Pointwise reconstruction of wave functions from their moments through weighted polynomial expansions: an alternative global-local quantization procedure

TL;DR

This paper presents a novel method for reconstructing quantum wave functions by combining global moment-based expansions with local Taylor series, using weighted orthogonal polynomial expansions within the OPPQ framework.

Contribution

It introduces an alternative quantization procedure that couples global moments and local structure through weighted polynomial expansions, enhancing wave function reconstruction.

Findings

Provides a convergent basis expansion for wave functions

Enables direct coupling of global moments and local series

Improves local structure reconstruction of quantum states

Abstract

Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrodinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define both an $L^2$ convergent, non-orthogonal, basis expansion with sufficient point-wise convergent behaviors enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed by Handy and Vrinceanu. The quantization approach pursued here defines an alternative strategy emphasizing the relevance OPPQ to the reconstruction of the local structure of the physical states.