A note on the validity of the Schrödinger approximation for the Helmholtz equation

Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δu + ω2u = 0 for (x, y, z) ∈ ℝ3. For the evolution of such waves along the z-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the z-axis.

---

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).