On Eigenfunction Decay for Two Dimensional�Magnetic Schr�dinger Operators

Cornean, Horia D.; Nenciu, G.

doi:10.1007/s002200050314

Cited by 37 publications

(45 citation statements)

References 17 publications

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“…Doing some very formal computations one can show that at T = 0 and =0, 12 ͑B , ͒ as given by ͑3.9͒ and ͑3.10͒ coincide with the formula for the quantized Hall conductivity ͓see, e.g., formulas ͑5͒ and ͑6͒ in Ref. 34͔ which in turn gives ͑again at the heuristic level͒ the well-known Widom-Streda formula.…”

Section: The Zero Frequency Limit At T = 0: a Rigorous Proof Of Tmentioning

confidence: 75%

“…Moreover, the function defined by R 3 x → s B ͑x͒ ª ͚ s=1 2 A s,s ϱ ͑x , x͒ C is continuous and periodic with respect to Z 3 . ͑iii͒ The thermodynamic limit exists, 12 ͑ϱ͒ ͑B,͒ ª lim…”

Section: Gauge Invariance and Existence Of The Thermodynamic Limitmentioning

confidence: 99%

“…This is the typical situation for semiconductors and/or isolators. In the absence of spin, the Widom-Streda formula roughly states that 12 …”

Section: The Zero Frequency Limit At T = 0: a Rigorous Proof Of Tmentioning

confidence: 99%

“…Accordingly, one can rewrite 12 Note that since the singularities of f FD ͑z͒ lie on ͑c + d͒ /2+iy , y ͑−ϱ , ϱ͒, one can take ⌫ ␤, j independent of ␤, i.e., one can take d = ͉I͉͑͒ /2 in ͑2.29͒. At this point we take the limit ␤ → ϱ.…”

Section: ͑4͒mentioning

confidence: 99%

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The Faraday effect revisited: General theory

Cornean

Nenciu

Pedersen

2006

Journal of Mathematical Physics

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This paper is the first in a series revisiting the Faraday effect, or more generally, the theory of electronic quantum transport/optical response in bulk media in the presence of a constant magnetic field. The independent electron approximation is assumed. At zero temperature and zero frequency, if the Fermi energy lies in a spectral gap, we rigorously prove the Widom-Streda formula. For free electrons, the transverse conductivity can be explicitly computed and coincides with the classical result. In the general case, using magnetic perturbation theory, the conductivity tensor is expanded in powers of the strength of the magnetic field $B$. Then the linear term in $B$ of this expansion is written down in terms of the zero magnetic field Green function and the zero field current operator. In the periodic case, the linear term in $B$ of the conductivity tensor is expressed in terms of zero magnetic field Bloch functions and energies. No derivatives with respect to the quasi-momentum appear and thereby all ambiguities are removed, in contrast to earlier work.Comment: Final version, accepted for publication in J. Math. Phy

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Section: The Zero Frequency Limit At T = 0: a Rigorous Proof Of Tmentioning

confidence: 75%

Section: Gauge Invariance and Existence Of The Thermodynamic Limitmentioning

confidence: 99%

“…This is the typical situation for semiconductors and/or isolators. In the absence of spin, the Widom-Streda formula roughly states that 12 …”

Section: The Zero Frequency Limit At T = 0: a Rigorous Proof Of Tmentioning

confidence: 99%

Section: ͑4͒mentioning

confidence: 99%

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The Faraday effect revisited: General theory

Cornean

Nenciu

Pedersen

2006

Journal of Mathematical Physics

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“…What counts is the magnetic field, which is hidden. Obviously, good spectral results should be expressed with no reference to any vector potential (see also [8] and [31] for related results concerning the discrete spectrum). On the other hand, it is clear that the magnetic field plays a very different role than a scalar potential.…”

Section: Introductionmentioning

confidence: 99%

Spectral and propagation results for magnetic Schrödinger operators; A C∗-algebraic framework

Măntoiu

Purice

Richard

2007

Journal of Functional Analysis

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We study generalized magnetic Schrödinger operators of the form H h (A, V ) = h(Π A ) + V , where h is an elliptic symbol, Π A = −i∇ − A, with A a vector potential defining a variable magnetic field B, and V is a scalar potential. We are mainly interested in anisotropic functions B and V . The first step is to show that these operators are affiliated to suitable C * -algebras of (magnetic) pseudodifferential operators. A study of the quotient of these C * -algebras by the ideal of compact operators leads to formulae for the essential spectrum of H h (A, V ), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same C * -algebras by other ideals give localization results on the functional calculus of the operators H h (A, V ), which can be interpreted as non-propagation properties of their unitary groups.

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Two Dimensional Magnetic Schrödinger Operators: Width of Mini Bands in the Tight Binding Approximation

Cornean¹,

Nenciu²

2000

Ann. Henri Poincaré

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On Eigenfunction Decay for Two Dimensional�Magnetic Schr�dinger Operators

Cited by 37 publications

References 17 publications

The Faraday effect revisited: General theory

The Faraday effect revisited: General theory

Spectral and propagation results for magnetic Schrödinger operators; A C∗-algebraic framework

Two Dimensional Magnetic Schrödinger Operators: Width of Mini Bands in the Tight Binding Approximation

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