Partition of the Potential of the One-dimensional Schr Odinger Equation

The one-dimensional Schrodinger equation is considered when the potential and its rst moment are absolutely integrable. The transmission coe cient vanishes at zero energy in the generic case, and it never vanishes in the exceptional case. It is shown that any nontrivial exceptional potential can always be fragmented into two generic potentials. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into all generic pieces in in nitely many ways. The results remain valid when Dirac delta functions are included in the potential, in which case even the trivial potential can be fragmented into generic pieces. PACS Numbers: 03.65.Nk, 03.80.+r