The one-dimensional Schrodinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schrodinger equation. For such Schrodinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem. (C) 1996 American Institute of Physics.