Non-Hermitian quantum mechanics approach for extracting and emulating continuum physics based on bound-state-like calculations: Detailed description

TL;DR

This work develops a reduced-basis, non-Hermitian framework to extract continuum physics from bound-state-like calculations by training on complex-energy solutions and projecting onto a small subspace. The core ideas are the complex-energy emulator (CEE) and the complex-energy real-parameter emulator (CERPE), which together allow interpolation and extrapolation of Green's-function amplitudes $\mathcal{A}(E,\bm{\theta})$ across complex $E$ and multiple physical inputs. The approach leverages a variational RBM formulation, spectrum compression, and connections to (near-)optimal rational approximations, with detailed two- and three-body demonstrations showing moved branch-cut lines, discretized branch-cut poles, and accurate reproduction of bound, resonance, and DC states. The results indicate substantial computational savings and potential compatibility with established continuum methods (CE, LIT, complex scaling), offering a path to efficient continuum-calculation workflows and uncertainty quantification in finite quantum systems.

Abstract

This work applies a reduced basis method to study the continuum physics of a finite quantum system -- either few or many-body. Specifically, I develop reduced-order models, or emulators, for the underlying inhomogeneous Schrödinger equation and train the emulators against the equation's bound-state-like solutions at complex energies. The emulators rapidly and accurately interpolate and extrapolate the matrix elements of the Hamiltonian resolvent operator (Green's function) across a parameter space that includes both complex energy and other real-valued physical inputs in the Schrödinger equation. The spectra, discretized and compressed as the result of emulation, and the associated resolvent matrix elements (or amplitudes), have the defining characteristics of non-Hermitian quantum mechanics calculations, featuring complex eigenenergies with negative imaginary parts and branch cuts moved below the real axis in the complex energy plane. Therefore, one now has a method that extracts continuum physics from bound-state-like calculations and emulates those extractions in the input parameter space. Building on a prior Letter [arXiv:2408.03309], this article provides the full theoretical details, a comprehensive analysis of the method's performance, and a brief discussion of how it can be coupled with existing continuum approaches to perform emulations in their input parameter spaces.

Figures (26)

• Figure 1: A resolvent matrix element $\mathcal{A}(E)$ in the complex $E$ plane with fixed sources. Here, the off-shell behavior of a $T$-matrix, specifically its non-Born term, is explored. The two rows show the real and imaginary parts of the amplitude. From left to right, three different calculations are shown. The first two are based on bound-state-like calculations, where I enforce a Dirichlet boundary condition on the relative motion, forcing the wave functions to be zero at $r = R_{IR}$; the third is the exact calculation, corresponding to the $R_{IR}\to \infty$ limit.
• Figure 2: Emulation in $(\mathrm{Re}\, E, \mathrm{Im}\, E, E_\mathrm{rel})$. The training wave functions are plotted in the top panel, and at the bottom, the emulated and the exact wave functions are compared for a particular point for on-shell scattering. Note that all the plotted function are $\langle r | \Psi \rangle$ multiplied by $r$. The information about training points is listed in the top panel, while the information about the testing point is provided in the title of the bottom panel.
• Figure 3: The relative error for emulated $\mathcal{A}$ in the complex $E$ plane. In each sub-figure, from left to right, $N_b$ increases from 3 to 23, while from top to bottom $\mathrm{Im}\, E^\mathrm{tr}_\alpha$ decreases from 30 to 3 MeV. As discussed in the main text, $E^\mathrm{tr}_\alpha$ are evenly separated along the black lines in each panel. The two sub-figures differ in the arrangement of the training points.
• Figure 4: The left panels show the compressed spectra. The right ones plot the residues of the poles (see Eq. \ref{['eq:emulated_amp_eigen_decomp']}) vs. the absolute values of the corresponding eigenvalues. Their insets show the absolute values of the eigenenergies in the $\log$ scale. The physical (bound) state is marked with a black "$+$" in the left panels and the insets of the right panels. From the top and bottom rows, $N_b$ increases from $3$ to $23$. Each row compares two emulation results with different $\mathrm{Im}\, E^\mathrm{tr}_\alpha$.
• Figure 5: The emulation errors vs $\sqrt{N_\mathrm{eff}}$. The left panels show the emulation errors for the bound state pole location and its residues. The right ones show the absolute errors for $\mathcal{A}$ (1) on the real energy axis with different $E$ values marked in the legends, and (2) averaged over the interpolation region around the training energies (marked as "int."). From the top to bottom row, $\mathrm{Im}\, E^\mathrm{tr}_\alpha$ reduces from $30$ to $3$ MeV.