Semiclassical Analysis of the Largest Gap of Quasi-Periodic Schrödinger Operators

H. Krüger

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 256-268
  • ISSN: 0973-5348

Abstract

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In this note, I wish to describe the first order semiclassical approximation to the spectrum of one frequency quasi-periodic operators. In the case of a sampling function with two critical points, the spectrum exhibits two gaps in the leading order approximation. Furthermore, I will give an example of a two frequency quasi-periodic operator, which has no gaps in the leading order of the semiclassical approximation.

How to cite

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Krüger, H.. "Semiclassical Analysis of the Largest Gap of Quasi-Periodic Schrödinger Operators." Mathematical Modelling of Natural Phenomena 5.4 (2010): 256-268. <http://eudml.org/doc/197639>.

@article{Krüger2010,
abstract = {In this note, I wish to describe the first order semiclassical approximation to the spectrum of one frequency quasi-periodic operators. In the case of a sampling function with two critical points, the spectrum exhibits two gaps in the leading order approximation. Furthermore, I will give an example of a two frequency quasi-periodic operator, which has no gaps in the leading order of the semiclassical approximation.},
author = {Krüger, H.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {gaps in the spectrum; Schrödinger operators; semiclassical analysis},
language = {eng},
month = {5},
number = {4},
pages = {256-268},
publisher = {EDP Sciences},
title = {Semiclassical Analysis of the Largest Gap of Quasi-Periodic Schrödinger Operators},
url = {http://eudml.org/doc/197639},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Krüger, H.
TI - Semiclassical Analysis of the Largest Gap of Quasi-Periodic Schrödinger Operators
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 256
EP - 268
AB - In this note, I wish to describe the first order semiclassical approximation to the spectrum of one frequency quasi-periodic operators. In the case of a sampling function with two critical points, the spectrum exhibits two gaps in the leading order approximation. Furthermore, I will give an example of a two frequency quasi-periodic operator, which has no gaps in the leading order of the semiclassical approximation.
LA - eng
KW - gaps in the spectrum; Schrödinger operators; semiclassical analysis
UR - http://eudml.org/doc/197639
ER -

References

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  8. J.P. Guillement, B. Helffer, P. Treton. Walk inside Hofstadter’s butterfly. J. Phys. France, 50 (1989), 2019–2058. 
  9. B. Helffer, P. Kerdelhué, J. Sjöstrand. Le papillon de Hofstadter revisité. Mém. Soc. Math. France (N.S.), No. 43 (1990), 87 pp.  Zbl0732.44004
  10. D. Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B, 14 (1976), 2239. 
  11. H. Krüger. Probabilistic averages of Jacobi operators, Comm. Math. Phys., 295 (2010), No. 3, 853–875. Zbl1192.47028
  12. H. Krüger. In preparation.  
  13. G. Teschl. Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon., 72, Amer. Math. Soc., Rhode Island, 2000.  

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