The Multidimensional Darboux Transformation

TL;DR

This paper generalizes the classical Darboux and Moutard transformations to multidimensional Riemannian manifolds, enabling new solutions for multidimensional matrix Schrödinger operators on curved spaces.

Contribution

It introduces an intrinsic framework for multidimensional Darboux transformations and extends the Moutard transformation beyond two dimensions.

Findings

Constructed a multidimensional Darboux transformation using twisted Hodge Laplacians.

Generalized the Moutard transformation to higher-dimensional non-compact manifolds.

Produced new quasi-exactly solvable multidimensional matrix Schrödinger operators.

Abstract

A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schr\"odinger operators on curved manifolds are obtained by applying the above results.