Localization Properties and Wavelet-Like Orthonormal Bases for the Lowest Landau Level

J.P. Antoine

Institut de Physique Théorique, Université Catholique de Louvain, Chemin du Cyclotron 2, B-1348 - Louvain-la-Neuve, Belgium
email: antoine@fyma.ucl.ac.be

F. Bagarello

Dipartimento di Matematica ed Applicazioni, Facoltà di Ingegneria, Università di Palermo, Viale delle Scienze, I-90128 Palermo, Italy
email: bagarel@unipa.it

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 $L^2({\mathbb R})$. 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.