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Layer-Resolved Impurity States Reveal Competing Pairing Mechanisms in Trilayer Nickelate Superconductor La$_4$Ni$_3$O$_{10}$

Trilayer Ruddlesden-Popper nickelate superconductor $\mathrm{La}_4 \mathrm{Ni}_3 \mathrm{O}_{10}$ has generated considerable interest due to its unconventional superconductivity and complex electronic structure. Notably, $\mathrm{La}_4 \mathrm{Ni}_3 \mathrm{O}_{10}$ features a mixed Ni valence state and an asymmetric trilayer configuration, leading to distinct quasiparticle distributions and local density of states (LDOS) between the inner and outer NiO$_2$ planes. In this work, we investigate impurity-induced states in $\mathrm{La}_4 \mathrm{Ni}_3 \mathrm{O}_{10}$ using a two-orbital model combined with $T$-matrix formalism, focusing on the contrasting roles of intra- and interlayer pairing channels. Our self-consistent mean-field analysis reveals that interlayer pairing results in partially gapless Fermi surfaces, with unpaired quasiparticles concentrated in the outer layers and a pronounced low-energy LDOS. We demonstrate that impurity effects vary significantly depending on both the pairing symmetry and impurity location: interlayer-dominant pairing produces sharp resonance states when impurities are in the inner layer, whereas impurities in the outer layer lead to in-gap enhancements without sharp resonances; in contrast, intralayer-dominant pairing generally yields increased in-gap LDOS without sharp impurity resonances, regardless of impurity position. These findings suggest that single-impurity spectroscopy can serve as a powerful probe to distinguish between competing superconducting pairing mechanisms in trilayer nickelates and highlight the rich physics arising from their multilayer structure.