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

TL;DR

This paper develops a reduced basis method to emulate continuum physics of quantum systems using bound-state-like solutions, enabling rapid interpolation of Green's functions across complex energies and physical parameters.

Contribution

It introduces a novel approach combining non-Hermitian quantum mechanics with reduced-order models to extract continuum physics from bound-state calculations.

Findings

Accurately interpolates Green's function matrix elements across parameter space.

Extracts continuum physics from bound-state-like solutions.

Provides a comprehensive theoretical framework and performance analysis.

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 Schr\"{o}dinger 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 Schr\"{o}dinger 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.