Let $\mathcal{F}$ be a Fourier transform on $L^2(\mathbb{R}^d)$. Then for any $f \in L^\infty(\mathbb{R}^d)$, we can define the (bounded) Fourier multiplier $f(-i\nabla):= \mathcal{F}^{-1} f \mathcal{F}$.
I would like to know when the following holds:
The orthonormal system of $L^2(\mathbb{R}^d)$ given as all eigenfunctions of $f(-i\nabla)$ is complete.
Observation: It is clear that when $f$ is a constant, then the above statement holds.
if $g$ is an eigenfunction with eigenvalue $\lambda$, then $f\hat g=\lambda \hat g$, meaning $f$ must equal $\lambda$ a.e. on the support of $\hat g$. Hence $f$ needs to be piecewise a.e. constant and its level sets form a partition of $\mathbb R^d$ up to the difference of a null set.