Non-Hermitian Quantum Mechanics Approach for Extracting and Emulating Continuum Physics Based on Bound-State-Like Calculations

This work 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 Schr\"{o}dinger equation that operate in a combined parameter space of complex energy ($E$) and other inputs ($\bm{\theta}$). 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 $\bm{\theta}$. This yields a small, non-Hermitian system whose properties (e.g., resonances and scattering observables) can be rapidly predicted for any $\bm{\theta}$. 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 $\bm{\theta}$-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.