A Singular Critical Potential for the Schrödinger Operator

http://dx.doi.org/10.4153/CMB-2007-004-3
Canad. Math. Bull. 50(2007), 35-47

  Published:2007-03-01
 Printed: Mar 2007

Consider a real potential $V$ on $\RR^d$, $d\geq 2$, and the Schr\"odinger equation: \begin{equation} \tag{LS} \label{LS1} i\partial_t u +\Delta u -Vu=0,\quad u_{\restriction t=0}=u_0\in L^2. \end{equation} 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 \eqref{LS1}. 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 $(\log|x|)^2/|x|^2$ 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.