Home / A. Sergeev

List of Publications

Unpublished Reports

Divergent Series, Summation, and Resonances

Overview of Research

A significant portion of my research is dedicated to the mathematical methods of quantum mechanics, specifically the summation of divergent perturbation series. In many quantum-mechanical problems, standard Rayleigh-Schrödinger perturbation theory yields divergent asymptotic expansions. By developing and applying advanced analytic continuation and summation techniques—such as Padé approximants and their multi-valued generalizations (algebraic/Padé-Hermite approximants)—it is possible to reconstruct physical quantities and transform divergent series into rapidly converging sequences.

These summation techniques are particularly powerful for calculating the complex energies of resonance (quasi-stationary) states. I have successfully applied these methods to a variety of physical systems, including:

• Atoms subjected to strong external electric and magnetic fields (Stark resonances and Zeeman effect).
• Anharmonic oscillators and double-well potentials.
• Unstable atomic anions undergoing ionization.
• Fermi resonances in molecular vibrational spectroscopy.
• Gradient expansions in density functional theory.

Selected Publications on this Topic

• Sergeev A. V. (1995). Summation of the eigenvalue perturbation series by multi-valued Padé approximants: application to resonance problems and double wells. Journal of Physics A: Mathematical and General, 28, 4157. (In this paper, quadratic Padé approximants are used to reconstruct the complex-valued energy of resonances by summing the real terms of the perturbation series, effectively modeling the function on multiple sheets of the Riemann surface).
• Sergeev A. V. and Goodson D. Z. (1998). Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: application to anharmonic oscillators. Journal of Physics A: Mathematical and General, 31, 4301. (Demonstrates how high-degree algebraic approximants improve the summation of divergent series by reproducing the correct singularity structure for cubic, quartic, sextic, and octic oscillators).
• Sergeev A., Jovanovic R., Kais S., and Alharbi F. H. (2016). On the divergence of gradient expansions for kinetic energy functionals in the potential functional theory. Journal of Physics A: Mathematical and Theoretical, 49, 285202. (Demonstrates that the gradient expansion of a fermionic system's density is factorially divergent—similar to quantum anharmonic oscillators—and utilizes Padé-Hermite approximants to successfully sum the series and establish agreement with the exact density).
• Goodson D. Z. and Sergeev A. V. (1999). On the use of algebraic approximants to sum divergent series in vibrational spectroscopy. Journal of Chemical Physics, 110, 8205. (Applies multi-valued algebraic approximants to sum divergent series for eigenstates involved in Fermi resonances, where standard single-valued Padé approximants converge too slowly).
• Popov V. S., Mur V. D., Sergeev A. V. and Weinberg, V. M. (1990). Strong-field Stark effect: perturbation theory and 1/n-expansion. Physics Letters A, 149, 418. (Calculates the Stark shifts and widths of hydrogen atom states using two independent methods, including the summation of divergent perturbation theory series).
• Sergeev A. V. and Kais S. (2001). Resonance states of atomic anions. International Journal of Quantum Chemistry, 82, 255. (Studies the destabilization of atoms by analytic continuation from bound to resonance states, obtaining the complex energies of unstable atomic anions as the nuclear charge decreases).
• Sergeev A. V. and Sherstyuk A. I. (1982). Higher orders and structure of perturbation-theory series for screened Coulomb potential. Soviet Physics - JETP, 55, 625. (Develops an effective method for calculating higher orders of perturbation theory and uses the Padé technique to calculate the real and imaginary parts of the energies of quasi-stationary states).