Examples of Absolutely Continuous Schrödinger Operators in Magnetic Fields

Iwatsuka, Akira

doi:10.2977/prims/1195179628

Cited by 64 publications

(78 citation statements)

References 6 publications

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“…Second, if we suppose that b ∈ C 1 (R), it follows from the above remark and Lemma 5.2 that the band functions are non constant with no constraint on . We mention that Iwatsuka [12] proves absolutely continuity of the spectrum provided the magnetic field b(x) is smooth b(x) ∈ C ∞ (R), it is bounded 0 < M − b(x) M + < ∞, and lim sup x→−∞ b(x) < lim inf x→∞ b(x) or the reverse inequality. Furthermore, under the additional condition that b(x) is monotone (without any regularity assumption), Dombrowski, Germinet, and Raikov [6, Corollary 2.3] proved the quantization of the edge current (see section 1.1).…”

Section: 21mentioning

confidence: 99%

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Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Hislop

Soccorsi

2015

Journal of Mathematical Analysis and Applications

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We study purely magnetic Schrödinger operators in two-dimensions $(x,y)$ with magnetic fields $b(x)$ that depend only on the $x$-coordinate. The magnetic field $b(x)$ is assumed to be bounded, there are constants $0 < b_- < b_+ < \infty$ so that $b_- \leq b(x) \leq b_+$, and outside of a strip of small width $-\epsilon < x < \epsilon$, where $0 < \epsilon < b_-^{-1/2}$, we have $b(x) = b_\pm x$ for $\pm x > \epsilon$. The case of a jump in the magnetic field at $x=0$ corresponding to $\epsilon=0$ is also studied. We prove that the magnetic field creates an effective barrier near $x=0$ that causes edge currents to flow along it consistent with the classical interpretation. We prove lower bounds on edge currents carried by states with energy localized inside the energy bands of the Hamiltonian. We prove that these edge current-carrying states are well-localized in $x$ to a region of size $b_-^{-1/2}$, also consistent with the classical interpretation. We demonstrate that the edge currents are stable with respect to various magnetic and electric perturbations. For a family of perturbations compactly supported in the $y$-direction, we prove that the time asymptotic current exists and satisfies the same lower bound

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Section: 21mentioning

confidence: 99%

“…Iwatsuka [12] studied the case when the magnetic field is not constant at infinity. In particular, he considered the case when b(x, y) = b(x) is a function of x only.…”

Section: Introduction: Magnetic Barriers and Edge Currentsmentioning

confidence: 99%

Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Hislop

Soccorsi

2015

Journal of Mathematical Analysis and Applications

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“…[54, Thm. XIII.86] and [28,Lemma 2.6]) that the nth band contributes to the absolutely continuous spectrum of H(b). In the other case, |β n | = 0, the nth band contributes to the pure-point spectrum of H(b) [54, Thm.…”

Section: Definition 21 (Umf) a Unidirectionally Constant Magnetic Fmentioning

confidence: 99%

“…The quantum analogue of this unbounded motion should manifest itself in the exclusive appearence of absolutely continuous spectrum of the underlying one-particle Schrödinger operator with a UMF (only), which is not globally constant. Although plausible from the (quasi-) classical picture, a mathematical proof of this conjecture is nontrivial and has been accomplished so far only for certain classes of UMF's [28,45]. From the same picture, the absolutely continuous spectrum should come with ballistic transport along the direction perpendicular to the gradient of the UMF.…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, the first rigorous work explicitly dealing with a model involving a random UMF (with zero mean-value) is one of Ueki [67]. Models for a single particle on the plane Ê 2 subjected to a non-random unidirectionally constant magnetic field (UMF) have been the object of various studies in the mathematics [28,13,45,42] and physics [46,49,37,56,61,39] literature. These models illustrate unadulteratedly that inhomogeneous magnetic fields have a tendency to delocalise charged particles along the direction perpendicular to the magnetic-field gradient.…”

Section: Introductionmentioning

confidence: 99%

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Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

Leschke

Warzel

Weichlein

2006

Ann. Henri Poincaré

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ABSTRACT. We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a "unidirectionally constant" magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schrödinger operator (without scalar potential) by analysing its "energy-band structure". In particular, for an ergodic random UMF we provide conditions which ensure that the operator's entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localisation. The conditions are verified, for example, for Gaussian and Poissonian random UMF's with non-zero mean-values. These results may be viewed as "random analogues" of results first obtained by A.

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Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

147

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Examples of Absolutely Continuous Schrödinger Operators in Magnetic Fields

Cited by 64 publications

References 6 publications

Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

Quantum Waveguides

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