Zero-curvature solutions of the one-dimensional Schrödinger equation

We discuss special k = √2m(E - V(x))/ℏ² = 0 (i.e. zero-curvature) solutions of the one-dimensional Schrödinger equation in several model systems which have been used as idealized versions of various quantum well structures. We consider infinite well plus Dirac delta function cases (where E= V(x) = 0) and piecewise-constant potentials, such as asymmetric infinite wells (where E = V(x) = V₀ > 0). We also construct supersymmetric partner potentials for several of the zero-energy solutions in these cases. One application of zero-curvature solutions in the infinite well plus δ-function case is the construction of 'designer' wavefunctions. namely zero-energy wavefunctions of essentially arbitrary shape, obtained through the proper placement and choice of strength of the δ-functions.