Structures and decay properties of extremely proton-rich nuclei $^{11,12}$O

${\rm \bf Background:}$ The recent observation of the unbound nucleus $^{11}$O offers the unique possibility to study how the structure and dynamics of two-proton ($2p$) decay is affected by the removal of one neutron from $^{12}$O, and provides important information on the Thomas-Ehrman effect in the mirror pairs $^{11}_{~8}$O$_3$-$^{11}_{~3}$Li$_8$ and $^{12}_{~8}$O$_4$-$^{12}_{~4}$Be$_8$, which involve the $2p$ emitters $^{11}$O and $^{12}$O. ${\rm \bf Purpose:}$ We investigate how continuum effects impact the structure and decay properties of $^{11}$O and $^{12}$O, and their mirror partners. ${\rm \bf Methods:}$ We solve the three-body core-nucleon-nucleon problem using the Gamow coupled-channel (GCC) method. The GCC Hamiltonian employs a realistic finite-range valence nucleon-nucleon interaction and the deformed cores of $^{9,10}$C, $^{9}$Li, and $^{10}$Be. ${\rm \bf Results:}$ We calculate the energy spectra and decay widths of $^{11}$O and $^{12}$O as well as those of their mirror nuclei. In particular, we investigate the dynamics of the $2p$ decay in the ground state of $^{12}$O by analyzing the evolution of the $2p$ configuration of the emitted protons as well as their angular correlations in the coordinate space. We also show how the analytic structure of the resonant states of $^{10}$Li and $^{10}$N impacts the low-lying states of $^{11}$Li and $^{11}$O. ${\rm \bf Conclusions:}$ We demonstrate that, in both nuclei $^{11}$O and $^{12}$O, there is a competition between direct and "democratic" $2p$ ground-state emission. The broad structure observed in $^{11}$O is consistent with four broad resonances, with the predicted $3/2^-_1$ ground state strongly influenced by the broad threshold resonant state in $^{10}$N, which is an isobaric analog of the antibound (or virtual) state in $^{10}$Li.