Integrability of the one dimensional Schroedinger equation

TL;DR

This paper defines a comprehensive notion of integrability for the one-dimensional Schrödinger equation, classifies rational integrable potentials, and discovers new physically relevant integrable cases.

Contribution

It introduces the concept of rigid functions and classifies all rational integrable potentials, expanding the set of known integrable systems.

Findings

Complete classification of rational integrable potentials.

Discovery of new integrable cases with physical relevance.

Introduction of rigid functions as a tool for integrability analysis.

Abstract

We present a definition of integrability for the one dimensional Schroedinger equation, which encompasses all known integrable systems, i.e. systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.