Algebraic Bethe ansatz for the sl(2) Gaudin model with boundary

TL;DR

This paper develops an algebraic Bethe ansatz approach for the sl(2) Gaudin model with boundary conditions, deriving the spectrum and Bethe equations through a quasi-classical expansion and Sklyanin's framework.

Contribution

It introduces a novel algebraic Bethe ansatz method for the boundary Gaudin model using the Sklyanin determinant and quasi-classical expansion.

Findings

Derived the generating function of Gaudin Hamiltonians with boundary terms.

Obtained the spectrum and Bethe equations for the model.

Established simple off-shell action of the generating function.

Abstract

Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.