2006 International Conference in Honor of Jacqueline Fleckinger.
Electronic Journal of Differential Equations,
Conference 16 (2007), pp. 35-58.
Title: Compactness for a Schrodinger operator
in the ground-state space over $\mathbb{R}^N$
Authors: Benedicte Alziary (Univ. Toulouse 1, France)
Peter Takac (Univ. Rostock, Germany)
Abstract:
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 $X$,
$$
X = \{ f\in L^2(\mathbb{R}^N): f / \varphi\in L^\infty(\mathbb{R}^N) \} ,\quad
\| f\|_X = \mathop{\rm ess\,sup}_{\mathbb{R}^N} (|f| / \varphi)\, ,
$$
$X\hookrightarrow L^2(\mathbb{R}^N)$, 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 $\mathrm{const}\cdot \varphi$.
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.