Finite Translations in Solid-State Physics

Żak, J.

doi:10.1103/physrevlett.19.1385

Cited by 256 publications

(154 citation statements)

References 7 publications

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“…Weil [184] defined a Zak transform for locally compact abelian groups, and this transform is often called the Weil-Brezin map in representation theory and abstract harmonic analysis, e.g., [163,4]. Zak rediscovered this transform, which he called the kq transform, in his work on quantum mechanics, e.g., [190,5]. The terminology "Zak transform" has become customary in applied mathematics and signal processing.…”

Section: The Balian-low Theoremmentioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

140

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ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

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Section: The Balian-low Theoremmentioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

140

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“…In order to explain this we introduce the Zak transform [9] on a function S(y) on a real line, defined as…”

Section: Examplementioning

confidence: 99%

Phase space methods for particles on a circle

Zhang

Vourdas

2003

Journal of Mathematical Physics

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The phase space S ×Z for a particle on a circle is considered. Displacement operators in this phase space are introduced and their properties are studied. Wigner and Weyl functions in this context are also considered and their physical interpretation and properties are discussed. All results are compared and contrasted with the corresponding ones for the harmonic oscillator in the R × R phase space.

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“…The demonstration follows similar steps as those presented by Zak in reference [27]. Hence a complete set of functions, for the subspace (Q 0 , P 0 ; Q 2 , P 2 ), is provided by the eigenfunctions of the operators…”

Section: Wave Function and Generalized Bloch Conditionsmentioning

confidence: 99%

Hofstadter spectrum in electric and magnetic fields

Kunold

Torres

2005

Annals of Physics

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The problem of Bloch electrons in two dimensions subject to magnetic and intense electric fields is investigated. Magnetic translations, electric evolution and energy translation operators are used to specify the solutions of the Schrödinger equation. For rational values of the magnetic flux quanta per unit cell and commensurate orientations of the electric field relative to the original lattice, an extended superlattice can be defined and a complete set of mutually commuting space-time symmetry operators is obtained. Dynamics of the system is governed by a finite difference equation that exactly includes the effects of: an arbitrary periodic potential, an electric field orientated in a commensurable direction of the lattice, and coupling between Landau levels. A weak periodic potential broadens each Landau level in a series of minibands, separated by the corresponding minigaps. The addition of the electric field induces a series of avoided and exact crossing of the quasienergies, for sufficiently strong electric field the 1 spectrum evolves into equally spaced discreet levels, in this "magnetic Stark ladder" the energy separation is an integer multiple of hE/aB, with a the lattice parameter.

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Finite Translations in Solid-State Physics

Cited by 256 publications

References 7 publications

History and Evolution of the Density Theorem for Gabor Frames

History and Evolution of the Density Theorem for Gabor Frames

Phase space methods for particles on a circle

Hofstadter spectrum in electric and magnetic fields

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