Bethe Ansatz in Quantum Mechanics. 1. The Inverse Method of Separation of Variables

TL;DR

This paper introduces a general inverse separation of variables method for constructing integrable quantum systems using multi-parameter spectral equations, linking solvability of spectral equations to integrable models.

Contribution

It develops a novel inverse procedure that connects multi-parameter spectral equations with integrable quantum models, enabling systematic construction of solvable systems.

Findings

Multi-parameter spectral equations can be reduced to integrable models.

Exactly or quasi-exactly solvable spectral equations lead to solvable integrable models.

The method provides a systematic way to build integrable quantum systems.

Abstract

In this paper we formulate a general method for building completely integrable quantum systems. The method is based on the use of the so-called multi-parameter spectral equations, i.e. equations with several spectral parameters. We show that any such equation, after eliminating some spectral parameters by means of the so-called inverse procedure of separation of variables can be reduced to a certain completely integrable model. Starting with exactly or quasi-exactly solvable multi-parameter spectral equations we, respectively, obtain exactly or quasi-exactly solvable integrable models.