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Modeling Schrödinger equation diffraction with generalized function potentials and initial values

Günther Hörmann, Ljubica Oparnica, Christian Spreitzer

TL;DR

The paper develops a rigorous framework for Schrödinger diffraction at planar patterns by modeling both the potential and the initial state with Colombeau regularizations $\{V_\varepsilon\}$ and $\{g_\varepsilon\}$. It shows that each $H_\varepsilon = -\Delta + V_\varepsilon$ is self-adjoint with spectrum contained in $[0,\infty)$, has no eigenvalues, and, under suitable assumptions, preserves the essential spectrum of the free operator; generalized eigenvalues are analyzed within the Colombeau setting. The work extends to scattering theory via short-range perturbation methods, yielding distorted Fourier representations for the regularized dynamics. It also connects these regularized models to common physics approximations that replace the potential by a source term or embed slit geometry into initial data, providing coherence checks and conditions under which such approximations are justified. Overall, the results offer a mathematically robust bridge between generalized-function regularizations and physically motivated diffraction analyses, with clear criteria for when approximations reproduce the expected interference patterns.

Abstract

We discuss spectral properties of a regularization approach to a Schrödinger 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.

Modeling Schrödinger equation diffraction with generalized function potentials and initial values

TL;DR

The paper develops a rigorous framework for Schrödinger diffraction at planar patterns by modeling both the potential and the initial state with Colombeau regularizations and . It shows that each is self-adjoint with spectrum contained in , has no eigenvalues, and, under suitable assumptions, preserves the essential spectrum of the free operator; generalized eigenvalues are analyzed within the Colombeau setting. The work extends to scattering theory via short-range perturbation methods, yielding distorted Fourier representations for the regularized dynamics. It also connects these regularized models to common physics approximations that replace the potential by a source term or embed slit geometry into initial data, providing coherence checks and conditions under which such approximations are justified. Overall, the results offer a mathematically robust bridge between generalized-function regularizations and physically motivated diffraction analyses, with clear criteria for when approximations reproduce the expected interference patterns.

Abstract

We discuss spectral properties of a regularization approach to a Schrödinger 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.
Paper Structure (13 sections, 9 theorems, 106 equations)

This paper contains 13 sections, 9 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Modeling of the Cauchy problem and of the Hamiltonian
  3. Regularizations representing the potential
  4. Regularizations representing the initial configuration
  5. Analysis of the Hamiltonian
  6. Spectral properties
  7. Generalized eigenvalues
  8. Aspects of scattering theory
  9. Alternative approximations and regularizations: Cauchy problems with source term or initial value replacing the potential
  10. The approximate solutions from theoretical physics
  11. A plausibility check
  12. Improving on the plausibility of \ref{['VuApprox']}
  13. Generalized function solutions and coherence properties

Key Result

Theorem 2.1

Let $V \in \mathcal{G}_{\infty}(\mathbb{R}^2)$ denote the potential defined via the regularizations potentialreg. For every $g \in \mathcal{G}_{2}(\mathbb{R}^2)$, there is a unique solution $u \in \mathcal{G}_{2}(\mathbb{R}^2 \times [0,T])$ to the Cauchy problem

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 10 more