Spectral edge regularity of magnetic Hamiltonians

Cornean, Horia D.; Purice, Radu

doi:10.1112/jlms/jdv019

Cited by 12 publications

(19 citation statements)

References 28 publications

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“…Continuity of the spectrum can be proved under quite general conditions on the Hamiltonians [1,3,4], while more refined properties like the Lipschitz behaviour of spectral edges were first proved by Bellissard [5] for discrete Hofstadter-like models [17]. Cornean, Purice and Helffer [7,8,9,11,12]…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

Cornean

Garde

Støttrup

et al. 2018

J. Pseudo-Differ. Oper. Appl.

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First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength b varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least 1/2-Hölder continuous with respect to b in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in b. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

Cornean

Garde

Støttrup

et al. 2018

J. Pseudo-Differ. Oper. Appl.

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“…We may extend the function λ ǫ from (3.24) as a Γ * -periodic function defined on all X * and thus as a symbol defined on Ξ, constant with respect to the variables in X . Using the natural unitary isomorphism L 2 (X ) → ℓ 2 (Γ) ⊗ L 2 (E) we may compute the integral kernel of Op ǫ,κ (λ ǫ ) as in [10] and obtain…”

Section: )mentioning

confidence: 99%

Low lying spectral gaps induced by slowly varying magnetic fields

Cornean

Helffer

Purice

2017

Journal of Functional Analysis

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We consider a periodic Schr\"odinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schr\"odinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.Comment: 55 pages, to appear in J. Funct. Ana

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“…Recalling the results in [17], let H ǫ be the self-adjoint extension of Op ǫ (h) with the domain given by the magnetic Sobolev space. Using the results in [1,19] or [2,6,7] we know that the resolvent set is stable for small variations of ǫ. More precisely, given I as in Hypothesis 1.5, there exists ǫ 0 > 0 small enough, such that σ(H ǫ )∩I = ∅ and dist I, σ(H ǫ )\I ≥ d 0 /2 for any ǫ ∈ [0, ǫ 0 ].…”

Section: Our Main Resultsmentioning

confidence: 99%

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

2019

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We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudodifferential calculus for weak magnetic fields with no spatial decay conditions, when the nonmagnetic symbols have a certain spatial periodicity. We show in great generality that the symbol of the magnetic band Hamiltonian admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian admits a localized composite Wannier basis, we show that the magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In addition, if the magnetic field perturbation is slowly variable, then the spectrum of this matrix is close to the spectrum of a Weyl quantized, minimally coupled symbol.

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Spectral edge regularity of magnetic Hamiltonians

Cited by 12 publications

References 28 publications

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

Low lying spectral gaps induced by slowly varying magnetic fields

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

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