Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields

Verdìère, Yves Colin de; Torki-Hamza, Nabila; Truc, Francoise

doi:10.5802/afst.1319

Cited by 42 publications

(55 citation statements)

References 14 publications

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“…This property for the magnetic Schrödinger operators on a locally finite graph was proved in [LL93], [CTT11], [HS01]. In particular, if the magnetic flux of α is zero for any cycle on Γ * , then the magnetic Schrödinger operator H α = ∆ α + Q is unitarily equivalent to the Schrödinger operator H 0 = ∆ 0 + Q without a magnetic field.…”

Section: Floquet Decomposition Of Schrödinger Operators We Introducementioning

confidence: 83%

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Magnetic Schrödinger operators on periodic discrete graphs

Korotyaev

Saburova

2017

Journal of Functional Analysis

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Abstract. We consider magnetic Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the spectrum in terms of the Betti numbers and show that these estimates become identities for specific graphs. We estimate a variation of the spectrum of the Schrödinger operators under a perturbation by a magnetic field in terms of magnetic fluxes. The proof is based on Floquet theory and a precise representation of fiber magnetic Schrödinger operators constructed in the paper.

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Section: Floquet Decomposition Of Schrödinger Operators We Introducementioning

confidence: 83%

“…Colin de Verdière, Torki-Hamza and Truc [CTT11] obtained a condition under which the magnetic Laplacian on an infinite graph is essentially self-adjoint.…”

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confidence: 99%

Magnetic Schrödinger operators on periodic discrete graphs

Korotyaev

Saburova

2017

Journal of Functional Analysis

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“…The paper [1] contains a proof of the discrete version of Kato's inequality and a study of asymptotic properties of the spectrum of a discrete magnetic Schrödinger operator. In the context of a not necessarily complete graph of bounded degree, a sufficient condition for essential selfadjointness of ∆ σ | C c (V ) in ℓ 2 w (V ) is given in [26]. A different model for discrete magnetic Laplacian was given in [27] and, for that model, the essential self-adjointness of a semi-bounded below discrete magnetic Schrödinger operator was proven.…”

Section: Background Of the Problemmentioning

confidence: 98%

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

Milatovic

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn the context of a weighted graph with vertex set V and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator ∆ σ + W , where ∆ σ is the magnetic Laplacian and W : V → R is a function satisfying W (x) ≥ −q(x) for all x ∈ V , with q: V → [1, ∞). The condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M.A. Shubin in the setting of non-compact Riemannian manifolds.

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“…Following [4], we understand by a magnetic potential on the set X a function θ : X × X → [−π, π] such that θ(x, y) = −θ(y, x), x, y ∈ X.…”

Section: Quadratic Formsmentioning

confidence: 99%

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

Güneysu

Keller

Schmidt

2015

Probab. Theory Relat. Fields

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In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödi-nger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman-Kac-Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman-Kac-Itô formula, we prove a Kato inequality, a Golden-Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials. Mathematics Subject Classification

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Essential self-adjointness for combinatorial Schrödinger operators III- Magnetic fields

Cited by 42 publications

References 14 publications

Magnetic Schrödinger operators on periodic discrete graphs

Magnetic Schrödinger operators on periodic discrete graphs

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

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