How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping

TL;DR

This paper demonstrates that certain spherically symmetric spacetimes, extending Bertrand's theorem, allow exact solutions to quantum many-body problems in the periodic table, linking classical geometry with quantum spectra.

Contribution

It proves that the topology of extended Bertrand spacetimes enables exact quantum solutions for atoms, connecting classical geometric theorems with quantum spectral analysis.

Findings

Exact quantum solutions for atoms via Bertrand spacetime topology

Recalculation of Tietz's Schrödinger equation results

Analytical proof of the Madelung rule

Abstract

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently extended Bertrand theorem. He designed new static spherically symmetric (Bertrand) spacetimes obeying Einsteins equations and supporting closed orbits. In this work we prove that the topology and geometry of these spacetimes permits to solve quantum many-body problem for any atom of periodic system exactly. The computations of spectrum for any atom of periodic system becomes analogous to that for hydrogen atom. Initially the exact solution of the Schr\"odinger equation for any multielectron atom was obtained by Tietz in 1956. However, neither himself nor others fully comprehended what actually was obtained. We recalculated Tietz results by applying the methodology consistent with new (different from that developed by Fock in 1936) way of solving Schro"odingers equation for hydrogen atom. In the light of this new result it had become possible to demonstrate rigorously that the Tietz-type Schro"odingers equation is in fact describing the quantum motion in Bertrand spacetime. As a bonus, we obtained the analytical proof of the Madelung rule defined in the text