Characterization of solutions of a generalized Helmholtz problem

In this article, we classify all distributional solutions of $f(-\mathrm{\Delta })u=f(1)u$ 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 $(-\mathrm{\Delta }{)}^{\sigma }$ (for $\sigma \in (0,1]$), $\log (1-\mathrm{\Delta })$ and $(-\mathrm{\Delta }{)}^{\frac{1}{2}}\text{tanh}((-\mathrm{\Delta }{)}^{\frac{1}{2}})$.