Quantization of the Gaudin System

TL;DR

This paper constructs a quantum version of the Gaudin integrable system using the Bethe subalgebra in Y(gl(n)), providing explicit quantum Hamiltonians and revealing quantum corrections at higher orders.

Contribution

It introduces explicit quantum Hamiltonians for the Gaudin system derived from the Bethe subalgebra, including quantum corrections at order four.

Findings

Explicit quantum Hamiltonians QI_k(u) for the Gaudin system

Quantum corrections identified at order four

Connection between classical and quantum Hamiltonians established

Abstract

In this article we exploit the known commutative family in Y(gl(n)) - the Bethe subalgebra - and its special limit to construct quantization of the Gaudin integrable system. We give explicit expressions for quantum hamiltonians QI_k(u), k=1,..., n. At small order k=1,...,3 they coincide with the quasiclassic ones, even in the case k=4 we obtain quantum correction.