Motion in periodic potentials

Asch, Joachim; Knauf, Andreas

doi:10.1088/0951-7715/11/1/011

Cited by 53 publications

(95 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The relation of the asymptotic velocity operator to the energy-band functions is similar to that for one-dimensional motion in a -periodic scalar potential [22,5]. In case of a globally constant magnetic field (cf.…”

Section: Lemma 211 (Properties Of the Asymptotic Velocity) Let B Bementioning

confidence: 72%

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

View full text Add to dashboard Cite

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.

show abstract

Section: Lemma 211 (Properties Of the Asymptotic Velocity) Let B Bementioning

confidence: 72%

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

View full text Add to dashboard Cite

show abstract

“…Using such a decomposition, it is shown in [2] that ballistic transport takes place for every localized initial condition at an asymptotic velocity characterized in terms of the spectral measure.…”

Section: Difficulties and Limitationsmentioning

confidence: 99%

Approximate spectral theory and wave propagation in quasi-periodic media

Benoit¹,

Duerinckx²,

Gloria³

et al. 2018

Journées Équations Aux Dérivées Partielles

View full text Add to dashboard Cite

In this article we make specific in the quasi-periodic setting the general Floquet-Bloch theory we have introduced for stationary ergodic operators together with the associated approximate spectral theory. As an application we consider the long-time behavior of the Schrödinger flow with a quasi-periodic potential (in the regime of small intensity of the discorder), and the long-time behavior of the wave equation with quasi-periodic coefficients (in the homogenization regime).

show abstract

“…Before we turn to formal statements and proofs, we shortly describe the main ideas, starting with the following observation. Two given quasimodes associated to different KAM tori give rise to different expectations of the sub-principal symbol k · D of the operator H (k) defined in (2). Thus they can be separated energetically by varying the quasimomentum k, and for typical k in the Brillouin zone T * one should not have too many near-degeneracies of energies.…”

Section: Heuristicsmentioning

confidence: 99%

“…A is λ-measurable and generates an image measure ν := λA −1 on R d+1 . On the other hand (see [2]) for almost all k ∈ T * the operator of asymptotic velocitȳ v (k) := lim…”

Section: Approximation Of Eigenfunctionsmentioning

confidence: 99%

Quantum Transport on KAM Tori

Asch

Knauf

1999

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the Schr\"{o}dinger equation of a particle in $\bR^d$ under the influence of a smooth periodic potential. In particular this implies that the distribution of quantum group velocities near energy $E$ converges to the distribution of the classical asymptotic velocities near $E$, up to a term of the order $\cO(1/\sqrt{E})$.Comment: 21 page

show abstract

Motion in periodic potentials

Abstract: We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.

Cited by 53 publications

References 21 publications

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

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

Approximate spectral theory and wave propagation in quasi-periodic media

Quantum Transport on KAM Tori

Contact Info

Product

Resources

About