Three-Dimensional Matrix Superpotentials

Abstract

We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map. We present а classification of matrix superpotentials that correspond to exactly solvable systems of Schrodinger equations. Superpotentials of the following form are considered: $W_k = kQ + P \frac 1k$, where $k$ is a parameter and $P, Q$ and $R$ are Hermitian matrices that depend on a variable $x$. The list of three-dimensional matrix superpotentials is explicitly presented.