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Interfacial thermal transport is critical for many thermal-related applications such as heat dissipation in electronics. While the total interfacial thermal conductance (ITC) can be easily measured or calculated, the ITC spectral mapping has been investigated only recently and is not fully understood. By combining nonequilibrium molecular dynamics simulations and atomistic Green's function method, we systematically investigate the ITC spectrum across an ideal interface, i.e., the argon-heavy argon interface. Our results show that the ITC spectrum increases gradually with temperature as more phonons and anharmonic scattering channels are activated, e.g., the vibrations with frequencies larger than 1 THz can contribute 5% (15%) to the total ITC at 2 K (40 K) through anharmonic phonon scatterings channels. We further find that the ITC spectrum from the left interfacial Hamiltonian is quite different from that of the right interfacial Hamiltonian, which stems from the asymmetry of anharmonic phonon scatterings caused by the dissimilar vibrational property of the two interfacial contacts. While all the phonons are involved in the anharmonic scatterings for the heavy argon interfacial Hamiltonian, these phonons involved in the anharmonic phonon scatterings from the argon interfacial Hamiltonians are mainly these vibrations with frequency smaller than 1 THz (i.e., the cut-off frequency of heavy argon). Finally, we find the quantum effect is important for the ITC spectrum at low temperatures, e.g., below 30 K in our systems. Our results here systematically investigate the influence of anharmonicity, non-homogeneity, and quantum effects on the ITC spectrum, which is critical for designing and optimizing the interfaces with better performance.