A Singular Critical Potential for the Schrödinger Operator

Duyckaerts, Thomas

Canadian mathematical bulletin, Tome 50 (2007) no. 1, pp. 35-47

Résumé

Consider a real potential $V$ on ${{\text{R}}^{d}}$ , $d\ge 2$ , and the Schrödinger equation: $$\left( \text{LS} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i{{\partial }_{t}}u+\Delta u-Vu=0,{{u}_{\upharpoonright }}_{t=0}={{u}_{0}}\in {{L}^{2}}$$ In this paper, we investigate the minimal local regularity of $V$ needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of $1/2$ derivative) on solutions of $\left( \text{LS} \right)$ . Prior works show some dispersive properties when $V$ (small at infinity) is in ${{L}^{d/2}}$ or in spaces just a little larger but with a smallness condition on $V$ (or at least on its negative part).In this work, we prove the critical character of these results by constructing a positive potential $V$ which has compact support, bounded outside 0 and of the order ${{\left( \log \left| x \right| \right)}^{2}}/{{\left| x \right|}^{2}}$ near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator $P:=-\Delta +V$ .The elementary construction of $V$ consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator $P$ admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.

Détail

DOI : 10.4153/CMB-2007-004-3

Mots-clés : 35B65, 35L05, 35Q40, 35Q55