Rigorous convergence theory for Padé-type approximants in interacting many-body systems

Establish a fully rigorous convergence theory for Padé-type approximants applied to interacting many-body systems, providing mathematically justified conditions under which such rational approximations converge.

Background

The paper demonstrates numerically that two-point Padé approximants yield accurate global approximations for correlation functions in lattice φ4 theory, and provides heuristic arguments for convergence based on analyticity and analytic continuation.

Despite these observations, the authors state that a complete mathematical framework proving convergence for Padé-type approximants in interacting many-body settings is lacking, highlighting the need for a rigorous convergence theory.

References

Theoretically, a fully rigorous convergence theory for Pad e-type approximants in interacting many-body systems is still missing , and it would be interesting to see how 2Pad e approximants behave near the thermodynamic limit where genuine non-analyticities start to arise.

— Strong-coupling expansion and two-point Padé approximation for lattice $φ^4$ field theory  (2604.00525 - Zhu et al., 1 Apr 2026) in Section 4, Conclusion

Although (diagrammatic) series extrapolations have been exploited in the context of various numerical methods, including diagrammatic Monte Carlo, the high-temperature expansion, and the numerical linked-cluster method, developing a stable extrapolation scheme for the continuous spectral function is an open problem. Finding a robust solution to this problem is especially important to self-consistent iteration schemes like $X$OA.

— Quantics tensor cross interpolation for high-order strong-coupling expansions  (2608.23160 - Kim et al., 24 Aug 2026) in Section Conclusion