Quantization of classical integrable systems. Part I: quasi-integrable quantum systems

We propose in this work a concept of integrability for quantum systems, which corresponds to the concept of noncommutative integrability for systems in classical mechanics. We determine a condition for quantum operators which can be a suitable replacement for the condition of functional independence for functions on the classical phase space. This condition is based on the properties of the main parts of the operators with respect to the momenta. We are led in this way to the definition of what we call a “quasi-integrable quantum system”. This concept will be further developed in a series of following papers.