# Extended Hamiltonians and shift, ladder functions and operators

## TL;DR

This paper explores extended Hamiltonians with high-degree constants of motion, focusing on their factorization and quantization via shift and ladder operators, applicable to classical and quantum systems of any finite dimension.

## Contribution

It introduces a new approach to constants of motion based on factorization, expanding the understanding of extended Hamiltonians and their quantization methods.

## Key findings

- Identifies a subclass of extended Hamiltonians with factorized constants of motion.
- Provides explicit expressions for these factorized constants.
- Demonstrates quantization using shift and ladder operators for systems of any finite dimension.

## Abstract

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended manifolds, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians admits factorized constants of motion and we determine their expression. The classical constants may be non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.