  We develop a technique for proving number rigidity (in the sense of
Ghosh-Peres) of the spectrum of general random Schr\"odinger operators (RSOs).
Our method makes use of Feynman-Kac formulas to estimate the variance of
exponential linear statistics of the spectrum in terms of self-intersection
local times.
  Inspired by recent results concerning Feynman-Kac formulas for RSOs with
multiplicative white noise by Gorin, Shkolnikov and the first-named author, we
use this method to prove number rigidity for a class of one-dimensional
continuous RSOs of the form $-\frac12\Delta+V+\xi$, where $V$ is a
deterministic potential and $\xi$ is a stationary Gaussian noise. Our results
require only very mild assumptions on the domain on which the operator is
defined, the boundary conditions on that domain, the regularity of the
potential $V$, and the singularity of the noise $\xi$.
