Stieltjes functions in physics

Stieltjes functions arise naturally in physics when linear differential operators with positive spectra lead to observable quantities that admit positive spectral representations.

This structure makes Padé approximants particularly powerful, enabling accurate physical predictions from perturbation series or spectral data, even when conventional methods fail.

Here are three examples of fields in physics where Stieltjes functions play an important role and Padé approximants provide powerful approximation techniques:

1. Quantum anharmonic oscillator: The perturbation series for the ground-state energy of the quartic anharmonic oscillator diverges factorially at large order. Nevertheless, its Stieltjes structure provides a rigorous foundation for Padé approximation, allowing highly accurate energy estimates to be obtained from perturbative coefficients, even at strong coupling. This is an example of how a divergent series can encode remarkably precise physical information.
2. Composite materials:
   Predicting the effective dielectric constant or electrical conductivity of heterogeneous materials is a challenging problem, since the macroscopic response depends on the microscopic structure of the constituents. Through the spectral representations developed in Bergman–Milton theory, these effective properties can, under appropriate assumptions, be expressed as Stieltjes functions of the contrast between the constituent materials. Padé approximants and related methods can then provide accurate estimates and rigorous bounds on the effective properties.
3. Atomic and molecular polarizabilities:
   The frequency-dependent dipole polarizability of atoms and molecules admits a spectral representation in terms of excitation energies and oscillator strengths. Under the usual assumptions, this gives rise to a Stieltjes structure in the squared frequency. The associated moment problem is the foundation of the Stieltjes imaging technique, which uses Padé-related approximations to reconstruct photoabsorption spectra and photoionization cross sections from discrete variational calculations.

Despite their different physical origins, these examples share a common mathematical structure: positive spectral representations make it possible to exploit the special properties of Stieltjes functions and the convergence behavior of Padé approximants.