Helium-like ions(Z, e, e)in Lagrange Mesh method, interpolating the highly-accurate energy spectravs.Z

This paper addresses the long-standing challenge of calculating the energy spectra for helium-like ions (two-electron systems with a central nuclear charge Z). The authors propose two complementary methods: a direct numerical approach using the Lagrange-mesh method (LMM) and an analytical interpolation formula. The LMM is used to solve the non-relativistic three-dimensional Schrödinger equation in perimetric coordinates, providing high-precision results with minimal computational cost. The interpolation formula is constructed by matching the 1/Z-expansion (valid at large Z) with the Puiseux expansion (valid near the second critical charge Z_B), creating a meromorphic function that bridges these two physical regimes.

The Lagrange-mesh method successfully computes the energy levels for the 1^1S, 2^1S, 2^3S, and 3^3S states with 14-15 significant digits, even for high nuclear charges (Z up to 50). These results serve as a benchmark for the proposed interpolation formula. The authors show that by using a generalized two-point Padé approximant, they can reproduce these highly accurate energies using only nine free parameters. This approach effectively captures the physics of the system, including the level crossings and the behavior near the critical charge where bound states cease to exist, providing a robust tool for predicting energies without the need for expensive, large-scale variational calculations.

Calculating the energy spectra of helium-like ions is a classic problem in quantum mechanics. While variational methods can reach extreme precision, they become computationally prohibitive as the nuclear charge Z increases or for higher excited states. This work provides a computationally efficient alternative that maintains high accuracy, offering a practical solution for researchers who need precise energy values for a wide range of ions. Furthermore, the use of Puiseux expansions to connect different states analytically provides deeper insight into the underlying structure of the helium-like sequence.