The uniqueness of the integrated density of states for the Schrödinger operators with magnetic fields

Doi, Satoshi; Iwatsuka, Akira; Mine, Takuya

doi:10.1007/pl00004872

Cited by 43 publications

(43 citation statements)

References 11 publications

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“…Thus we generalize a result of [5], where the scalar potential was non-negative. Moreover, we prove the existence of IDS for the case of periodical magnetic field and scalar potential.…”

Section: Introductionsupporting

confidence: 70%

“…The solution of problem a) is the main result of this paper: This theorem was proved in [5] in the case where V ≥ 0. The proof in §4 uses some ideas of [5], along with a property of comparison of resolvents, essentially proved in [4], and which requires the hypothesis V − ∈ K n .…”

Section: Definition 13mentioning

confidence: 89%

“…Usually, one works with another Dirichlet realization on Ω (see [5], for instance). More exactly, one considers the operatorH Ω , self-adjoint on L 2 (Ω), associated with the sesqui-linear formh Ω , which is the closure on We shall see in §3 that H Ω =H Ω if Ω is a "Lipschitz domain" (or, following Stein [15], a domain with "minimally smooth boundary").…”

Section: D(h) = U ∈ D(h); −(∇ − Ia)mentioning

confidence: 99%

“…The proof in §4 uses some ideas of [5], along with a property of comparison of resolvents, essentially proved in [4], and which requires the hypothesis V − ∈ K n . Remark 1.5.…”

Section: Definition 13mentioning

confidence: 99%

“…An analysis of the proof of Theorem 1.2 in [5] shows that if F ⊂ LM (r, A, B) (see the notation in [5]), Theorem 1.4 remains true if the Dirichlet boundary conditions are replaced by Neumann boundary conditions. The problem b) will be solved only in a special case.…”

Section: Definition 13mentioning

confidence: 99%

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Uniqueness and Existence of the Integrated Density of States for Schrödinger Operators with Magnetic Field and Electric Potential with Singular Negative Part

Iftimie¹

2005

Publ. Res. Inst. Math. Sci.

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We prove the coincidence of the two definitions of the integrated density of states (IDS) for Schrödinger operators with strongly singular magnetic fields and scalar potentials: the first one using the counting function of eigenvalues of the induced operator on a bounded open set with Dirichlet boundary conditions, the second one using the spectral projections of the whole space operator. Thus we generalize a result of [5], where the scalar potential was non-negative. Moreover, we prove the existence of IDS for the case of periodical magnetic field and scalar potential. §1. Introduction One considers the vector potential a = (a 1 , . . . , a n ) :(which is identified to the differential form 1≤j≤n a j dx j ) and the scalar potential V : R n → R satisfying the following hypotheses:loc (R n ) and V − := max(0, −V ) belongs to the Kato class K n , that is, one has:

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“…Thus we generalize a result of [5], where the scalar potential was non-negative. Moreover, we prove the existence of IDS for the case of periodical magnetic field and scalar potential.…”

Section: Introductionsupporting

confidence: 70%

Section: Definition 13mentioning

confidence: 89%

Section: D(h) = U ∈ D(h); −(∇ − Ia)mentioning

confidence: 99%

“…The proof in §4 uses some ideas of [5], along with a property of comparison of resolvents, essentially proved in [4], and which requires the hypothesis V − ∈ K n . Remark 1.5.…”

Section: Definition 13mentioning

confidence: 99%

Section: Definition 13mentioning

confidence: 99%

See 3 more Smart Citations

Uniqueness and Existence of the Integrated Density of States for Schrödinger Operators with Magnetic Field and Electric Potential with Singular Negative Part

Iftimie¹

2005

Publ. Res. Inst. Math. Sci.

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The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

Hislop¹,

Klopp²

2002

Journal of Functional Analysis

127

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We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Ho¨lder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L p -theory of the spectral shift function for p51 (Comm. Math. Phys. 218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys. 167 (1995), 553-569). We discuss the application of this result to Schro¨dinger operators with random magnetic fields and to band-edge localization. # 2002 Elsevier Science (USA)

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A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids

Leschke

Müller

Warzel

Interacting Stochastic Systems

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Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schrödinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.

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The uniqueness of the integrated density of states for the Schrödinger operators with magnetic fields

Cited by 43 publications

References 11 publications

Uniqueness and Existence of the Integrated Density of States for Schrödinger Operators with Magnetic Field and Electric Potential with Singular Negative Part

Uniqueness and Existence of the Integrated Density of States for Schrödinger Operators with Magnetic Field and Electric Potential with Singular Negative Part

The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids

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