Abstract
As an illustration of the phase space localization problem of quantum mechanical systems, we study a two-dimensional electron gas in a magnetic field, such as encountered in the Fractional Quantum Hall Effect (FQHE). We discuss a general procedure for constructing an orthonormal basis for the lowest Landau level (as well as for the other Landau levels), starting from an arbitrary orthonormal basis in L 2(ℝ). After some remarks concerning localization properties of the wave functions of any orthonormal basis in the Landau levels, we discuss in detail some relevant examples stemming from wavelet analysis like the Haar, the Littlewood—Paley, the Journé and the splines bases. We also propose a toy model, which indicates how the use of wavelets in the analysis of the FQHE may help in the search for the correct ground state of the system. Finally, we exhibit an intriguing equivalence between FQHE states generated by the so-called magnetic translations and vectors of an orthonormal basis derived from an arbitrary multiresolution analysis.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S.T. Ali, J-P. Antoine, and J-P. GazeauCoherent States Wavelets and Their Generalizations. Springer-Verlag, New York, 2000.
J-P. Antoine, Ph. Antoine, and B. Piraux, Wavelets in atomic physics and in solid state physics, inWavelets in Physics pp. 299–338. J.C. van den Berg (ed.), Cambridge Univ. Press, Cambridge, 1999.
J-P. Antoine and F. Bagarello, Wavelet-like orthonormal bases for the lowest Landau levelJ. Phys. A: Math. Gen.27: 2471–2481, 1994.
J-P. Antoine, A. Coron, and J-M. Dereppe, Water peak suppression: Time-frequency vs. time-scale approachJ. Magn. Reson.144: 189194, 2000.
M. Abramowitz and I.A. StegunHandbook of Mathematical Functions.Dover, New York, 1965.
D. Barache, J-P. Antoine, and J-M. Dereppe, The continuous wavelet transform, a tool for NMR spectroscopyJ. Magn. Reson.128: 1–11, 1997.
F. Bagarello, More wavelet-like orthonormal bases for the lowest Landau level: Some considerationsJ. Phys. A: Math. Gen.27: 5583–5597, 1994.
F. Bagarello, Applications of wavelets in quantum mechanics: A pedagogical exampleJ. Phys. A: Math. Gen.29: 565–576, 1996.
F. Bagarello, Multi-resolution analysis and Fractional Quantum Hall Effect: An equivalence result J. Math. Phys.42: 5115–5129, 2001.
G. Battle, Phase space localization theorem for ondelettesJ. Math. Phys.30: 2195–2196, 1989.
H. Bacry, A. Grossmann, and J. Zak, Proof of the completeness of lattice states inkqrepresentation,Phys. Rev. B, 12: 1118–1120, 1975.
[12] J.J. Benedetto, C. Heil, and D.F. Walnut, Gabor systems and the Balian—Low Theorem,in [22], Chap. 2, pp. 85–122.
B.H. Bransden and C.J. JoachainPhysics of Atoms and Molecules.Longman, London and New York, 1983.
F. Bagarello, G. Morchio, and F. Strocchi, Quantum corrections to the Wigner crystal: A Hartree—Fock expansionPhys. Rev. B48: 53065314, 1993.
M. Boon, Coherent states and Pippard networks, inGroup-Theoretical Methods in Physics (Proc. Nijmegen 1975) pp. 282–288. Lecture Notes Phys. 50, A. Janner, T. Janssen, M. Boon (eds.), Springer-Verlag, Berlin, Heidelberg, 1976.
V. Bargmann, P. Butera, L. Girardello, and J.R. Klauder, On the completeness of coherent statesReports Math. Phys.2: 221–228, 1971.
C.K. ChuiAn Introduction to Wavelets.Academic Press, New York and London, 1992.
I. DaubechiesTen Lectures on Wavelets.SIAM, Philadelphia, 1992.
I. Dana and J. Zak, Adams representation and localization in a magnetic fieldPhys. Rev. B28: 811–820, 1983.
G. Fano, Comments on the mathematical structure of the Laughlin wave function for the anomalous quantum Hall effect. A pedagogical exposition. Lectures given at the International School for Advanced Studies, Trieste, May 1984.
R. Ferrari, Two-dimensional electrons in a strong magnetic field: A basis for single particle statesPhys. Rev. B42: 4598–4609, 1990.
H.G. Feichtinger and T. Strohmer (eds.)Gabor Analysis and Algorithms — Theory and Applications.Birkhäuser, Boston-Basel-Berlin, 1998.
D. Gabor, Theory of communicationJ. Inst. Electr. Eng. (London)93: 429–457, 1946.
M. Greiter and I.A. McDonald, Hierarchy of quantized Hall states in double-layer electron systemsNucl. Phys. B [FS]410: 521–534, 1993.
I.S. Gradshteyn and I.M. RyzhikTables of Integrals Series and Products. Academic Press, New York and London, 1980.
F.D.M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid statesPhys. Rev. Lett.51: 605–608, 1983.
G. KlambauerReal Analysis.Elsevier, Amsterdam, 1973.
R.B. Laughlin, Quantized motion of three two-dimensional electrons in a strong magnetic fieldPhys. Rev. B27: 3383–3389, 1983.
G. MorandiQuantum Hall Effect.Bibliopolis, Napoli, 1988.
G. MorandiThe Role of Topology in Classical and Quantum Physics.Lecture Notes Phys. m7, Springer-Verlag, Berlin, Heidelberg, 1992.
Jean-Pierre Antoine and Fabio Bagarello
M. Moshinsky and C. Quesne, Linear canonical transformations and their unitary representationsJ. Math. Phys.12: 1772–1780, 1971.
K. Maki and X. Zotos, Static and dynamic properties of a two-dimensional Wigner crystal in a strong magnetic fieldPhys. Rev. B28: 4349–4356, 1983.
A.M. Perelomov, On the completeness of a system of coherent statesTheor. Math. Phys.6: 156–164, 1971.
A.M. PerelomovGeneralized Coherent States and Their Applications.Springer-Verlag, Berlin, Heidelberg, 1986.
D. Yoshioka and P.A. Lee, Ground-state energy of a two-dimensional charge-density-wave state in a strong magnetic fieldPhys. Rev. B27: 4986–4996, 1983.
J. Zak, Dynamics of electrons in solids in external fieldsPhys. Rev.168: 686–695, 1968.
J. Zak, The kg-representation in the dynamics of electrons in solidsSolid State Physics27: 1–62, 1972.
J. Zak, Orthonormal sets of localized functions for a Landau levelJ. Math. Phys.39: 4195–4200, 1998.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Antoine, JP., Bagarello, F. (2003). Localization Properties and Wavelet-Like Orthonormal Bases for the Lowest Landau Level. In: Feichtinger, H.G., Strohmer, T. (eds) Advances in Gabor Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0133-5_10
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0133-5_10
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6627-3
Online ISBN: 978-1-4612-0133-5
eBook Packages: Springer Book Archive