  Let $\eta$ be a Poisson random measure (defined on some Polish space), and
let $F(\eta)$ be a square-integrable functional of $\eta$. In this paper we
define and study a new notion of {\it spectral point process} associated with
$F(\eta)$, and use such an object to study sharp noise instability and
sensitivity properties of planar critical continuum percolation models under
spatial birth-death (Ornstein-Uhlenbeck) dynamics -- the notion of sharp noise
instability being a natural strengthening of the absence of noise stability.
The concept of spectral point process is defined by exploiting the Wiener-It\^o
chaos expansion of $F$, and represents a natural continuum counterpart to the
notion of {\it spectral sample}, as introduced in Garban, Pete and Schramm
(2010), in the context of discrete percolation models. In the particular case
where $\eta$ is a marked Poisson measure, we use Hoeffding-ANOVA decompositions
to establish an explicit connection with the notion of {\it annealed spectral
sample}, introduced in Vanneuville (2021) in the context of Poisson-Voronoi
percolation. We also relate spectral processes with an appropriate notion of
{\it pivotal processes}. As applications, we show sharp noise instability of
crossing events in the critical Poisson Boolean model with unit-radius balls
and, using an observation of Vanneuville, we obtain sharp noise sensitivity (as
well as sharp noise instability) for crossing events in the Poisson Voronoi
percolation model. As an important ingredient, we prove quasi-multiplicativity
of the $4$-arm probabilities in the critical Poisson Boolean percolation model.
