# Absorbing boundary conditions for the time-dependent Schr\"odinger-type equations in $\mathbb R^3$

## TL;DR

This paper develops and analyzes absorbing boundary conditions for 3D time-dependent Schrödinger equations, improving computational efficiency and accuracy in quantum simulations with arbitrary geometries.

## Contribution

It introduces a semi-discrete derived boundary condition compatible with finite-difference schemes and provides stability proofs and practical tests.

## Key findings

- Boundary conditions are stable for zeroth and first order.
- Effective in reducing boundary reflections in benchmark problems.
- Accurate in energy and nucleon density for Hartree-Fock models.

## Abstract

Absorbing boundary conditions are presented for three-dimensional time-dependent Schr\"odinger-type of equations as a means to reduce the cost of the quantum-mechanical calculations. The boundary condition is first derived from a semi-discrete approximation of the Schr\"odinger equation with the advantage that the resulting formulas are automatically compatible with the finite-difference scheme and no further discretization is needed in space. The absorbing boundary condition is expressed as a discrete Dirichlet-to-Neumann (DtN) map, which can be further approximated in time by using rational approximations of the Laplace transform to enable a more efficient implementation. This approach can be applied to domains with arbitrary geometry. The stability of the zeroth order and first order absorbing boundary conditions is proved. We tested the boundary conditions on benchmark problems. The effectiveness is further verified by a time-dependent Hartree-Fock model with Skyrme interactions. The accuracy in terms of energy and nucleon density is examined as well.