Quantum Bi-Hamiltonian Systems

TL;DR

This paper introduces quantum bi-Hamiltonian systems as operator algebra derivations compatible with two associative structures, using Nijenhuis tensors, and provides explicit examples like the harmonic oscillator.

Contribution

It defines quantum bi-Hamiltonian systems using associative Nijenhuis tensors and offers explicit examples, advancing the understanding of compatible algebraic structures in quantum mechanics.

Findings

Defined quantum bi-Hamiltonian systems as inner derivations in operator algebras.

Developed a method to find compatible associative structures using Nijenhuis tensors.

Provided explicit examples, including the harmonic oscillator.

Abstract

We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.