The inverse scattering is one of the most useful devices to obtain information about quantum mechanical forces. The solutions of the radial Schrödinger equation, describing mathematically the situation, behave asymptotically like sine functions with shifted phase. The sequence of these phase shifts is a central notion in scattering theory. In this talk some classical and recent results about phase shifts will be presented and some open questions will be discussed.
In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. The grid points are constructed as the image of an equidistant grid under a smooth deformation map. We show that for all strongly stable linear multistep methods, there is an $N^*$ such that a condition of zero stability is always fulfilled for $N > N^*$ under a smoothness condition. Examples are given for Adams and BDF type methods.
Non-polynomial series solution of a coupled first order partial and ordinary differential equation (PDE-ODE) system for a discontinuous initial and boundary condition has been developed. Linear equation systems are constructed to calculate the constant coefficients of the series solution. Explicit expressions have been found to the solution of these linear equation systems. Different forms of the solution have been compared to the numerical solution of the PDE-ODE system and the rate of the convergence is also investigated. The studied first order PDE-ODE system describes an unsteady convection dominated heat transfer process induced by a buoyant plume entrainment.