Multi-resolution analysis and fractional quantum Hall effect: An equivalence result

Abstract

In this paper we prove that any multi-resolution analysis of L 2 (R R) produces, for some values of the filling factor, a single-electron wave function of the lowest Landau level (LLL) which, together with its (magnetic) translated, gives rise to an orthonormal set in the LLL. We also give the inverse construction. Moreover, we extend this procedure to the higher Landau levels and we discuss the analogies and the differences between this procedure and the one previously proposed by J.-P. Antoine and the author.

Multi-resolution analysis and fractional quantum Hall effect: more results

“…a . This result extends, as expected, the one for the square lattice, [2], and explains why we call T j translation operators.…”

“…Here we use the suffix t to emphasize the shape of the lattice, triangular in this paper, while the index 0 means that we are working in the LLL. Just to compare this result with the one obtained for the square lattice, [2], we recall that K (s) 0 (r, s) = e ixy/2 √ 2π 3/4 e iys e −(x+s) 2 /2 . The extensions to higher Landau levels (see again [2] for K (s) 1 ), can be found simply replacing f 0 in (3.10) with the excited harmonic oscillator eigenstates: using f l will produce, clearly, a wave function in the l-th Landau level.…”

“…We have already discussed in [2] the relevance of the ortonormality of differently localized single electron wave functions when the thermodynamical limit of the model is considered. For this reason we look for o.n.…”

“…This is due to the fact that the triangular lattice minimizes, for a fixed electron density, the Coulomb energy. Therefore, in order to have a deeper understanding about the concrete relevance of wavelets in FQHE, it is absolutely necessary to extend the results in [2] to the triangular lattice. Moreover, the analysis of this new geometry, gives a new insight about the procedure and, in particular, clarifies the role of the kq-representation, which was an essential mathematical tool in our original approach.…”

“…We also show that the same procedure can be adapted to lattices of arbitrary shapes, provided that the so-called rationality condition is satisfied. Incidentally, this extension reduces the relevance of the kq-representation with respect to [2].…”