Classical and quantum Kummer shape algebras

TL;DR

This paper explores a class of integrable classical and quantum systems of coupled oscillators, revealing their symmetry via Kummer shape algebras and providing new reproducing kernel identities and illustrative examples.

Contribution

It introduces Kummer shape algebras as a symmetry framework for integrable oscillator systems and derives new reproducing kernel identities with multiple examples.

Findings

Established the integrability of coupled oscillators through Kummer shape algebras.

Derived resolution of identity for reproducing kernels in these systems.

Provided illustrative examples demonstrating the theory.

Abstract

We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras called here Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, is also presented.