Resonances and Spectral Shift Function near the Landau levels

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

doi:10.5802/aif.2270

Cited by 29 publications

(53 citation statements)

References 22 publications

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“…We deduce also an asymptotic expansion of the SSF near 2bq + ; in the case of v 0 = 0, this expansion is given in Ref. 4. For a positive potentials V which decay slowly enough as ʈX Ќ ʈ → ϱ, this expansion yields a remainder estimate for the corresponding asymptotic relations obtained in Ref.…”

Section: Ssf and Resonancessupporting

confidence: 52%

“…Among them, we can mention the analytic dilation ͑see Aguilar and Combes 1 ͒ or the analytic distortion ͑see Hunziker 10 ͒ and meromorphic continuation of the resolvent or of the scattering matrix ͑see Lax and Philips 14 and Vainberg 20 ͒. For Schrödinger operators with constant magnetic field, the resonances can be defined by analytic dilation ͑only͒ with respect to the variable along the magnetic field ͑see Avron et al, 3 Wang, 21 and Astaburuaga et al 2 ͒ and by meromorphic continuation of the resolvent ͑see Bony et al 4 ͒.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. 4, Bony et al obtained a Breit-Wigner approximation of the SSF near a Landau level for the three-dimensional ͑3D͒ Schrödinger operator with constant magnetic field. For the last operator, under more general assumptions, Fernández and Raikov 9 studied the singularities of the SSF at a Landau level.…”

Section: Introductionmentioning

confidence: 99%

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Resonances and spectral shift function for a magnetic Schrödinger operator

Khochman

2009

Journal of Mathematical Physics

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We consider the three-dimensional Schrödinger operator H 0 with a constant magnetic field and subject to an electric potential v 0 depending only on the variable along the magnetic field x 3 . The operator H 0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H 0 byWe assume also that V and v 0 have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of H = H 0 + V as the eigenvalues of the nonself-adjoint operator obtained from H by analytic distortions of R x 3 . We study their distribution near any fixed real eigenvalue of H 0 , 2bq + for q N. In a ring centered at 2bq + with radii ͑r ,2r͒, we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions ͑x 1 , x 2 ͒. Finally, we deduce a representation of the derivative of the spectral shift function for the operator pair ͑H 0 , H͒ in terms of resonances. This representation justifies the Breit-Wigner approximation and implies a local trace formula.

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Section: Ssf and Resonancessupporting

confidence: 52%

Section: Introductionmentioning

confidence: 99%

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Resonances and spectral shift function for a magnetic Schrödinger operator

Khochman

2009

Journal of Mathematical Physics

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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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“…A lot of results, often related to some dynamical assumptions, are obtained by microlocal arguments (see for instance [4], [22], [17] and [18]). For the magnetic Schrödinger operator, we also mention [5] where localization of resonances is obtained by perturbation methods. If H θ (α) is not non-negative, we can apply Corollary 5 to obtain an estimate for moments of resonances.…”

Section: Corollary 5 Under the Assumptions And Notation Of Theorem 4mentioning

confidence: 99%

Lieb–Thirring estimates for non-self-adjoint Schrödinger operators

Bruneau

Ouhabaz

2008

Journal of Mathematical Physics

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Abstract. For general non-symmetric operators A, we prove that the moment of order γ ≥ 1 of negative real-parts of its eigenvalues is bounded by the moment of order γ of negative eigenvalues of its symmetric part H = 1 2 [A + A * ]. As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schrödinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer [11]. We also discuss moment of resonances of Schrödinger self-adjoint operators.

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Resonances and Spectral Shift Function near the Landau levels

Cited by 29 publications

References 22 publications

Resonances and spectral shift function for a magnetic Schrödinger operator

Resonances and spectral shift function for a magnetic Schrödinger operator

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

Lieb–Thirring estimates for non-self-adjoint Schrödinger operators

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