Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials

TL;DR

This paper develops an extended trigonometric Cherednik algebra framework to explicitly solve nonstationary Schrödinger equations with delta-potentials using Bethe ansatz methods, linking algebraic structures to quantum many-body systems.

Contribution

It introduces an extended version of the trigonometric Cherednik algebra with explicit Dunkl operators and constructs solutions to related Schrödinger equations via Bethe wave functions.

Findings

Explicit nonstationary Schrödinger equation with delta-potential derived

Solutions constructed using coordinate Bethe ansatz methods

Connections established to quantum Bose gas with delta interactions

Abstract

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schr\"odinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schr\"odinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with pairwise delta-function interactions is indicated.