Modeling Schr\"odinger equation diffraction with generalized function potentials and initial values

TL;DR

This paper investigates a regularization method for solving the Schr"odinger equation in quantum diffraction, analyzing spectral properties and comparing with traditional physics approximations.

Contribution

It introduces a regularization framework for Schr"odinger equation solutions using generalized functions, providing spectral and scattering analysis.

Findings

Spectral properties of regularized solutions are established.

Comparison with physics-based approximations shows consistency.

Solutions can be interpreted as generalized functions.

Abstract

We discuss spectral properties of a regularization approach to a Schr\"odinger equation set-up for the diffraction of a quantum particle at almost planar patterns. Physically meaningful initial values and potentials are modeled in terms of regularizing families and the solutions can be interpreted as generalized functions. We establish spectral and scattering theoretical properties of the regularizing solution families and provide some comparison with the more direct approximations and simplifications used in physics.