

2020

Ann. Henri Poincaré

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We consider the 3D Schrödinger operator H 0 with constant magnetic field B of scalar intensity b > 0, and its perturbations H + (resp., H − ) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain Ω in ⊂ R 3 . We introduce the Krein spectral shift functions ξ(E; H ± , H 0 ), E ≥ 0, for the operator pairs (H ± , H 0 ), and study their singularities at the Landau levels Λ q := b(2q + 1), q ∈ Z + , which play the role of thresholds in the spectrum of H 0 . We show that ξ(E; H + , H 0 ) remains bounded as E ↑ Λ q , q ∈ Z + , being fixed, and obtain three asymptotic terms of ξ(E; H − , H 0 ) as E ↑ Λ q , and of ξ(E; H ± , H 0 ) as E ↓ Λ q . The first two terms are independent of the perturbation while the third one involves the logarithmic capacity of the projection of Ω in onto the plane perpendicular to B.

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Counting Function of Characteristic Values and Magnetic Resonances

Cited by 21 publications

References 34 publications

Mathematical study of scattering resonances

Mathematical study of scattering resonances

On the index of meromorphic operator-valued functions and some applications

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

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