We present an exact solution of the two-electron problem on the Lieb lattice and analyze how the combination of a flat band, sublattice structure, and local interactions shapes the two-particle excitation spectrum. The separable form of the on-site interaction reduces the problem to a $3\times 3$ secular equation whose roots yield a triplet of discrete modes. At the $\mathrm{\Gamma }$ point these modes form bonding, nonbonding, and antibonding combinations of sublattice amplitudes, while at the $M$ point they become strictly site selective due to the vanishing of intersublattice correlations. Along high-symmetry lines, destructive interference suppresses the bandwidth of one branch to order ${\left(\right|t|/U)}^4$, providing a clear signature of flat-band locality. The interaction-induced redistribution of spectral weight is reflected in the two-electron density of states, where resonances evolve into discrete bound states that constitute the low-density precursors of the upper Hubbard band. The $\omega =0$ flat-band peak remains pinned by symmetry, but virtual transitions into the dispersive bands deplete its intensity and generate an asymmetric infrared continuum with a logarithmic divergence. Together, these results provide a complete and analytically controlled description of two-electron physics on the Lieb lattice and illustrate how flat-band singularities and local interactions conspire to produce discrete excitations, anomalously weak dispersion, and marginal infrared behavior.

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• Flat bands
• Electron-correlation calculations
• Exact solutions for many-body systems
• Green's function methods
• Hubbard model