Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

J. Cruz-Sampedro

doi:10.3934/dcds.2013.33.1061

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2013, Volume 33, Issue 3: 1061-1076. Doi: 10.3934/dcds.2013.33.1061

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Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

J. Cruz-Sampedro^1,

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

Received: March 31, 2011

Revised: September 30, 2011

Published: February 2013

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Abstract

Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$. For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.

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

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