  In the framework of a nonrelativistic potential quark model, we investigate
the mass spectrum of the $1S$-wave charmed-strange tetraquark states of
$cn\bar{s}\bar{n}$ and $cs\bar{n}\bar{n}$ ($n=u$ or $d$) systems. The
tetraquark system is solved by a correlated Gaussian method. With the same
parameters fixed by the meson spectra, we obtained the mass spectra for the
$1S$-wave tetraquark states. Furthermore, based on the predicted tetraquark
spectra we estimate their rearrangement decays in a quark-exchange model. We
find that the rearrangement decays of the tetraquarks may be mainly driven by
the spin-spin interactions. The resonances $X_0(2900)^0$ and
$T^a_{c\bar{s}0}(2900)^{++/0}$ reported from LHCb may be assigned to be the
lowest $1S$-wave tetraquark states $\bar{T}_{cs0}^f(2818)$ and
$T^{a}_{c\bar{s}0}(2828)$ classified in the quark model, respectively. It also
allows us to extract the couplings for the initial tetraquark states to their
nearby $S$-wave interaction channels. We find that some of these couplings turn
out to be sizeable. Following the picture of the wavefunction renormalization
for the near-threshold strong $S$-wave interactions, the sizeable coupling
strengths can be regarded as an indication of their dynamic origins as
candidates for hadronic molecules. Furthermore, our predictions suggest that
signals for the $1S$-wave charmed-strange tetraquark states can also be
searched in the other channels, such as $D^0K^+$, $D^+K^+$, $D^{*+}K^-$,
$D^{*+}K^+$, $D^{*0}K^+$, $D^0\bar{K}^{*0}$, $D_s^+\rho^0$, etc.
