### Schrödinger equation - The Perfectly Matched Layer

- Schrödinger equation:
- one-dimensional, in suitably chosen units
- potential V(x) must be constant outside
- in the following for simplicity V = 0
- plane wave solutions
- with

- simple absorbing layer doesn't work

- Basic idea:
- geometry in the following
- in the boundary x → x + i σ with σ
> 0 ⇒ e
^{ikx} decreases exponentially
- more specific
- wave function with new coordinate
- transformed Schrödinger equation
- has plane wave solutions
- exponentially decreasing

- Reflections at the boundary:
- disappear only for σ = 0
- remedy: introduce σ(x) with σ(0) = 0,
increasing with x
- new coordinate
- Schrödinger equation becomes
- result
- waves with exponential decrease
- no reflections