The spectral dimension controls the decay of the quantum first detection probability
We consider a quantum system that is initially localized at $x_{in}$ and that is repeatedly projectively probed with a fixed period $τ$ 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 $τ_c$ 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.