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