# Adiabatic projection method with Euclidean time subspace projection

## TL;DR

This paper explores an improved adiabatic projection method using Euclidean time subspace projection to efficiently analyze low-energy spectra and wave functions in quantum scattering systems, reducing computational costs.

## Contribution

It introduces a novel approach to construct adiabatic Hamiltonians in smaller subspaces, maintaining accuracy in low-energy spectrum analysis.

## Key findings

- Adiabatic Hamiltonian in smaller subspaces can accurately reproduce low-lying spectra.
- The method is tested on fermion systems with zero-range and Gaussian potentials.
- Results guide improvements for large-scale nuclear scattering calculations.

## Abstract

Euclidean time projection is a powerful tool that uses exponential decay to extract the low-energy information of quantum systems. The adiabatic projection method, which is based on Euclidean time projection, is a procedure for studying scattering and reactions on the lattice. The method constructs the adiabatic Hamiltonian that gives the low-lying energies and wave functions of two-cluster systems. In this paper we seek the answer to the question whether an adiabatic Hamiltonian constructed in a smaller subspace of the two-cluster state space can still provide information on the low-lying spectrum and the corresponding wave functions. We present the results from our investigations on constructing the adiabatic Hamiltonian using Euclidean time projection and extracting details of the low-energy spectrum and wave functions by diagonalizing it. In our analyses we consider systems of fermion-fermion and fermion-dimer interacting via a zero-range attractive potential in one dimension, and fermion-fermion interacting via an attractive Gaussian potential in three dimensions. The results presented here provide a guide for improving the adiabatic projection method and for reducing the computational costs of large-scale calculations of \emph{ab initio} nuclear scattering and reactions using Monte Carlo methods.