Schrödinger-like operators and the eikonal equation

Jaime Cruz-Sampedro

doi:10.3934/cpaa.2014.13.495

Article Contents

2014, Volume 13, Issue 2: 495-510. Doi: 10.3934/cpaa.2014.13.495

This issue Previous Article Nontrivial solutions for Kirchhoff type equations via Morse theory Next Article Liouville type theorem to an integral system in the half-space

Schrödinger-like operators and the eikonal equation

Jaime Cruz-Sampedro^1,

1.
Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reynosa, Mèxico D. F. 02200

Received: June 2012

Revised: August 2013

Published: March 2014

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Abstract

Let $V$ be a real-valued function of class $C^5$ on $\mathbb{R}^n$, $n \geq 2$, and suppose that $\partial^\alpha V(x)=O(|x|^{-|\alpha|})$, as $|x| \to \infty$, for $|\alpha| \leq 5$. For $\lambda > 0$ we set $W_\lambda(x) = 1-(V(x)/\lambda)$ and consider the Schrödinger-like operator $\mathcal{H}_\lambda=W_\lambda^{-{1/2}} H_0 W_\lambda^{-{1/2}}$ acting on $L^2(\mathbb{R}^n)$, where $H_0=-\Delta$ is the classical laplacian on $\mathbb{R}^n$. Using properties of the maximal solution to the eikonal equation $|\nabla S_\lambda|^2=W_\lambda$, for $\lambda$ sufficiently large we establish the behavior of $(\mathcal{H}_\lambda-z^2)^{-1}$ as Im $z\downarrow 0$ in the framework of Besov Spaces $B(\mathbb{R}^n)$. For $k\in \mathbb{R}\setminus\{0\}$ and $f\in B(\mathbb{R}^n)$ we find the unique solution to $-\Delta u-k^2 W_\lambda u = f $ on $\mathbb{R}^n$ that satisfies a certain radiation condition. These results can be applied to the study of the scattering theory of the Schrödinger operator $H=-\Delta+V$.

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Mathematics Subject Classification: Primary: 35P25, 81U99, 47A40; Secondary: 46C99.

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