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Sommerfeld Radiation Condition for Helmholtz Equations with long-range Potentials

Eric Ströher

Abstract

We study the electric Helmholtz equation $Δu + Vu + λu =f$ and show that, for certain potentials, the solution $u$ given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical approach based on the solution $K$ of the eikonal equation $|\nabla K|^2=1 + \frac{p}λ$ to improve previous results in that area and extend them to long-range potentials which decay like $|x|^{-2-α}$ at infinity, with $α> 0$.

Sommerfeld Radiation Condition for Helmholtz Equations with long-range Potentials

Abstract

We study the electric Helmholtz equation and show that, for certain potentials, the solution given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical approach based on the solution of the eikonal equation to improve previous results in that area and extend them to long-range potentials which decay like at infinity, with .
Paper Structure (5 sections, 8 theorems, 95 equations)

This paper contains 5 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Properties of the eikonal equation
  3. Key Results
  4. Main result
  5. Outlook

Key Result

Theorem 1

For $d \geq 3$, let $V$ be a potential such that $|V| \leq C|x|^{-3-\alpha}$ when $|x| \geq 1$, $C, \alpha > 0$. Then, for $u$ given by res_sol, we have

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • Corollary 1