Relations between multi-resolution analysis and quantum mechanics

TL;DR

This paper presents a model-independent method to construct multi-resolution analyses of square-integrable functions, inspired by quantum mechanics, particularly the fractional quantum Hall effect, providing a clearer framework for generating examples.

Contribution

It introduces a new, general procedure for building multi-resolution analyses from seed functions, linking quantum mechanics concepts to wavelet theory.

Findings

Method is model independent.

Simplifies the construction formulas.

Provides a framework for generating MRA examples.

Abstract

We discuss a procedure to construct multi-resolution analyses (MRA) of $\Lc^2(\R)$ starting from a given {\em seed} function $h(s)$ which should satisfy some conditions. Our method, originally related to the quantum mechanical hamiltonian of the fractional quantum Hall effect (FQHE), is shown to be model independent. The role of a canonical map between certain canonically conjugate operators is discussed. This clarifies our previous procedure and makes much easier most of the original formulas, producing a convenient framework to produce examples of MRA.