A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

Details A counterexample to dispersive estimates for Schrödinger operators in higher dimensions A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

2005-08-11

arxiv.org/abs/math/0508206v1

Comm. Math. Phys. 266 no. 1 (2006), 211-238.

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s00220-006-0013-5

In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^\alpha$, $\alpha < \frac{n-3}2$, for which the solutions to the linear Schr\"odinger equation in $\R^n$, $$ -i\partial_t u = - \Delta u + Vu, \quad u(0)=f, $$ do not obey the usual $L^1\to L^{\infty}$ dispersive estimate. This contrasts with known results in dimensions $n \leq 3$, where a pointwise decay condition on $V$ is generally sufficient to imply dispersive bounds.

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