  We consider a quantum system that is initially localized at $x_{in}$ and that
is repeatedly projectively probed with a fixed period $\tau$ at position $x_d$.
We ask for the probability that the system is detected in $x_d$ for the very
first time, $F_n$, where $n$ is the number of detection attempts. We relate the
asymptotic decay and oscillations of $F_n$ with the system's energy spectrum,
which is assumed to be absolutely continuous. In particular $F_n$ is determined
by the Hamiltonian's measurement spectral density of states (MSDOS) $f(E)$ that
is closely related to the density of energy states (DOS). We find that $F_n$
decays like a power law whose exponent is determined by the power law exponent
$d_S$ of $f(E)$ around its singularities $E^*$. Our findings are analogous to
the classical first passage theory of random walks. In contrast to the
classical case, the decay of $F_n$ is accompanied by oscillations with
frequencies that are determined by the singularities $E^*$. This gives rise to
critical detection periods $\tau_c$ at which the oscillations disappear. In the
ordinary case $d_S$ can be identified with the spectral dimension found in the
DOS. Furthermore, the singularities $E^*$ are the van Hove singularities of the
DOS in this case. We find that the asymptotic statistics of $F_n$ depend
crucially on the initial and detection state and can be wildly different for
out-of-the-ordinary states, which is in sharp contrast to the classical theory.
The properties of the first detection probabilities can alternatively be
derived from the transition amplitudes. All our results are confirmed by
numerical simulations of the tight-binding model, and of a free particle in
continuous space both with a normal and with an anomalous dispersion relation.
We provide explicit asymptotic formulae for the first detection probability in
these models.
