On the Regularity of the Hausdorff Distance Between Spectra of Perturbed Magnetic Hamiltonians

J. Pseudo-Differ. Oper. Appl.

Garde

Støttrup

et al. 2018

Self Cite

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.

Section: Introduction and Main Resultsmentioning

confidence: 99%

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

J. Pseudo-Differ. Oper. Appl.

Garde

Støttrup

et al. 2018

Self Cite

“…This paper is dedicated to the proof of (1.10). The estimate (1.11) is a direct consequence of the results of [9], but we included it here in order to make a comparison with previous results obtained for constant magnetic fields (i.e. κ = 0), where the values of the a k 's and b k 's from (1.10) are found with much better accuracy, see Theorem 2.2.…”

Section: More On the State Of The Artmentioning

confidence: 96%

“…Proof. The first point follows from the spectral stability (see Corollary 1.2 in [24] or Theorem 1.4 in [2] or Theorem 3.1 in [9] for a more precise result).…”

Section: The Mapmentioning

confidence: 99%

Low lying spectral gaps induced by slowly varying magnetic fields

Journal of Functional Analysis

Helffer

Purice

2017

Self Cite

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

“…Continuity of spectral gaps has been proven in some cases in the past. The problem occurred, in particular, with the dependence of the spectrum as a function of an external, uniform magnetic field [47,13,14]. It also occurred for the Schrödinger operator on the line R, with almost periodic potentials in which the frequency module is varying [19].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, to the best of their knowledge, the authors believe that the Lipschitz/Hölder continuity dependence of the spectrum in terms of the underlying atomic configuration, described in the present article, is new. A bit of explanation for the method used here is in order, a method which employs and adapts a numbers of ideas from [13,14] as well as [4]. Consider the simplest case, the Schrdinger operator defined in (1.4) by the discrete Laplacian plus a potential on Z d .…”

Section: Introductionmentioning

confidence: 99%

Hölder Continuity of the Spectra for Aperiodic Hamiltonians

Beckus

Bellissard

2019

Ann. Henri Poincaré

We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.