This Letter introduces a unified emulation framework for studying continuum physics in finite quantum systems. Using a reduced basis method, we construct powerful emulators for the inhomogeneous Schrödinger equation that operate in a combined parameter space of complex energy ($E$) and other inputs ($\mathit{\theta }$). Within the space, the emulators simultaneously perform analytical continuation in $E$—extracting continuum physics from numerically simpler bound-state-like calculations—and interpolate this entire process across $\mathit{\theta }$. This yields a small, non-Hermitian system whose properties (e.g., resonances and scattering observables) can be rapidly predicted for any $\mathit{\theta }$. Crucially, the complex-$E$ emulation provides a pathway to compute continuum observables for complex systems where advanced bound-state methods exist but direct continuum calculations are yet to be developed, while the $\mathit{\theta }$ emulation enables rapid parameter-space exploration and can be adapted to accelerate other existing continuum calculations. Demonstrations with two- and three-body systems highlight the method’s effectiveness and suggest its connection to (near-)optimal rational approximation. This Letter presents the key results, with further details reserved for a companion paper.

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• Nuclear reactions
• Ab initio calculations
• Few-body systems
• Many-body techniques