Intertwined isospectral potentials in an arbitrary dimension

The method of intertwining with n-dimensional (nD) linear intertwining oper ator L is used to construct nD isospectral, stationary potentials. It has b een proven that the differential part of L is a series in Euclidean algebra generators. Integrability conditions of the consistency equations are inve stigated and the general form of a class of potentials respecting all these conditions have been specified for each n=2, 3, 4, 5. The most general for ms of 2D and 3D isospectral potentials are considered in detail and constru ction of their hierarchies is exhibited. The followed approach provides coo rdinate systems which make it possible to perform separation of variables a nd to apply the known methods of supersymmetric quantum mechanics for 1D sy stems. It has been shown that in choice of coordinates and L there are a nu mber of alternatives increasing with n that enlarge the set of available po tentials. Some salient features of higher dimensional extension as well as some applications of the results are presented. (C) 2001 American Institute of Physics.