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A Singular Critical Potential for the Schrödinger Operator

Published online by Cambridge University Press:  20 November 2018

Thomas Duyckaerts
Thomas Duyckaerts*
Affiliation:
Département de mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France e-mail: Thomas.Duyckaerts@u-cergy.fr
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Abstract

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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.

Keywords

35B6535L0535Q4035Q55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 1 , 01 March 2007 , pp. 35 - 47
Copyright
Copyright © Canadian Mathematical Society 2007

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