Explicit higher-dimensional Darboux transformations for the time-dependent Schr?dinger equation

摘要

We construct Darboux transformations for time-dependent Schr?dinger equations in arbitrary spatial dimensions. The Darboux operator that connects a pair of Schr?dinger equations and the corresponding potential difference are obtained in explicit form. An example in (5 + 1) dimensions is presented and the representation of the Darboux operator in different coodi-nate systems is discussed.