Perfectly matched layers for the stationary Schrodinger equation in a periodic structure

TL;DR

This paper develops a perfectly matched absorbing layer for the stationary Schrödinger equation in periodic structures with decaying potentials, ensuring accurate approximation with exponentially decreasing error as layer length increases.

Contribution

It introduces a new PML construction for the Schrödinger equation in periodic media and proves its effectiveness and exponential convergence.

Findings

Unique solvability of the PML problem is established.

Solution approximation error decreases exponentially with layer length.

The method effectively simulates the original problem with high accuracy.

Abstract

We construct a perfectly matched absorbing layer for stationary Schrodinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity.