The Snapshot Problem for the Euler-Poisson-Darboux Equation

The generalized Euler-Poisson-Darboux (EPD) equation with complex parameter $\alpha$ is given by $${\mathrm{\Delta }}_{x}u=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}+\frac{n-1+2\alpha }{t}\frac{\mathrm{\partial }u}{\mathrm{\partial }t},$$ where $u(x,t)\in \mathcal{E}({\mathbb{R}}^n\times \mathbb{R})$, with $u$ even in $t$. For $\alpha =0$ and $\alpha =1$ the solution $u(x,t)$ represents a mean value over spheres and balls, respectively, of radius $\mathrm{\mid }t\mathrm{\mid }$ in ${\mathbb{R}}^n$. In this paper we consider existence and uniqueness results for the following two-snapshot problem: for fixed positive real numbers $r$ and $s$ and smooth functions $f$ and $g$ on ${\mathbb{R}}^n$, what are the conditions under which there is a solution $u(x,t)$ to the generalized EPD equation such that $u(x,r)=f(x)$ and $u(x,s)=g(x)$? The answer leads to a discovery of Liouville-like numbers related to Bessel functions, and we also study the properties of such numbers.