Matrix approximations of operators

TL;DR

This paper analyzes the accuracy and convergence of finite matrix representations of operators, highlighting slow convergence and residual oscillations, and compares different bases and formal approaches.

Contribution

It provides a detailed analysis of operator approximation by matrices, including formulas for oscillations and a comparison of basis choices and formal methods.

Findings

Slow convergence of matrix approximations with residual oscillations.

Separable potentials are better represented than local interactions.

Alternative bases can influence approximation quality.

Abstract

The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak approximation of the $\delta$ function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of $-\delta''$ is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.