Hidden $sl(2)-$Symmetry of the Generalized Landau-Zener Vibronic Model

TL;DR

This paper reveals a hidden $sl(2)$ symmetry in a generalized Landau-Zener vibronic model, simplifying the spectrum analysis and deriving eigenfunctions through algebraic and Bethe ansatz methods.

Contribution

It uncovers a hidden $sl(2)$ algebra in the generalized Landau-Zener model and provides explicit spectrum and eigenfunctions using algebraic techniques.

Findings

Identification of hidden $sl(2)$ symmetry in the model

Simplified expression for the exceptional spectrum

Eigenfunctions obtained via Bethe ansatz

Abstract

The one-dimensional harmonic vibronic model, which is a generalization of the so-called linear Landau-Zener model and appears in the form of coupled Schr\"{o}dinger equations, is revisited. After decoupling the components, the resulting fourth-order equation is shown to have a hidden $sl(2)$ algebra. The so-called exceptional part of the spectrum is then expressed in a rather simple way. For completeness, the eigenfunctions are obtained via the Bethe ansatz approach directly in position space.