The kinetic energy operator in the subspaces of wavelet analysis

TL;DR

This paper critiques the standard kinetic energy operator in wavelet analysis, proposing a near-optimal matrix construction that improves numerical calculations by addressing artificial periodicity effects and enabling use with non-differentiable basis functions.

Contribution

It introduces an explicit method to construct a more accurate kinetic energy matrix in wavelet analysis, surpassing the canonical projection approach and applicable even when derivatives of basis functions are undefined.

Findings

The canonical kinetic energy matrix is suboptimal due to artificial periodicity.

A new method for near-optimal matrix construction improves numerical results.

Effective kinetic energy can be viewed as a renormalization with a momentum-dependent mass.

Abstract

At any resolution level of wavelet expansions the physical observable of the kinetic energy is represented by an infinite matrix which is ``canonically'' chosen as the projection of the operator $-\Delta/2$ onto the subspace of the given resolution. It is shown, that this canonical choice is not optimal, as the regular grid of the basis set introduces an artificial consequence of periodicity, and it is only a particular member of possible operator representations. We present an explicit method of preparing a near optimal kinetic energy matrix which leads to more appropriate results in numerical wavelet based calculations. This construction works even in those cases, where the usual definition is unusable (i.e., the derivative of the basis functions does not exist). It is also shown, that building an effective kinetic energy matrix is equivalent to the renormalization of the kinetic energy by a momentum dependent effective mass compensating for artificial periodicity effects.