Matrix approximations of operators

TL;DR

The work analyzes how well fundamental quantum operators can be represented by finite matrices in truncated basis sets, focusing on the identity, kinetic energy, and various potentials. It demonstrates that convergence is slow and plagued by oscillations tied to orthogonal-polynomial structures, and shows that the choice of basis—including symmetry-adapted or translated Gaussians—significantly affects accuracy. By deriving compact, CD-like formulas for kernels and extending them to operators like r^2, it provides a framework to understand and mitigate oscillations. The study also applies Feshbach projection to illustrate how effective, energy-dependent interactions can improve eigenvalues but introduce nonlocality, highlighting practical limits and the need for careful subspace selection in finite-matrix approximations.

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 $δ$ 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 $-δ''$ 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.

Figures (25)

• Figure 1: Contour plot of the "approximate $\delta$-function" $D_N\left( r, s \right)$ obtained with $N=50$ harmonic oscillator components.
• Figure 2: Crest profile $D_{50}\left( r, r \right)$ of the ridge in Fig. \ref{['cnt50']}.
• Figure 3: As for Fig. \ref{['crst50']} but for $N = 200$.
• Figure 4: Shape $D_{50}\left( r, 0 \right)$ of the $s=0$ cut of the ridge when $N=50$.
• Figure 5: Shape $D_{50}\left( r, 3 \right)$ of the $s=3$ cut of the same ridge.