2006 International Conference in honor of Jacqueline Fleckinger. Electron. J. Diff. Eqns., Conference 16 (2007), pp. 35-58.

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

Benedicte Alziary, Peter Takac

We investigate the compactness of the resolvent $(\mathcal{A} - \lambda I)^{-1}$ of the Schrodinger operator $\mathcal{A} = - \Delta + q(x)\bullet$ acting on the Banach space $X$,
  X = \{ f\in L^2(\mathbb{R}^N): f / \varphi\in L^\infty(\mathbb{R}^N) \} ,\quad
  \| f\|_X = \hbox{\rm ess\,sup}_{\mathbb{R}^N} (|f| / \varphi)\, ,
$X\hookrightarrow L^2(\mathbb{R}^N)$, where $\varphi$ denotes the ground state for $\mathcal{A}$ acting on $L^2(\mathbb{R}^N)$. The potential $q: \mathbb{R}^N\to [q_0,\infty)$, 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 $|x|\to \infty$. If $\Lambda$ is the ground state energy for $\mathcal{A}$, i.e. $\mathcal{A}\varphi = \Lambda\varphi$, we show that the operator $(\mathcal{A} - \lambda I)^{-1} : X\to X$ is not only bounded, but also compact for $\lambda\in (-\infty, \Lambda)$. 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 $\hbox{\rm const}\cdot \varphi$.

Published May 15, 2007.
Math Subject Classifications: 47A10, 35J10, 35P15, 81Q15.
Key Words: Ground-state space; compact resolvent; Schrodinger operator; monotone radial potential; maximum and anti-maximum principle; comparison of ground states; asymptotic equivalence.

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Bénédicte Alziary
Université Toulouse 1 (Sciences Sociales)
21 Allées de Brienne, F-31000 Toulouse Cedex, France
email: alziary@univ-tlse1.fr
Peter Takác
Institut für Mathematik, Universität Rostock
Universitätsplatz 1, D-18055 Rostock, Germany
email: peter.takac@uni-rostock.de

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