Professor Silverstone studies and publishes about expansions as tools in quantum chemistry. The question with some expansions is how fast do they converge, as with configuration interaction for atomic and molecular wave functions. (One answer: like l–4, where l is the highest angular momentum orbital used in the expansion.) With divergent expansions, questions are how fast do they diverge and how to decode the encrypted information, as with the power-series-in-electric-field-strength for of the energy of an atom in an electric field – Rayleigh-Schrödinger perturbation theory. (One set of answers: factorially fast; decode with Borel summation via Padé approximants. A second: decode via hyperasymptotic summation.) With divergent power series there are often additional divergent power series multiplied by an exponentially small factor. In the electric field case, this exponentially small series gives the tunneling field-ionization rate. The Jeffries-Wentzel-Kramers-Brillouin (JWKB) method expands the logarithm of the wave function in a divergent power series in ħ, and its associated exponentially small series are especially important for tunneling.
Recent open issues involve the 1/r12 Coulomb interaction in molecular wave functions: evaluation of multicenter integrals using numerical contour integration; the conditional convergence of the bipolar expansion when charge distributions overlap.