On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers

Jory Griffin

doi:10.1063/1.4960469

On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers

Jory Griffin

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Abstract

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (to
appear); [e-print arXiv:1409.6878 (2014)] who show momentum localisation for zero quasi-momentum in 2-dimensions, and is the first result in this direction in 3-dimensions.

Original language	English
Article number	082101
Number of pages	13
Journal	Journal of Mathematical Physics
Volume	57
DOIs	https://doi.org/10.1063/1.4960469
Publication status	Published - 9 Aug 2016

Bibliographical note

16 pages, 2 figures

Keywords

math-ph
math.MP

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10.1063/1.4960469

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This is the accepted author manuscript (AAM). The final published version (version of record) is available online via AIP Publishing at http://dx.doi.org/10.1063/1.4960469. Please refer to any applicable terms of use of the publisher.
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title = "On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers",

abstract = "Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (toappear); [e-print arXiv:1409.6878 (2014)] who show momentum localisation for zero quasi-momentum in 2-dimensions, and is the first result in this direction in 3-dimensions.",

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N2 - Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Ueberschaer and Kurlberg, J. Eur. Math. Soc. (JEMS) (toappear); [e-print arXiv:1409.6878 (2014)] who show momentum localisation for zero quasi-momentum in 2-dimensions, and is the first result in this direction in 3-dimensions.

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JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

M1 - 082101

ER -

On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers

Abstract

Bibliographical note

Keywords

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