Harris J. Silverstone
Johns Hopkins University
3400 North Charles St.
Baltimore, MD 21218
PhD - California Institute of Technology
NSF Post Doctoral Fellow - Yale University
Professor Silverstone is known for his work on expansions in quantum chemistry. The standard (Rayleigh-Schrödinger) perturbation theory for the energy levels of atoms in external electric and magnetic fields provides prime examples of expansions that diverge but are summable. Some are term-by-term real but have complex sums. Others are explicitly complex but have real sums. All these divergent expansions have associated with them exponentially small subseries that play a key role in their asymptotics. The exponentially small subseries are also important for “connection formulas” in semiclassical approximations of quantum mechanics.
The so-called JWKB or semiclassical expansion in powers of hbar in quantum theory is asymptotic and divergent. By applying “Borel summation” to the JWKB expansion, exact, convergent, numerical values can be obtained, and historical confusion about the directionality of JWKB “connection formulas” set to rest unambiguously. When applied to the JWKB expansion for a quadratic barrier, analytic solutions can be found term-by-term, and their Borel sum proves a conjecture of Sato about the ratio of the JWKB solution and the Weber function.
Other earlier work provides a detailed feature-by-feature analysis of the photoionization cross section of hydrogen in an electric field in terms of expansions over resonances. Extension to infinite fields leads to a complete understanding of the “Bender-Wu” branch cuts of the anharmonic oscillator, a somewhat serendipitous result. Earlier work used perturbation expansions to elucidate the transition between classical, semiclassical and quantum mechanics. Still earlier work concerned electron correlation (how electrons avoid each other) in atoms and molecules, the evaluation of molecular integrals using expansions generated from Fourier-transforms, the use of piecewise-polynomial expansions for electronic wave functions, the rate of convergence of the configuration interaction expansion, asymptotic behavior of atomic and molecular wave functions, and simulation of electron magnetic resonance spectra for high-spin iron in heme proteins and in enzymes.