  In this article, we classify all distributional solutions of
$f(-\Delta)u=f(1)u$ where $f$ is a non-constant Bernstein function.
Specifically, we show that the Fourier transform of $u$ is a single-layer
distribution on the unit sphere. Examples of such operators include
$(-\Delta)^\sigma$ (for $\sigma \in (0,1]$), $\log(1-\Delta)$ and
$(-\Delta)^\frac{1}{2}\text{tanh}((-\Delta)^\frac{1}{2})$.
