Calculating bound states resonances and scattering amplitudes for arbitrary 1D potentials with piecewise parabolas

Even for one-dimensional (1D) potentials, the calculation of highly excited bound states and resonances (in particular broad and overlapping ones) often requires heavy numerical tools. The method presented here is based on representing an arbitrary 1D potential as a set of piecewise parabolas, where the solutions of the Schrodinger equation within each parabolic region are analytic. Outgoing, incoming, or zero-valued boundary conditions are imposed to solve for bound and resonance states, while asymmetric boundary conditions are enforced to calculate scattering amplitudes. The developed method is also applicable for complex potentials, as for example found in non-Hermitian PT symmetric systems (such as optical waveguides, where the spectrum varies from real to complex as a function of gain and loss). We demonstrate for several model Hamiltonians that calculations of energies, decay rates, scattering amplitudes, and wave functions are obtained in high accuracy and with minimal computational effort in comparison to other widely used methods. We believe our approach will be useful in many areas of physics and chemistry, and for example can be used to calculate various types of resonances, to solve scattering problems in cold molecular collisions, and to create analytical basis functions for multidimensional potentials.