  We perform a theoretical examination of the local electronic structure in the
recently discovered bilayer high-temperature superconductor
${\mathrm{La}_3\mathrm{Ni}_2\mathrm{O}_7}$. Our method begins with a bilayer
two-orbital tight-binding model, incorporating various pairing interaction
channels. We determine superconducting order parameters by self-consistently
solving the real-space Bogoliubov-de Gennes (BdG) equations, revealing a robust
and stable extended s-wave pairing symmetry. We investigate the single impurity
effect using both self-consistent BdG equations and non-self-consistent
T-matrix methods, uncovering low-energy in-gap states that can be explained
with the T-matrix approach. Additionally, we analyze magnetic vortex states
using a self-consistent BdG technique, which shows a peak-hump structure in the
local density of states at the vortex center. Our results provide identifiable
features that can be used to determine the pairing symmetry of the
superconducting ${\mathrm{La}_3\mathrm{Ni}_2\mathrm{O}_7}$ material.
