Approximations of potentials through the truncation of their inverses

TL;DR

This paper introduces a novel finite-dimensional approximation method for potential operators using matrix inversion and truncation, enabling accurate low-rank representations that preserve spectral properties, demonstrated with nucleon-nucleon potentials.

Contribution

It proposes a new approach to approximate potential operators by combining basis enlargement, matrix inversion, and truncation, maintaining spectral accuracy in finite dimensions.

Findings

The method produces low-rank potential representations with preserved spectral features.

Application to nucleon-nucleon potentials demonstrates high accuracy.

The approach offers a computationally efficient way to approximate infinite-dimensional operators.

Abstract

The inverse of an $\infty \times \infty$ symmetric band matrix can be constructed in terms of a matrix continued fraction. For Hamiltonians with Coulomb plus polynomial potentials, this results in an exact and analytic Green's operator which, even in finite-dimensional representation, exhibits the exact spectrum. In this work we propose a finite dimensional representation for the potential operator such that it retains some information about the whole Hilbert-space representation. The potential should be represented in a larger basis, then the matrix should be inverted, then truncated to the desired size, and finally inverted again. This procedure results in a superb low-rank representation of the potential operator. The method is illustrated with a typical nucleon-nucleon potential.