Multi-Resolution Analysis and Fractional Quantum Hall Effect: an Equivalence Result

TL;DR

This paper establishes a mathematical equivalence between multi-resolution analysis of square-integrable functions and the construction of orthonormal bases in the lowest Landau level for fractional quantum Hall systems, extending to higher levels.

Contribution

It proves that multi-resolution analysis naturally generates orthonormal wave functions in the Landau levels and provides an inverse construction, linking wavelet theory with quantum Hall physics.

Findings

Constructs orthonormal sets in the LLL from multi-resolution analysis.

Extends the construction to higher Landau levels.

Discusses the relation to previous methods by Antoine and the author.

Abstract

In this paper we prove that any multi-resolution analysis of $\Lc^2(\R)$ 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.