Exact and approximate solutions to the Helmholtz, Schr\"odinger and wave equation in $\mathbf{R}^3$ with radial data

Abstract

We derive simple-to-evaluate, closed-form solutions to the inhomogeneous Helmholtz equation, $\Delta u + k^2 u = \chi_{B_{x_0,r}} $, the Schr\"odinger equation, $i\hbar \partial_t u + \frac{\hbar^2}{2m}\Delta u = 0$ with initial data ${u(x,0) = \chi_{B_{x_0,r}} }$, and the Cauchy problem for the linear wave equation, ${\partial_t^2 u - c^2 \Delta u = 0 }$ with initial data $\left(u(x,0),\partial_t u(x,0)\right) = \left(\chi_{B_{x_0,r}},\chi_{B_{x_0,r}} \right). $ The function $\chi_{B_{x_0,r}}$ is the characteristic function on the ball $B_{x_0,r} = \{x \in \mathbf{R}^3 : |x_0 - x| \leq r \} $. Furthermore, we use these solutions to construct explicit approximate solutions when the data are radial functions on $B_{x_0,r}$, and give various error estimates on these approximations.