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Compactness for a Schrodinger operator in the ground-state space over $mathbb{R}^N$

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Author(s): Benedicte Alziary | Peter Takac

Journal: Electronic Journal of Differential Equations
ISSN 1072-6691

Volume: Conference;
Issue: 16;
Start page: 35;
Date: 2007;
Original page

Keywords: Ground-state space | compact resolvent | Schrodinger operator | monotone radial potential | maximum and anti-maximum principle | comparison of ground states | asymptotic equivalence.

ABSTRACT
We investigate the compactness of the resolvent $(mathcal{A} - lambda I)^{-1}$ of the Schrodinger operator $mathcal{A} = - Delta + q(x)ullet$ acting on the Banach space $X$,$$ X = { fin L^2(mathbb{R}^N): f / varphiin L^infty(mathbb{R}^N) } ,quad | f|_X = mathop{m ess,sup}_{mathbb{R}^N} (|f| / varphi), ,$$ $Xhookrightarrow L^2(mathbb{R}^N)$, where $varphi$ denotes the ground state for $mathcal{A}$ acting on $L^2(mathbb{R}^N)$. The potential $q: mathbb{R}^No [q_0,infty)$, bounded from below, is a "relatively small" perturbation of a radially symmetric potential which is assumed to be monotone increasing (in the radial variable) and growing somewhat faster than $|x|^2$ as $|x|o infty$. If $Lambda$ is the ground state energy for $mathcal{A}$, i.e. $mathcal{A}varphi = Lambdavarphi$, we show that the operator $(mathcal{A} - lambda I)^{-1} : Xo X$ is not only bounded, but also compact for $lambdain (-infty, Lambda)$. In particular, the spectra of $mathcal{A}$ in $L^2(mathbb{R}^N)$ and $X$ coincide; each eigenfunction of $mathcal{A}$ belongs to $X$, i.e., its absolute value is bounded by $mathrm{const}cdot varphi$.
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