Exact and approximate solutions to the Helmholtz, Schrödinger 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 Schrödinger equation, $i\hslash {\partial }_{t}u+\frac{{\hslash }^2}{2m}\Delta u=0$ 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 $B_{x_0,r}=\{x\in {\mathbf{R}}^3:|x_0-x|\le r\}$. 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.