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 Benedicte Alziary, Peter Takac

Abstract:
We investigate the compactness of the resolvent of the Schrodinger operator acting on the Banach space ,

, where denotes the ground state for acting on . The potential , 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 as . If is the ground state energy for , i.e. , we show that the operator is not only bounded, but also compact for . In particular, the spectra of in and coincide; each eigenfunction of belongs to , i.e., its absolute value is bounded by .

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 CEREMATH - UMR MIP 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