Quantum Integrals from Coalgebra Structure

TL;DR

This paper constructs numerous quantum integrals for non-Euclidean quantum systems like the hydrogen atom and harmonic oscillator using coalgebra structures, confirming previous conjectures and extending superintegrability proofs.

Contribution

It introduces a multi-dimensional factorization method to generate integrals and extends systems via coalgebra, advancing understanding of quantum superintegrability.

Findings

Constructed 2N-1 integrals of arbitrary order for quantum systems.

Confirmed the conjecture of Riglioni (2013) regarding integrals.

Applied dimensional reduction to prove superintegrability of known Hamiltonians.

Abstract

Quantum versions of the hydrogen atom and the harmonic oscillator are studied on non Euclidean spaces of dimension N. 2N-1 integrals, of arbitrary order, are constructed via a multi-dimensional version of the factorization method, thus confirming the conjecture of D Riglioni 2013 J. Phys. A: Math. Theor. 46 265207. The systems are extended via coalgebra extension of sl(2) representations, although not all integrals are expressible in these generators. As an example, dimensional reduction is applied to 4D systems to obtain extension and new proofs of the superintegrability of known families of Hamiltonians.