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

TL;DR

The paper introduces a non-Hermitian quantum mechanics framework that uses a reduced-basis method to build emulators for the inhomogeneous Schrödinger equation in the joint space of complex energy $E$ and parameters $\bm{\theta}$. This approach analytically continues in $E$ and interpolates in $\bm{\theta}$ to produce a small, non-Hermitian emulator that rapidly predicts resonances and scattering observables from bound-state-like calculations. Demonstrations in two- and three-body systems show accurate replication of spectra, branch-cut structure, and real-energy observables, with indications of connections to near-optimal rational approximation and potential integration with CE and LIT approaches. The method enables continuum calculations for complex systems by leveraging advanced bound-state solvers and offers fast parameter-space exploration, potentially accelerating existing continuum calculations and providing a new link between bound-state methods and continuum physics.

Abstract

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 Schrödinger equation that operate in a combined parameter space of complex energy ($E$) and other inputs ($\bmθ$). 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 $\bmθ$. This yields a small, non-Hermitian system whose properties (e.g., resonances and scattering observables) can be rapidly predicted for any $\bmθ$. 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 $\bmθ$-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.

Figures (4)

• Figure 1: The relative error (in its logarithm) of an emulated resolvent matrix element for a two-body system in the complex-$E$ plane. The training energies are evenly separated on the black solid line(s) which are 10 MeV away from the real axis.
• Figure 2: Emulation in $E$ at fixed $\bm{\theta}$ for a three-body system with (top rows) and without (bottom rows) a bound dimer. The left panels show $\mathcal{A}$'s eigenvalues ($\diamond$). Note the different scales on the two sides of the x-axis. The exact locations of bound or resonance states are also marked (see the legends). The insets zoom in on the regions around the branch points. The right panels compare the emulated and the exact calculations of $-\mathrm{Im}\, \mathcal{A}$ at real energies.
• Figure 3: Emulated spectra (marked as $\diamond$s) by a single emulator. The insets zoom in on the regions around the branch points. Two testing results are plotted, with the parameter values shown in the panel titles. The exact locations of the physical states (no resonance here) are marked by "$\times$" in both panels.
• Figure 4: The non-Born term of the particle-dimer scattering $T$-matrix are emulated by two emulators with different $\mathrm{Im}\, E^\mathrm{tr}_\alpha$. For each curve, $\lambda$ and thus $B_2$ are fixed (see the legend), but a sample of $\lambda_4$ values is chosen to test the emulators. The means of the sample of relative errors are plotted.