Particular Integrability and (Quasi)-exact-solvability

TL;DR

The paper introduces the concept of particular integrability, demonstrating how certain operators commute on subspaces, and finds particular integrals for specific quantum and classical systems, highlighting special trajectories and constants of motion.

Contribution

It defines particular integrability and derives particular integrals for (quasi)-exactly-solvable quantum and classical systems, expanding understanding of their structure.

Findings

Particular integrals for 1D (quasi)-exactly-solvable Schrödinger operators.

Particular integrals for Calogero-Sutherland Hamiltonians for all roots.

Identification of special classical trajectories with constants of motion.

Abstract

A notion of a particular integrability is introduced when two operators commute on a subspace of the space where they act. Particular integrals for one-dimensional (quasi)-exactly-solvable Schroedinger operators and Calogero-Sutherland Hamiltonians for all roots are found. In the classical case some special trajectories for which the corresponding particular constants of motion appear are indicated.