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Approximations of potentials through the truncation of their inverses

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.