The Multidimensional Darboux Transformation

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 ≥ 2. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results. † Supported in part by DGICYT Grants PB92–0197 and PB96–0197. ‡ Supported in part by an NSERC Grant. On sabbatical leave from the Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6. February 7, 2008